PREFACE This volume contains the proceeding of the National Research Symposium on Geometric Analysis and Applications held at the Centre for Mathematics and its Applications, Australian National University, Canberra, from June 26 – 30, 2000. The Symposium celebrated the many significant contributions of Professor Derek W. Robinson to mathematics, on the occasion of his 65th birthday. The first day, Monday June 26, in particular was devoted to Derek; speakers with a particularly close connection to Derek, including A. Carey, M. Cowling, D. Evans, P. Jorgensen and T. ter Elst recalled and elaborated on important aspects of Derek’s work, and the day ended with a banquet in Derek’s honour. The Symposium brought together researchers working in harmonic analysis, linear and nonlinear partial differential equations, quantum mechanics and mathematical physics, and included researchers from North America, Europe and Asia as well as Australasia. We gratefully acknowledge the support of the contributors to this volume. We would like especially thank to Derek W. Robinson for his participation. A short synopsis of Derek’s career, and a full list of his publications to date are included in this volume. Contributions for the proceedings were sought from all participants and all papers received were carefully refereed by peer referees.
Alexander Isaev, Andrew Hassell, Alan McIntosh, Adam Sikora (Editors)
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At the end of 2000 Derek Robinson retired from his position as Professor of Mathematics in the Institute of Advanced Studies at ANU, a position he had held for 19 years. Robinson obtained his early training at Oxford with a Bachelor’s degree in mathematics in 1957 and a Doctorate of Philosophy in theoretical nuclear physics in 1960. Subsequently he held various postdoctoral positions in Switzerland, the United States, Germany and France before being appointed Professor of Mathematical Physics at the University of Aix–Marseille in 1968. He remained in this position until 1978 at which point he accepted appointment as Professor of Mathematics at the University of New South Wales. Soon after he moved to Australia he became a Fellow of the Australian Academy of Science. In his early years Robinson worked in a wide range of areas of mathematical physics, quantum field theory, statistical mechanics, operator algebras, etc. Operator algebras were introduced into quantum field theory in the 1960s as a means to describe the macroscopic observables but their most fruitful application was to the characterization of the equilibrium states of statistical mechanics. The observables of classical statistical mechanics form an abelian C ∗ -algebra and Robinson realized that the corresponding quantum algebra was ‘asymptotically abelian’. This observation was of fundamental importance since the states of asymptotically abelian algebras could be shown to share many of the good properties of abelian algebras. Therefore extremal states of the algebra could be identified with pure phases of the system and decomposition theory of the states could be applied to describe the separation of the phases. Although these results were of a general abstract nature Robinson showed that they could be applied to a broad class of realistic models of quantum spin systems. In particular he demonstrated that a number of these systems exhibited a phase transition at low temperatures and the pure phases were indeed described by extremal invariant states. Much of this is described in the two volume monograph Operator Algebras and Quantum Statistical Mechanics which Derek Robinson co-authored with Ola Bratteli. Both volumes of this book appeared in a second edition and continue to be used by research workers in these areas twenty years later. The majority of applications of operator algebras to quantum statistical mechanics concerned the equilibrium theory. Robinson and Bratteli realized that the description of non-equilibrium phenomena required a dynamical theory and this in turn required a theory of unbounded derivations of the algebras. The foundations of this theory were laid in a series of joint papers which are also described in their book. As a direct result of this latter work Robinson’s interests then developed to evolution equations and semigroup ii
theory in a much broader context. This change of direction coincided with his move to Australia and most of his subsequent work has been on evolution equations, in particular equations involving elliptic operators on Lie groups. One of the issues at that time was the integrability of a Lie algebra representation. Together with Ola Bratteli, George Elliott and Palle Jørgensen, Robinson gave a sufficient condition in terms of a dissipativity condition and an estimate on the semigroup generated by a Laplacian. Using Lipschitz spaces Robinson could not only weaken the assumptions in the above mentioned paper, but he could also prove that the analytic vectors associated with any representation of a Lie group coincide with the analytic elements of the Poisson semigroup. By this stage Robinson became more interested in elliptic operators on Lie groups and decided to write the monograph Elliptic Operators and Lie Groups. After completion of the book Robinson began a collaboration with Tom ter Elst and their continuing work can be divided into three parts. First, they developed the theory of complex, weighted, higher order subcoercive operators on Lie groups, in particular proving Gaussian bounds for the kernel, and its derivatives, of the semigroup generated by such an operator. They established that the Gaussian bounds are equivalent with a subcoercivity condition on the operator, under weak additional assumptions. Secondly, they studied second-order divergence form subelliptic operators on Lie groups with complex bounded measurable coefficients and proved, under a variety of conditions, optimal smoothness properties of the kernel. Thirdly, in joint work with Adam Sikora or Nick Dungey, they studied asymptotic properties of the semigroup and its kernel. They showed that on a Lie group with polynomial growth the second-order Riesz transforms associated with the Laplacian are bounded if, and only if, the group is a direct product of a compact and a nilpotent Lie group. Their analysis of the asymptotics of higher order operators continues. Robinson has maintained an active involvement in University affairs, having been Chairman of the Board of the Institute for Advanced Studies at ANU (1988-1992), and a member of ANU Council (1997-2000), where he never resiled from robust and provocative debate. He has also been an active member of the mathematical community, in particular being president of the Australian Mathematical Society (1994-1996) and continuing as Vicepresident. He also served four years as a member of the Research Training and Careers Committee of the ARC (1996-2000) and as Chair of the National Committee for Mathematics (1997-2001).
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LIST OF PUBLICATIONS Books [1] The Thermodynamic Pressure in Quantum Statistical Mechanics - Springer Verlag Lecture Notes in Physics 9, (1971) 115 pages. [2] Operator Algebras and Quantum Statistical Mechanics Vol I (with O Bratteli) 500 pages, Springer-Verlag (1979), 2nd edition 505 pages, Springer-Verlag (1979). Russian translation of first edition (1981). [3] Operator Algebras and Quantum Statistical Mechanics Vol II (with O Bratteli) 501 pages, Springer-Verlag (1981), 2nd edition 517 pages, Springer-Verlag (1996). [4] Basic Semigroup Theory Proceedings of the Centre for Mathematical Analysis, ANU, Canberra Volume 2, (1983) 138 pages. [5] Elliptic Operators and Lie Groups 556 pages, Oxford University Press (1991).
Research papers [1] Multiple Coulomb Excitations of Vibrational Nuclei Nuclear Physics 45 (1961) 459–470. [2] Zero-Mass Representations of the Inhomogeneous Lorentz Group Helv Phys Acta 35 (1962) 98–112. [3] Support of a Field in Momentum Space Helv Phys Acta 35 (1962) 403–413. [4] Multiple Coulomb Excitations of Deformed Nuclei Helv Phys Acta 36 (1963) 140–154. [5] On a Soluble Model of Relativistic Field Theory Physics Letters 9 (1964) 189–191. [6] A Theorem Concerning the Positive Metric Commun Math Phys 1 (1965) 89–95. [7] The Ground State of the Bose Gas Commun Math Phys 1 (1965) 159–174. [8] Conserved Currents and Associated Symmetries (with D Kastler and A Swieca) Commun Math Phys 2 (1966) 108–120. [9] Covariance Algebras in Field Theory (with S Doplicher and D Kastler) Commun Math Phys 3 (1966) 1–28. [10] Invariant States in Statistical Mechanics (with D Kastler) Commun Math Phys 3 (1966) 151–180. [11] Extremal Invariant States (with D Ruelle) Ann Inst Henri Poincar´ e 6 (1967) 299-310. [12] Mean Entropy of States of Classical Statistical Mechanics (with D Ruelle) Commun Math Phys 5 (1967) 288–300. [13] Mean Entropy of States of Quantum Statistical Mechanics (with O E Lanford) J Math Phys 9 (1968) 1120–1125. [14] Statistical Mechanics of Quantum Spin Systems Commun Math Phys 6 (1967) 151–160. [15] Asymptotic Abelian Systems (with S Doplicher, R Kadison and D Kastler) Commun Math Phys 6 (1967) 101–120. [16] Analyticity Properties of a Lattice Gas (with G Gallavotti and S Miracle-Sole) Physics Letter 25A (1967) 443–444. [17] Statistical Mechanics of Quantum Spin Systems II Commun Math Phys 7 (1968) 337–346. [18] Statistical Mechanics of Quantum Spin Systems III (with O E Lanford) Commun Math Phys 9 (1968) 327–338. [19] Analyticity Properties of the Anisotropic Heisenberg Model (with G Gallavotti and S Miracle-Sole) Commun Math Phys 10 (1968) 311–324. [20] Proof of the Existence of Phase Transitions in the Anisotropic Heisenberg Model Commun Math Phys 14 (1969) 196–204. [21] Physical States of Fermi Systems (with S Miracle-Sole) Commun Math Phys 14 (1969) 235–270. [22] Statistical Mechanics of Quantum Mechanical Particles with Hard Cores I (with S MiracleSole) Commun Math Phys 16 (1970) 290–309. [23] Statistical Mechanics of Quantum Mechanical Particles with Hard Cores II (with S MiracleSole) Commun Math Phys 19 (1970) 204–218. [24] Normal and Locally Normal States Commun Math Phys 19 (1970) 219–234. [25] Normal States and Representations of the Canonical Commutation Relations (with M Courbage and S Miracle-Sole) Ann Inst Henri Poincare XIV (1971) 171–178. iv
[26] Approach to Equilibrium of Free Quantum Systems (with O E Lanford) Commun Math Phys 24 (1972) 193–210. [27] The Finite Group Velocity of Quantum Spin Systems (with E H Lieb) Commun Math Phys 28 (1972) 193–210. [28] Return to Equilibrium Commun Math Phys 31 (1973) 171-189. [29] Scattering Theory with Singular Potentials I Ann Inst Henri Poincare XXI (1974) 185–215. [30] Scattering Theory with Singular Potentials II (with P Ferrero and O Depazzis) Ann Inst Henri Poincare XXI (1974) 216–231. [31] A Characterisation of Clustering States Commun Math Phys 41 (1975) 79–88. [32] Dynamical Stability and Pure Thermodynamic Phases (with H Narnhofer) Commun Math Phys 41 (1975) 89–97. [33] Unbounded Derivations of C ∗ -Algebras I (with O Bratteli) Commun Math Phys 42 (1975) 253–268. [34] Unbounded Derivations of C ∗ -Algebras II (with O Bratteli) Commun Math Phys 46 (1976) 11–30. [35] Unbounded Derivations of Von Neumann Algebras (with O Bratteli) Ann Inst Henri Poincare XXV (1976) 139–164. [36] Unbounded Derivations and Invariant Trace States (with O Bratteli) Commun Math Phys 46 (1976) 31–35. [37] The Approximation of Flows J Funct Anal 24 (1977) 280–290. [38] Green’s Functions, Hamiltonians and Modular Automorphisms (with O Bratteli) Commun Math Phys 50 (1976) 53–59. [39] Bose-Einstein Condensation with Attractive Boundary Conditions Commun Math Phys 50 (1976) 53–59. [40] Properties of Propagation of Quantum Spin Systems J Aust Math Soc Vol XIX (Series B) 19 (1976) 387–399. [41] Quasi Analytic Vectors and Derivations of Operator Algebras (with O Bratteli and R Herman) Math Scan 39 (1976) 371–38. [42] Perturbations of Flows on Banach Spaces and Operator Algebras (with O Bratteli and R Herman) Commun Math Phys 59 (1978) 167–196. [43] Stability Properties and the KMS Condition (with A Kishimoto and O Bratteli) Commun Math Phys 61 (1978) 209–238. [44] Ground States of Quantum Spin Systems (with A Kishimoto and O Bratteli) Commun Math Phys 64 (1978) 41–48. [45] Propagation Properties in Scattering Theory J Aus Math Soc (Series B) 21 (1979) 474–485. [46] Lie and Jordan Structure in Operator Algebras (with E Stormer) J Aus Math Soc (Series A) 29 (1980) 129–142. [47] Equilibrium States of a Bose Gas with repulsive interactions (with O Bratteli) J Aus Math Soc (Series B) 22 (1980) 129–147. [48] Positivity and Monotonicity Properties of C 0 -Semigroups I (with O Bratteli and A Kishimoto) Commun Math Phys 75 (1980) 67–84. [49] Positivity and Monotonicity Properties of C 0 -Semigroups II (with A Kishimoto) Commun Math Phys 75 (1980) 85–101. [50] Subordinate Semigroups and Order Properties (with A Kishimoto) J Aus Math Soc (Series A) 31 (1981) 69–76. [51] Positive C0 -Semigroups on C ∗ -Algebras (with O Bratteli) Math Scand 49 (1981) 259–274. [52] Strictly Positive and Strongly Positive Semigroups (with A Majewski) J Aus Math Soc (Series A) 34 (1983) 36–48. [53] Strongly Positive Semigroups and Faithful Invariant States Commun Math Phys 85 (1982) 129–142. [54] On Unbounded Derivations Commuting with a Compact Group of ∗-Automorphisms (with A Kishimoto) Publ RIMS Kyoto Univ 18 (1982) 1121–1136. [55] Positive Semigroups on Ordered Banach Spaces (with O Bratteli and T Digernes) J Op Theor 9 (1983) 371–400. [56] Addition of an Identity to an Ordered Banach Space (with S Yamamuro) J Aus Math Soc (Series A) 35 (1983) 200–210. [57] Hereditary Cones, Order Ideals, and Half-norms (with S Yamamuro) Pac J Math 110 (1984) 335–343. v
[58] The Jordan Decomposition and Half-norms (with S Yamamuro) Pac J Math 110 (1984) 345–353. [59] The Canonical Half-norm, Dual Half-norms, and Monotonic Norms (with S Yamamuro) Tohoku J Math 35 (1983) 375–386. [60] Continuous Semigroups on Ordered Banach Spaces J Funct Anal 5 (1983) 268–284. [61] On Positive Semigroups Publ RIMS Kyoto Univ 20 (1984) 213–224. [62] Extending Derivations (with C J K Batty, A Carey and D E Evans) Publ RIMS Kyoto Univ 20 (1984) 119–139. [63] Positive One-parameter Semigroups on Ordered Banach Spaces (with C J K Batty) Acta Appl Math 2 (1984) 221–296. [64] A C ∗ -algebraic Schoenberg Theorem (with O Bratteli, P Jørgensen and A Kishimoto) Ann Inst Fourier (Grenoble) XXXIV (1984) 155–187. [65] Relative Locality of Derivations (with O Bratteli and T Digernes) J Funct Anal 59 (1984) 12–40. [66] Derivations, Dynamical Systems, and Spectral Restrictions (with A Kishimoto) Math Scand 56 (1985) 83–95. [67] Dissipations, Derivations, Dynamical Systems and Asymptotic Abelianness (with A Kishimoto) J Op Theor 13 (1985) 237–253. [68] Derivations of Simple C ∗ -algebras Tangential to Compact Automorphism Groups (with E Stormer and M Takesaki) J Op Theor 13 (1985) 189–200. [69] Strong Topological Transitivity and C ∗ -dynamical Systems (with O Bratteli and G A Elliott) J Math Soc Japan 37 (1985) 115–133. [70] The Characterisation of Differential Operators by Locality; Classical Flows (with O Bratteli and G A Elliott) Comp Math 58 (1985) 279–319. [71] The Characterisation of Differential Operators by Locality; Abstract Derivations (with C J K Batty) Erg Theor and Dyn Syst 5 (1985) 171–183. [72] The Characterisation of Differential Operators by Locality; Dissipations and Ellipticity (with O Bratteli and G A Elliott) Publ RIMS Kyoto Univ 21 (1985) 1031–1049. [73] The Characterization of Differential Operators by Locality: C ∗ -algebras of Type I (with O Bratteli and G A Elliott) J Op Theor 16 (1986) 213–233. [74] Integration in Abelian C ∗ -dynamical Systems (with O Bratteli, T Digernes and F M Goodman) Publ RIMS Kyoto Univ, 21 (1985) 1001–1030. [75] Smooth Cores of Lipschitz Flows Publ RIMS Kyoto Univ 22 (1986) 659–669. [76] Smooth Derivations on Abelian C ∗ -dynamical Systems J Aus Math Soc (Series A) 42 (1987) 247–264. [77] Commutators and Generators (with C J K Batty) Math Scand 62 (1988) 303–326. [78] Commutators and Generators II Math Scand 64 (1989) 87–108. [79] Return to Equilibrium in the X −Y Model (with L R Hume) J Stat Phys 44 (1986) 829–848. [80] Embedding Product Type Actions into C ∗ -dynamical Systems (with O Bratteli and A Kishimoto) J Funct Anal 75 (1987) 188–210. [81] Fractional Powers of Generators of Equicontinuous Semigroups and Fractional Derivatives (with O E Lanford) J Aus Math Soc (Series A) 46 (1989) 1–32. [82] Commutator Theory on Hilbert Space Can J Math 5 (1987) 1235–1280. [83] The Differential and Integral Structure of Continuous Representations of Lie Groups J Op Theor 19 (1988) 95–128. [84] The Heat Semigroup and Integrability of Lie Algebras (with O Bratteli, F M Goodman and P E T Jørgensen) J Funct Anal 79 (1988) 351–397. [85] An Index for Continuous Semigroups of ∗ -endomorphisms of B(H) (with R T Powers) J Funct Anal 83 (1989) 1–12. [86] Lie Groups and Lipschitz Spaces Duke Math Jour 57 (1988) 357–395. [87] The Heat Semigroup, Derivations, and Reynold’s Identity (with O Bratteli and C J K Batty) Operator algebras and applications. London Math Soc Lecture Notes Series 136 (1988) 22–47. [88] The Heat Semigroup and Integrability of Lie Algebras: Lipschitz Spaces and Smoothness Properties Commun Math Phys 132 (1990) 217–243. [89] Lipschitz Operators J Funct Anal 85 (1989) 179–211. [90] Comparison of Commuting One-parameter Groups of Isometries (with O Bratteli and H Kurose) Trans Amer Math Soc 320 (1990) 677–694. vi
[91] Unitary Representations of Lie Groups and G˚ arding’s Inequality (with O Bratteli, F M Goodman and P E T Jørgensen) Proc Amer Math Soc 107 (1989) 627–632. [92] Positive Semigroups Generated by Elliptic Operators on Lie Groups (with W Arendt and C J K Batty). J Op Theor 23 (1990) 369–407. [93] Elliptic Differential Operators on Lie Groups J Funct Anal 97 (1991) 373–402. [94] 2nd order Elliptic Operators and Heat Kernels on Lie Groups (with O Bratteli) Trans Amer Math Soc 325 (1991) 683–713. [95] Subelliptic Operators on Lie Groups: Regularity (with A F M ter Elst) J Aus Math Soc 57 (1994) 179–229. [96] Subelliptic Operators on Lie Groups: Variable Coefficients (with O Bratteli) Acta Appl Math 42 (1996) 1–104. [97] Subcoercivity and Subelliptic Operators on Lie Groups I: Free Nilpotent Groups (with A F M ter Elst) Pot An 3 (1994) 283–387. [98] Subcoercivity and Subelliptic Operators on Lie Groups II:The General Case (with A F M ter Elst) Pot An 4 (1995) 205–243. [99] Subcoercive and Subelliptic Operators on Lie Groups: Variable Coefficients (with A F M ter Elst) Publ RIMS Kyoto Univ 5 (1993) 745–801. [100] Lp -regularity of Subelliptic Operators on Lie Groups (with R J Burns and A F M ter Elst) J Op Theor 31 (1994) 165–187. [101] Functional Analysis of Subelliptic Operators on Lie Groups (with A F M ter Elst) J Op Theor 31 (1994) 277–301. [102] Asymptotics of Periodic Subelliptic Operators (with C J K Batty, O Bratteli and P E T Jørgensen) J Geom Anal 5 (1995) 427–443. [103] Weighted Strongly Elliptic Operators on Lie Groups (with A F M ter Elst) J Funct Anal 125 (1994) 548–603. [104] On Positive Rockland Operators (with P Auscher and A F M ter Elst) Colloq Math LXVII (1994) 197–216. [105] Spectral Estimates for Positive Rockland Operators (with A F M ter Elst) Algebraic groups and Lie groups (Ed G I Lehrer), Aus Math Soc Lecture Series 9. Cambridge Univ Press (1997). [106] Reduced Heat Kernels on Nilpotent Lie Groups (with A F M ter Elst) Commun Math Phys 173 (1995) 475–511. [107] Semigroup Kernels, Poisson bounds and Holomorphic Functional Calculus (with X T Duong) J Funct An 142 (1996) 89–129. [108] Weighted Subcoercive Operators on Lie Groups (with A F M ter Elst) J Funct Anal 156 (1998) 88–163. [109] On Kato’s Square Root Problem (with A F M ter Elst) Hokkaido Math J 26 (1997) 1–12. [110] Analytic Elements of Lie Groups (with A F M ter Elst) Helv Phys Acta 69 (1996) 655–678. [111] Abundance of Invariant and Almost Invariant Pure States of C ∗ -dynamical Systems (with O Bratteli and A Kishimoto) Commun Math Phys 187 (1997) 491–507. [112] Second-order Subelliptic Operators on Lie Groups I: Complex Uniformly Continuous Principal Coefficients (with A F M ter Elst) Acta Appl Math 59 (1999) 299–331. [113] High Order Divergence-form Elliptic Operators on Lie Groups (with A F M ter Elst) Bull Aus Math Soc 55 (1997) 335-348. [114] Second-order Subelliptic Operators on Lie Groups II: Real Measurable Coefficients (with A F M ter Elst) Progess in Nonlinear Differential Equations 42 Birkh¨ auser Verlag (2000) 103–124. [115] Second-order Strongly Elliptic Operators on Lie Groups with H¨ older Continuous Coefficients. (with A F M ter Elst) J Aus Math Soc (Series A) 63 (1997) 297–363. [116] Heat Kernels and Riesz Transforms on Nilpotent Lie Groups. (with A F M ter Elst and A Sikora) Colloq Math LXXIV (1997) 191–218. [117] Second-order Subelliptic Operators on Lie Groups III: H¨ older Continuous Coefficients (with A F M ter Elst) Calc Var P D E 8 (1999), 327–363. [118] Spectral Asymptotics of Periodic Elliptic Operators (with O Bratteli and P E T Jørgensen) Math Z 232 (1999), 621–650. [119] Local Lower Bounds on Heat Kernels (with A F M ter Elst) Positivity 2 (1998) 123–151. [120] Riesz Transforms and Lie Groups of Polynomial Growth. (with A F M ter Elst and A Sikora) J Funct Anal 162 (1999), 14–51. vii
[121] Asymptotics of Semigroups on Nilpotent Lie Groups (with N Dungey, A F M ter Elst and A Sikora) J Op Theor (to appear) [122] Asymptotics of Sums of Subcoercive Operators (with N Dungey and A F M ter Elst) Colloq Math 82 (1999) 231–260. [123] On second-order Periodic Elliptic Operators in Divergence Form (with A F M ter Elst and Adam Sikora) Math Z (to appear) [124] On Anomalous Asymptotics of Heat Kernels (with A F M ter Elst) Evolution equations and their applications to Physical and Life Sciences, Lecture Notes in Pure and Applied Mathematics, Vol. 215, (2001) 89–103, Eds. G Lumer, L Weis, Marcel Dekker, New York [125] On Second-order Almost-periodic Elliptic Operators (with N Dungey and A F M ter Elst) J London Math Soc (submitted) [126] On Anomalous Asymptotics of Heat Kernels on Groups of Polynomial Growth (with N Dungey and A F M ter Elst) Comp Math (submitted)
Lecture notes, reviews and conference papers [1] Algebraic Aspects of Relativistic Field Theory - Brandeis Lectures, Gordon and Breach, New York, Vol 1 (1966). [2] Symmetries, Broken Symmetries, Charges and Currents - Istanbul Lecture, Freeman and Co, San Francisco, (1967) 463–513. [3] Conserved Currents and Broken Symmetries - Colloque du CNRS (1966), Eds du CNRS 15 (1968) 131–135. [4] Physical States and Finiteness Restrictions - Proceeding of the CNRS Conference at Gifsur-Yvette, CNRS (1969). [5] Existence Theorems in Quantum Statistical Mechanics - Proceedings of the Cargese Summer School (notes by R Lima and S Miracle) Gordon and Breach (1970). [6] Approach and Return to Equilibrium of Free Fermi States - Colloquium of the American Math Soc 1971 - Siam AMS Proceeding 5 55 (1972). [7] C ∗ -Algebras and Quantum Statistical Mechanics - Int School of Physics Enrico fermi Varenna-Editrice Comp Bologna (1975) 235–252. [8] Time Dependent Scattering Theory III - Int Colloq on Group Theoretical Methods, Marseille (1974). [9] Unbounded Derivations of C ∗ -Algebras - Int Symp on Math Problems in Theor Phys, Springer-Verlag (1975). [10] Dynamics in Quantum Statistical Mechanics - Int Summer School Bielefeld 1976, Plenum Press (1978). [11] Review of Derivations - Int Conference on Mathematical Physics, Lausanne 1979, SpringerVerlag (1980). [12] Commutator Theory and Partial Differential Operators on Hilbert Space - Proceedings of the Centre of Mathematical Analysis, ANU, Canberra 14 (1986) 295–302. [13] Differential Operators on C ∗ -algebras - Contemporary Mathematics, Amer Math Soc 62 (1987) 367–384. [14] C ∗ -algebras and a Single Operator - Surveys of some recent results in operator theory II Pitman Research 171 (1988) 235–266. [15] Integration of Lie Algebras - Proceedings of the Centre of Mathematical Analysis, ANU, Canberra 15 (1987) 255–278. [16] Schauder Estimates on Lie Groups- Proceedings of the Centre of Mathematical Analysis, ANU, Canberra 24 (1989) 1–9. [17] Subelliptic Operators on Lie Groups (with A F M ter Elst) - Proceedings of the Centre of Mathematical Analysis, ANU, Canberra 29 (1992) 63–72. [18] Strongly Elliptic and Subelliptic Operators on Lie Groups - Quantum and NonCommutative Analysis, Eds H Araki et al., Kluwer Academic Publishers (1993) 435–453 [19] Basic Semigroup Theory - Proceedings of the Centre of Mathematical Analysis, ANU, Canberra 34–Part III (1996) 1–33. [20] Elliptic Operators on Lie Groups (with A F M ter Elst) Acta Appl. Math. 44 (1996) 133– 150. [21] C ∗ -Algebras and Mathematical Physics - Fields Institute Monograph volume, Eds R Bhat, G Elliott and P Fillmore, American Mathematical Society, Providence, Rhode Island. viii
ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO NONLINEAR PDE HILARY BOOTH Abstract. The Maxwell-Dirac equations give a model of an elec-
tron in an electromagnetic (e-m) eld, in which neither the Dirac or the e-m elds are quantized. The two equations are coupled via the Dirac current which acts as a source in the Maxwell equation, resulting in a nonlinear system of partial di erential equations (PDE's). In this way the self- eld of the electron is included. We review our results to date and give the four real consistency conditions (one of which is conservation of charge) which apply to the components of the wavefunction and its rst derivatives. These must be met by any solutions to the Dirac equation. These conditions prove to be invaluable in the analysis of the nonlinear system, and generalizable to higher dimensional supersymmetric matter. In earlier papers, we have shown analytically that in an isolated stationary system, the surrounding electon eld must be equal and opposite to the central (external) eld. The nonlinearity forces electric neutrality, at least in the static case. We illustrate these properties with a numerical family of orbits which occur in the (static) spherical and cylindrical ODE cases. These solutions are highly localized and die o exponentially with increasing distance from the central charge.
1. The Maxwell-Dirac Equations and QED The coupled Maxwell-Dirac equations can be written as follows:
(@ , ieA ) + im = 0 where = 0; : : : ; 3 F = A ; , A ; @ F = ,4ej (1) where j = : Note that 2 C is the Dirac wave-function or 4-spinor and the Dirac conjugate. These are acted upon by which are the usual gamma matrices (representations of a Cli ord algebra), and A is the 4-potential. These equations model an electron in an electromagnetic eld. The two equations are coupled via the Dirac current j i.e. we include 4
1
2
HILARY BOOTH
the nonlinearity of the self- eld, see Equation (1). These equations form the foundation of quantum electrodynamics (QED), the theory of electrons interacting with elds. QED is one of the most successful physical theories, explaining the Lamb shift and the anomalous magnetic moment of the electron. The calculations of QED are acheived by quantizing the elds and using perturbation theory. In so doing, wellknown mathematical problems occur, which have yet to be resolved at the fundamental level. It is possible that the full nonlinearized equations must be analysed rigorously before we can hope to resolve these deep problems. For example, in [23], Lieb and Loss observe that the stability of matter requires that the electron be de ned with a Dirac operator with the magnetic vector potential A(x), instead of the free Dirac operator (without A(x)). That perturbation theory must start from the dressed electrons (including their own self- eld) might be \fundamentally important in a non-perturbative QED". This view is shared by many analysts working on this problem including Flato, Simon and Ta in who established global existence for the M-D equations as recently as 1997 [18], following many years of sustained interest [21] [9] [20] [19] in the problem. In [18], Flato et al showed that the nonlinear representation is integrable to a global nonlinear representation of the Poincare group on a di erential manifold, U1 of small initial conditions. This established the existence of global solutions for initial data in U1 at t = 0. They go on to show that the asymptotic representations are also nonlinear and draw conclusions for the infrared tail of the electron. Their results show that \in the classical case (also), one obtains infrared divergencies, if one requires free asymptotic elds as it is needed in QED". In other words, the elds must remain coupled via the self- eld if we are to resolve the infrared problem. In the case where we assume that the system is static and/or stationary (see Section 2), we can make some simpli cations. The system is elliptic in the stationary case (no time dependence). Esteban, Georgiev and Sere showed existence of soliton-like solutions (that is, solutions which are spatially localized) in this case [16]. Furthermore, the wavefunction, together with all its derivatives decreases exponentially at in nity. 2. Some simplified versions of the problem The full 4-dimensional nonlinear problem is somewhat intractable { as stated in Section 1, global existence has only been established recently [18] (and references therein). If we want to get some idea of
ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO NONLINEAR PDE 3
the types of behaviour we might expect in the 4-dimensional problem, we might begin by looking at various subcases. Some simpli ed versions of the problem are as follows: 1. The static case in which we assume that there exists a Lorentz frame in which there is no current \ ow" i.e. j = j [29] [4], 2. The stationary case in which we assume (x ; x) = ei!x0 (x) [16], 3. The static spherically symmetric case [29], 4. The static cylindrically symmetric case [4] [8], 5. The static case with z dependence only [6], 6. The 1 + 1 case which was solved exactly for a massless electron by Schwinger in [31], 7. The static axi-symmetric case, 8. The circular current case in which we assume (in spherical coordinates) that j = (j ; 0; j ; 0): It appears that the static and stationary assumptions are rather strong, since the resulting system becomes elliptic rather than hyperbolic. In [5] and [30] we proved electric neutrality in the static case (for an isolated system). While this is interesting in that it implies that solutions of this type must consist of an inner charge, say, surrounded by an equal and oppositely charged electron eld i.e. the solutions must be atom-like, it raises the problem of nding a solution that represents a single charged particle. We need to nd a weaker ansatz, perhaps the circular current assumption, with which we have sucient simpli cation without losing the important properties of the 4-dimensional system. It is possible of course, that electric neutrality could be shown to be fully general, which would supporting the conjecture that the total charge of the universe is zero. The larger (quantum cosmological) problem here is the Einstein-Maxwell-Dirac problem, and related problems such as Einstein-Dirac [17] and Einstein-Yang-Mills [3]. Following the methods of the analysts working in relativity theory, our aim in [29] [4] etc was to enumerate the subcases, starting with all possible ODE cases (see Section 4), and then progressing to (static) axi-symmetric and other two-dimensional cases. Meanwhile, the three dimensional static case proved to be somewhat tractable. 0
0
0
0
3. Using the Clifford Algebra to derive some constraints
In this section, we make use of the properties of the Cli ord algebra (which the represent as 4 4 matrices), to solve the Dirac equation for the potential and to show that there are some useful consisitency
4
HILARY BOOTH
conditions upon the wavefunction and its rst derivatives. We want to write the potential in terms of the wavefunction, so that we can substitute it into the Maxwell equation. A complex Cli ord algebra A(n) consists of all possible products of the n basis vectors (of an n-dimensional vector space), ei, which obey the following: (ei ) = ,1; if i 2 f0; : : : ; n , 1g; (2) eiej + ej ei = 0; if i 6= j: Using the isomorphism [11] (3) A(n + 2) = A(n) A(2) we can construct the representation of A(4), the gamma matrices, from the Pauli matrices which represent A(2). For the construction of higher dimensional Cli ord algebras from lower dimensions see for example [14]. One possible representation of A(4) is: 2
2
0 0 6 0
= 64 0 ,10 ,1 0 0
0 1 0 0
2
1 0 0 0
3 7 7 5
2
0 6 0
= i 64 0 ,1 1
3
0 0 1 0
3
0 ,1 1 0 77 0 05 0 0
2
3
1 0 0 0 0 0 ,1 0 6 0 1 7 6 7 (4) = i 64 0 0 ,10 00 75 = i 64 ,01 00 00 ,10 75 0 0 0 ,1 0 ,1 0 0 We note that in this representation, is anti-symmetric (a-s), and
i; i = 1; 2; 3 are symmetric. Whatever representation we use, we can always invert the Dirac equation to express the spatial components of the potential, Ai ; i = 1; 2; 3 in terms of A , the wavefunction, and its rst derivatives. Multiplying (1) on the left t i (where t is the transpose of ) we have t i A = 1 t i ( @ + im) : (5) ie But the product i j ; i 6= j is (a-s), i i = ,1, and i is symmetric, so that, using the argument following (7) we can write: t A = t i A + 1 t i ( @ + im) : (6) i ie In a similar way (by multiplying the Dirac equation by on the left, and subtracting the conjugate equation premultiplied by 2
3
0
0
0
0
0
0
1
2
3
ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO NONLINEAR PDE 5
) we can solve for A . There is a condition to be met here see [29] i.e. that j is not a null vector which has the physical interpretation that the electron is not travelling faster than the speed of light. In 1957, one of Dirac's students, Eliezer solved for the potential in this way [15]. He went on to show that when we solve for the potential, we must also adopt a consistency condition which applies to . In that paper there was a contribution by Dirac who streamlined some of Eliezer's calculations. Although they used a di erent representation of the algebra, the argument was essentially the following. If , is an antisymmetric (a-s) (4 4) matrix then t , = 0; since this quantity is an a-s scalar. (7) If we can nd a , such that , are all a-s, then t , ( A ) = 0; since A is a scalar, so that if satis es (1) then t , ( @ + im) = 0; (8) which gives us a consistency condition on and its rst derivatives. The same is true of the complex Dirac equation and we have t (9) , ( @ , im) = 0: We can extend Dirac's argument one step further by premultiplying the Dirac equation by and the complex equation by and noting that since , is a-s, t t , A = , t , A : This gives us another condition on namely t t (10) , ( @ + im) + t , ( @ , im) = 0: There is only one possible element of the Cli ord algebra which when premultiplying all of the yields an a-s matrix. In the representationused here, this is the product ,= : Using this in (8) (9) and (10), we have two complex conditions (or four real conditions) on the components of . As is well-known (see [11] for example), even dimensional complex Cli ord algebras are simple, that is, they can not be decomposed into the direct sum of two nontrivial subspaces which obey closure under algebraic multiplication. If we want to decompose the 4-dimensional into a 2-dimensional algebra and use 2-spinors, then when must accept 0
1
2
3
0
1
2
3
6
HILARY BOOTH
the additional structure in which the two 2-spinor spaces are conjugate dual spaces. See for example [28] in which the 2-spinor formalism is given in terms of In eld-van der Waerden symbols. An argument, equivalent to Dirac/Eliezer's but giving all of the consistency conditions, was given by Radford using the 2-spinor formalism in 1996 [29], and subsequently in [4], although Radford referred to them as \reality conditions", in keeping with the conventions in [28]. We can think of the Dirac equation and its conjugate equations as eight equations in four unknowns (the four real scalars, A ). If we solve for these A then we must have four additional (real) constraints, which correspond to the two complex consistency conditions, con rming that there are no further constraints upon the system. In [7] we outline these conditions and then go on to generalize to higher dimensional cases. Allowing these higher Cli ord algebras enables us to pursue the same arguments when applied to supersymmetric matter [7]. See [22] and references therein. 4. ODE solutions Within the static system there are three interesting ODE cases, spherically symmetric, cylindrically symmetric and dependence on z only. The spherical and cylindrical cases were examined extensively in [29] [4]. The case where dependence is on z only, is similar in some respects [5]. We rst apply our consistency conditions to the electromagnetic potential A which has been expressed in terms of Dirac spinors and their rst derivatives (by solving the Dirac equations for the potential as outlined in Section 3). These reality conditions allow us some simpler expressions which are then inserted into the Maxwell equation, resulting in fourth order ODE's. We will also note here that in the 1 + 1 case [12] [13] [6] the system was also reduced to fourth order ODE's, which in some cases are solved explicitly [12] [13], whilst in others we are currently developing more numerical results [6]. When we assume that the Dirac current is static, we lose three (real) degrees of freedom in , since three components of j are set to zero. See [29] and [4] in which we also x the gauge by choosing 0
(11)
1
= ,Y e 2i , = Xe 2i (
( + )
)
= ,Y e, 2i = Xe, 2i , ;
( + )
2 3
(
)
where , , X and Y are real functions. These expressions were substituted into the potential (which was solved for in terms of Dirac spinors
ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO NONLINEAR PDE 7
and their rst derivatives), yielding: , Y ) @ + (r ):l (12) A = cos + ((X X + Y ) @t (X + Y ) 1 @ (13) A = (X + Y ) @t l + (X , Y )r , r l : We now use the consistency conditions, one of which is conservation of charge, which is obeyed automatically as stated in [29]. The other three conditions, in the variables required for the static case, are given below. @ (X + Y ) = 0 (14) @t (15) r:l = ,(X + Y ) sin @ l + (r ) l = 0: (16) @t 0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
where l = (2XY cos ; 2XY sin ; X , Y ). The Maxwell equations act upon A, as de ned above, and the current vector becomes j = (2(X + Y ); 0; 0; 0) : We showed in [5] that the static equations, in the gauge given by @X @Y Eq. 11, are stationary if and only if @ @t = 0 and @t = 0 (or @t = 0). In @l the stationary case, @ @t = 0 and @t = 0. Now in the stationary case, the third reality condition Eq. 16 tells us that r is proportional to l and we choose the function such that l = r r : Substituting this into the expression for the potential Eq. 12 and noting that X + Y = jlj then s (17) A = cos r :l = cos r + + : (X + Y ) r r sin From here, it is a straightforward calculation to apply symmetry arguments and calculate the resulting ODE's [29] [4] [5]. Then, in dimensionless variables [29] [4], the equations reduce to: d = A , cos dx dF = Z dx dA = F dx f (x) dZ = ,Z sin ; (18) dx 2
2
2
2
sin
2
2
2
2
0
2
2
2
2
2
2
8
HILARY BOOTH
where
8 <
x in the spherical case f (x) = : x in the cylindrical case 1 in the z dependent case
and
2
8 <
r in the spherical case x = : in the cylindrical case z in the z dependent case, r being the distance from the origin and the distance from the central axis. It is easily shown that these four rst order equations can be written as the fourth order, d f (x) d , sin d dx dx dx d d d (19) sin = 0: + dx f (x) dx , sin dx 2
2
2
2
2
2
In the cylindrical case, Z () is the charge per unit ring radius , F () is the charge within a radius , and A is the scalar potential, A . See [4]. Similar physical quantities are represented in the spherical and z cases. Most importantly, we are looking for solutions whose charge density Z () decreases rapidly towards in nity, so that we can nd solutions which are localized or particle-like. Our zero total charge result [5] tells us to expect that F ! 0 as ! 1 and likewise, A ! const as ! 1. As pointed out by Chris Cosgrove (private communication), equation (19) has non-integer resonance numbers [1] [2] and we do not expect to nd an integrable system (in the soliton sense) here. Instead, we show that there are a family of orbits (in the sense of [10]) all of which approach the trivial (constant) solution at in nity. As an example, we look at the cylindrical case, noting that similar results hold in the spherical and z case. The numerical orbits in the Figure 1 complete the results in [4] in which a single member (analytic in ) of these families was shown to exist and calculated numerically. 0
1
5. Vacuum Maxwell singularities near the origin In [4] it was shown that in the cylindrical case Lemma 1. Suppose ( ; F; A; Z ) is a solution to Equation(18) on I = (0; ), for some ; 0 < < 1. Suppose also that Z 0 is continuous and bounded on I . Then, 1
1
1
ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO NONLINEAR PDE 9
(i) F is C on I and has a well de ned, nite limit as ! 0. Z has a well-de ned limit as ! 0. (ii) if F (0) 6= 0 then A is unbounded as ! 0. In particular, A =
() ln(), where is C and bounded on I, ! F (0) as ! 0. Also, is bounded as ! 0. Lemma 2. Suppose ( ; F; A; Z ) is a solution to Equation(18) on 2 (0; 1). Suppose also that Z 0 with F continous and bounded on the interval. Then (i) If F ( ) 0 for some 2 [0; 1), then ! 1, A ! 1, and F ! 1 as ! 1. (ii) If F < 0 on (0; 1) then F ! 0 as ! 1. In addition, if A and Z have well-de ned limits as ! 1 then Z ! 0 and A ! A1 as ! 1, with ,1 A1 1. Similar results were established for the spherical case in [29]. In [29][4], Radford and the current author have shown that the behaviour of solutions near the origin resembles the vacuum solutions of the Maxwell equations. Given that the Maxwell equation can be written as the square of a Dirac operator (see for example [26]), it should be possible to formulate these results, together with exponential decrease in terms of the properties of k-monogenic functions | those functions which are solutions to Dk = D + k = 0; where is a basis vector (corresponding to the time coordinate) of a complex Cli ord algebra. The fundamental solution of the k-Dirac operator has the same singularity at the origin and decreases exponentially as x ! 1. In this case, the operator has coecients in , the subspace of vectors . We are currently working on a clari cation of this point. 1
2
1
1
0
0
1
6. Numerical Solutions In [29][4] [8] we showed examples of numerical solutions exhibiting the characteristics referred to in Section 5. Earlier attempts at nding numerical solutions[32] [25] were marred by a \simplifying assumption" which proved to be invalid { solutions did not exist in that case. See [29]. In [8] we noted that there are families of orbits parameterized by the constant cjxj where we assume that solutions are of the form e,cjxjg(jxj), with g(jxj) an analytic function. By varying the value of c, the boundary conditions are perturbed to neighboring solutions
10
HILARY BOOTH
(orbits), all decreasing exponentially at in nity. See Figure 1. All values of c yield solutions which satisfy the two Lemmas, and the zero total charge result. That is, all solutions surround a central wire along the axis of symmetry. Similar numerical solutions can be found in the spherical case [6]. The spherical solutions surround a central Coulomb eld, but the static condition forces a monopole at the origin [29] [5] [30], that is, an unbounded A component. Further ODE solutions occur in the z case [5] and 1 + 1 case, neither of which have been fully examined in the previous literature [12] [13]. These results will be forthcoming also in [6]. Similar spherically-symmetric solutions were found in [27]. This time the Schrodinger-Newton equations provided an identical coupling as in the static Maxwell-Dirac case, which is essentially an elliptic system. More work must be done to investigate the stability of both of these systems. In [27] implications have been developed in the context of quantum gravity, but work in this area is far from complete. We are currently considering the problem in this context. The solutions found in [27] blow-up at larger distances from the origin. However, in the Maxwell-Dirac case, this behaviour appeared only as a numerical anomaly. When the total charge became slightly positive (due to the step-size of the numerical solution), we entered a regime described in Lemma 2 of [4] in which all solutions become unbounded as ! 1. These solutions were illustrated numerically in [4], but discarded as being of less interest than the bounded solutions. The solution shown in Figure 1 was calculated using the MATLAB ODE solver ODE113. The relative error tolerance was set at 1e , 4 and the absolute error tolerance at 1e , 8. The same behaviour was observed when the tolerances were decreased to 1e , 6 and 1e , 12 or increased to 1e , 3 and 1e , 6. All solutions remained stable whether calculated as a function of increasing or decreasing radius (i.e. shooting away from or towards the central axis). The two dimensional cases (static axi-symmetric, circular current axi-symmetric, the massive 1 + 1 case) still require reliable numerical results. Those available to date [32] [25] have been awed by the imposition of \approximations" which were shown in [29] to be possible only in trivial cases (in which the equations are no longer coupled). 7. Vanishing Total Charge In the solutions in Figure 1 the variable F () representing the total charge within a ring, radius , tends towards zero at 1. Lemma 1 also states that the potential must tend towards a solution which is
ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO NONLINEAR PDE 11 Orbits of charge on ring radius r
chi vs r 0
18 −50
16
14 −100 12
chi
Z
−150 10
8
−200
6 −250 4 −300
2
0
0
5
10
15
0
5
r
10
15
10
15
r Total charge within radius r
Potential vs log r 0
700 −10
600
−20
−30
500
−40
A
F
400 −50
300 −60
200
−70
−80
100
−90
0 −1 10
0
1
10
10
0
5 r
Figure 1. A family of orbits exhibiting the properties
of Lemma 1 and Lemma2.
logarithmic in , near the central axis. This corresponds to a central charged wire (along the z axis) which is a solution to the vacuum Maxwell equations. As such, we can think of this part of the solution as representing an external eld. (The scalar potential could be separated at this point into Aexternal + Afermion interaction , since the Aexternal , as the homogenous solution, contibutes nothing to the coupling between the Maxwell and Dirac equations.) Physically, this means that the (inner) external eld must be surrounded by an equal and oppositely charged eld. The total electric charge of the system Z , 1 r A :dS ; with S the unit ring of radius ; lim !1 4 S must vanish. +
0
12
HILARY BOOTH
In [5], we were able to show that this is the case for all stationary solutions, given that they are in a \reasonable" function space. If we de ne an isolated system as one for which all sources are contained in some ball Bk (k < 1) and for which the elds die o as jxj = r ! 1, then we showed in [5] and [30] that an isolated, stationary, static Maxwell-Dirac system is electrically neutral. A similar result, for the total electric charge per unit length in the z direction (or unit ring), holds in the cylindrical case. This shows mathematically, that a (stationary) solution must be atom-like (in the sense that any central charge must be surrounded by an equal and opposite charge. In addition it shows that there cannot be a stationary solution representing an isolated electron (or even one that rotates around the axis with constant velocity). Note also, that the signs of the charges can be reversed, giving us negatively charged singularites and positively charged \electron" elds. All results remain unchanged under such a reversal of sign. The total charge must vanish overall. A more dicult problem presents itself if we are to allow for two fermion elds of opposite charge, for example, the solution corresponding to positronium. We are currently considering ways in which we can have have solutions of opposite charge interacting together. 8. Discussion and Conclusions Until the mathematical problems of QED have been resolved, all methods of addressing the equations governing the interaction between electrons and elds are signi cant. A non-perturbative QED must consist of classical and/or semi-classical arguments which aim to justify the Feynman integral method, and the quantized eld approach. For the coupled M-D system, we have existence (for small initial data), existence for the stationary case, and descriptions of various subcases. But analysis of the system is far from complete. We have indicated, within this review paper, the directions we are currently pursuing. References [1] M.J.Ablowitz,A.Ramani,H.Segur A connection between nonlinear evolution equations and ordinary di erential equations of P-type.I J.Math.Phys. 21 (4), 715-721 (1980) [2] M.J.Ablowitz,A.Ramani,H.Segur A connection between nonlinear evolution equations and ordinary di erential equations of P-type.II J.Math.Phys. 21 (5), 1006-1018 (1980)
ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO NONLINEAR PDE 13 [3] R.Bartnik, J McKinnon, Particlelike solutions of the Einstein-Yang-Mills equations Phys. Rev. Lett., 61 141-144, 1988. [4] H.S.Booth, C.J.Radford, The Dirac-Maxwell equations with cylindrical symmetry J. Math. Phys. 38 (3), 1257-1268 (1997). [5] H.S.Booth, The Static Maxwell-Dirac Equations PhD Thesis, University of New England (1998). [6] H.S.Booth Various ODE Solutions to the static and 1+1 Maxwell-Dirac Equations (in preparation). [7] H.S.Booth,P.Jarvis,G.Legg, Algebraic solution for the vector potential in the Dirac equation (in preparation). [8] H.S.Booth, Nonlinear electron solutions and their characteristics at in nity The ANZIAM Journal (formerly J.Aust.M.S (B)) (to appear). [9] J.Chadam Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac system in one space dimension J.Funct.Anal. 13, 173-184 (1973). [10] K.Cieliebak and E.Sere, Pseudoholomorphic curves and multiplicity of homoclinic orbits, Duke Mathematical Journal 77, (2) 483-518 (1995). [11] R.Coquereaux, Modulo 8 periodicity of real Cli ord algebras and particle physics, Phys Lett 115B (5), 389-395 (1982). [12] A.Das, D.Kay, A class of exact plane wave solutions of the Maxwell-Dirac equations, J.Maths.Phys. 30 (10) (1989) 2280-2284. [13] A.Das, General Solutions of Maxwell-Dirac equations in 1+1 dimensional space-time and a spatially con ned solution, J.Maths.Phys. 34 (9) (1993) 39863999. [14] R.Delanghe,F.Sommen,V.Soucek, Cli ord Algebra and Spinor-Valued Functions, KLUWER (1992). [15] C.J. Eliezer, A Consistency Condition for Electron Wave Functions, Camb.Philos. 54, 2 (1957). [16] M.Esteban, V.Georgiev, E. Sere, Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations, Calc. Var. 4, 265-281, (1996). [17] Felix Finster, Niky Kamran, Joel Smoller, Shing-Tung Yau, Nonexistence of time-periodic solutions of the Dirac equation in an axisymmetric black hole geometry CPAM 53,7, 902-929 (2000). [18] M. Flato, J.C.H. Simon, E Ta in, Asymptotic Completeness, Global Existence and the Infrared Problem for the Maxwell-Dirac Equations Memoirs of the American Mathematical Society, (May 1997). [19] V.Georgiev, Small amplitude solutions of the Maxwell-Dirac equations Indiana Univ. Math. J. 40 (3), 845-883 (1991). [20] R.T.Glassey, W.A.Strauss Conservation Laws for the Maxwell-Dirac and Klein-Gordon-Dirac equations, J.Math.Phys., 20, (1979). [21] L.Gross The Cauchy problem for the coupled Maxwell and Dirac equations Comm. Pure Appl. Math. 19 1-5 (1966). [22] P.S.Howe, G.Sierra, P.K.Townsend Supersymmetry in six dimensions Nuclear Phys B221 331-348 (1983). [23] E.H. Lieb and M. Loss, Self-Energy of Electrons in Non-perturbative QED , in Di erential Equations and Mathematical Physics, University of Alabama, Birmingham, 1999, R. Weikard and G. Weinstein, eds. 255-269 Internat. Press (1999). arXiv math-ph/9908020, mp arc 99-305.
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[24] E.H. Lieb and M. Loss, Remarks about the ultraviolet problem in QED, (in preparation). [25] A.G. Lisi, A solitary wave solution of the Maxwell-Dirac equations, J.Phys.A 28 No.18, 5385-5392 (1995). [26] A.McIntosh, Cli ord algebras, Fourier theory, singular integrals, and harmonic functions on Lipschitz domains in: Cli ord Algebras in Analysis and Related Topics (John Ryan ed), 33-88 CRC Press Boca Raton, FL, 1996. [27] I.M.Moroz,R.Penrose,P.Tod, Spherically-symmetric solutions of the Schrodinger-Newton equations, Class. Quantum Grav. 15 2733-2742 (1998). [28] R.Penrose,W.Rindler, Spinors and Space-Time I (Two Spinor Calculus and Relativistic Fields), CAMBRIDGE UNIVERSITY PRESS, (1984). [29] C.J. Radford, Localised Solutions of the Dirac-Maxwell Equations, J. Math. Phys. 37 (9), 4418-4433 (1996). [30] C.J.Radford, H.S.Booth Magnetic Monopoles, electric neutrality and the static Maxwell-Dirac equations J.Phys.A:Math.Gen. 32, 5807-5822 (1999). [31] J.Schwinger Gauge Invariance and Mass. II Phys. Rev. 128 (5), 2425-2429 (1962). [32] M. Wakano, Intensely Localized Solutions of the Dirac-Maxwell Field Equations, Prog.T.Phys. 35, 1117-1141, (1966). Centre for Mathematics and it Applications, Australian National University, Canberra ACT 0200 Australia
E-mail address :
[email protected]
QUANTUM MECHANICS AS AN INTUITIONISTIC FORM OF CLASSICAL MECHANICS JOHN V. CORBETT AND MURRAY ADELMAN Abstract. Intuitionistic real numbers are constructed as sheaves
on the state space of the Schrodinger representation of a CCRalgebra with a nite number of degrees of freedom. These numbers are used as the values of position and momentum variables that obey Newton's equations of motion. Heisenberg's operator equations of motion are shown to give rise to numerical equations that, on a family of open subsets of state space, are local approximations to Newton's equations of motion for the intuitionistically valued variables.
Introduction
Do sub-atomic particles have positions and momenta at all times? Can the numerical value of the position of such a particle always be given a triplet of real numbers? What real numbers should be used? In this paper we argue that there is a class of real numbers that can be used to label the positions and momenta of sub-atomic particles at all instances of time. Furthermore if these numbers, as values of positions and momenta, are assumed to satisfy the equations of motion of classical mechanics, then these classical equations are approximated locally by the quantum mechanical operator equations of motion, restricted to act on certain subsets of state space. The real numbers that we use are real numbers in the topos of sheaves on the state space of the Schrodinger representation of a CCR-algebra with a nite number of degrees of freedom. In the standard Hilbert space framework for quantum mechanics[1] the algebra of the canonical commutation relations (CCR) is generated by the operators Qj = Qj , Pj = Pj, j = 1; : : : ; n, and the identity operator I which satisfy the relations [Pj ; Qj ] = ,iI , [Pj ; Qk ] = 0 if k is di erent from j , [Qj ; Qk ] = 0, [Pj ; Pk ] = 0 for all j; k. We have put Planck's constant divided by 2 equal to one. We will further simplify the notation by taking n = 1. 15
16
JOHN V. CORBETT AND MURRAY ADELMAN
1. The Schrodinger representation The Schrodinger representation of the CCR-algebra M is the representation in which the Hilbert space is L (R). Q is represented by multiplication by the real variable x and P by (1=i) times the operator of di erentiation with respect to x. Let S (R ) denote the Schwartz space of in nitely di erentiable functions of rapid decrease on R . Then the physical quantities are represented by self-adjoint elements in the closure M of the CCR-algebra , M , where M is the smallest closed extension of M . M = X X 2 M and X , is the restriction to S (R ) of the Hilbert space closure of X . Following the de nitions of Powers[7], M is essentially self-adjoint because the adjoint M of M equals the closure M of M . 2. The State Space ES We give only a resume of the results, for more details see Inoue[8]. A linear functional f on M is strongly positive i f (X ) 0 for all X 0 in M . De nition 1. The states on M are the strongly positive linear functionals on M that are normalised to take the value 1 on the element I of M . Theorem 1 (Inoue[8]). Every strongly positive linear functional on M is given by a trace functional. De nition 2. The state space ES of the Schrodinger representation of M is the set of all strongly positive linear functionals on M that are normalised. The state space ES is contained in the convex hull of projections P onto one-dimensional subspaces spanned by unit vectors u 2 S (R ). Recall that each state 2 ES is a trace class positive bounded operator P with trace 1, which can be written as = PnPn, where the sum over n may go from 1 to 1. For all n, n 0, n = 1 and the Pn are orthogonal projections. The following result demonstrates that all states in ES must satisfy the further condition that as n approaches in nity the sequence fng converges to zero faster than any power of 1=n. Theorem 2. Given = P nPn 2 ES , where the projections Pn project onto one-dimensional subspaces spanned by unit vectors un 2 S (R ). tr A is nite for any self-adjoint operator A in the Schrodinger representation of the CCR-algebra M if and only if lim nk n = 0, for all integers k > 0. 2
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Proof. Let A be a self-adjoint operator in M and is a state on M which belongs to ES . Then for any orthonormal basis of P H, fumg, m = 1 to 1, which is composedPfrom elements in S , tr A = (um ; Aum) . Now using = nPn the trace becomes a double sum X X, um ; nPnAum : tr A = Because the trace is independent of the basis used to calculate it, we can choose the ortho-normal basis fumg to be in the ranges of the Pn so that the double sum reduces to X , tr A = m um ; Aum : Since A is in the algebra M , it is a polynomial of some nite degree k in the self-adjoint operators P , Q. By Lemma 18 in Ja e[10], A is majorized P 2 by a2 polynomial of degree k in the self-adjoint operator 1 H = 2 (P + Q ), , , um ; Aum 2 um ; pk (H )um : When the fumg are the eigenfunctions of H , pk (H )um = pk (m)um, where m = 12 (2m + 1) are the eigenvalues of H , so that the absolute P th value of the m term of the series m (um; Aum) satis es 0 m (um; Aum) , 1 m um; H (m)um 2 = m pk (m ) 21 : Therefore if lim nk n = 0, for all k > 0, then 9 a positive integer N and a positive constant C such that, for all m > N ,
mpk (m) 12 < mC P and so the series m (um; Aum) converges absolutely by the comparison test. On the otherPhand, if tr A is nite for all self-adjoint operators A then the series m (um; Aum) must converge. If A is positive then the convergence is absolute so that when A is a monomial of degree k in the Hamiltonian operator H ,then lim m(m) 12 k = 0 for all positive k which shows the converse is true. The same results hold even when the fumg are not the eigenfunctions of H . The minimax principle for eigenvalues of symmetric operators, see for example, Kato[6] section I.6.10, implies that , um; pk (H )um pk (m) , because it states that the maximum of v; pk (H )v , when v is a unit vector and (v; ei) = 0, i = 1; ; n , 1, for any orthonormal set of vectors fei g, is the eigenvalue pk (m) obtained when the feig are the 2
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JOHN V. CORBETT AND MURRAY ADELMAN
rst n , 1 eigenvalues of H . This result is applicable here to the set of vectors fuig, i = 1; ; m, with v = um. We have shown that tr A is nite for any self-adjoint operator A in this representation of the CCR-algebra M if and only if lim nk n = 0, for all integers k > 0. The topology on ES is chosen to make the functions A^, where A^() = tr A, continuous when A is an essentially self-adjoint continuous operator on S (R ). 3. The topos Shv(ES ) A sheaf Y on a topological space X can be described[3] by a rule which assigns to each point x of X a set Y (x) consisting of the germs of a prescribed class of functions, where the germs of the functions are de ned in neighborhoods of the point x. The collection of sets Y (x) which are labelled by points x in X can be glued together to form a space Y in such a way that the projection from Y onto X is a local homeomorphism; that is, for each x in X and each y on the ber above x (i.e. for each such y the projection of y onto X is x) there is a neighborhood N of y such that the projection of N onto X is a neighborhood of x. A section of the sheaf Y over the open subset U of X is a function s from U to Y that belongs to the prescribed class of functions and satis es the condition that, for all x in U , the projection of s(x) onto X is x. The sheaf construction allows a section f de ned on the open set U to be restricted to sections f V on open subsets V contained in the open set U and, conversely, the section f on U can be recovered by patching together the sections f V where V 0 belongs to an open cover of U . A spatial topos is a category of sheaves on a topological space. The objects of this category are sheaves over the topological space and the arrows are sheaf morphisms, that is, an arrow is a continuous function that maps a sheaf Y to a sheaf Y 0 in such a way that it sends bers in Y to bers in Y 0, equivalently, sections of Y over U to sections of Y 0 over U , where U is an open subset of X . The topos Shv(ES ) of sheaves on the topological space ES is constructed in this way. In 1970, Lawvere[2] showed that toposes can be viewed as a \variable" set theory whose internal logic is intuitionistic. The propositional calculus of the logic of the spatial topos of sheaves over X is the Heyting Algebra[3] of the open subsets of X . This means that as well as 0
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being true or false, propositions in this logic can be true to intermediate extents which are given by open subsets of X . True corresponds to the whole set X . False corresponds to the empty set. There exist Boolean algebras in which propositions can be true to varying extents but, in addition, the Heyting algebra of open sets does not satisfy all the laws of classical logic. Two of the most striking di erences between classical and intuitionistic logics are that the law of the excluded middle and the Axiom of Choice do not hold for intuitionistic logic. We have argued elsewhere[5] that aspects of quantum mechanics, such as the two slit experiment,that are dicult to understand with Boolean logic are better described using intuitionistic logic. There is an analogy between the language of toposes and that of sets which makes it easier to work in toposes. Sheaves in a topos correspond to sets, subsheaves of a sheaf to subsets and local sections to elements of a set. Then as long as a proof in set theory does not use the law of excluded middle or the Axiom of Choice then it can be translated into a proof in topos theory.
4. Real Numbers in Spatial Toposes Dedekind numbers are de ned to be the completion of the rational numbers obtained by using cuts, and Cauchy numbers are de ned as the completion of the rationals obtained by using Cauchy sequences. These di erent constructions can only be shown to be equivalent by using either the Axiom of Choice or the law of the excluded middle.[3] Therefore, when intuitionistic logic is assumed, we expect that these two types of real numbers are not equivalent. It has been shown[4] that in a spatial topos the sheaf of rational numbers Q is the sheaf whose sections over an open set U are given by locally constant functions from U with values in the standard rationals while the sheaf of Cauchy reals R is the sheaf whose sections over an open set U are given by locally constant functions from U with values in the standard reals. A function is locally constant if it is constant on each connected open subset of its domain. On the other hand the sheaf of Dedekind reals R is the sheaf whose sections over U are given by continuous functions from U to the standard reals. The Cauchy reals form a proper sub-sheaf of the Dedekind reals unless the underlying topological space X is the one point space. C
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5. The Quantum Reals By the construction of the topology on ES for any self-adjoint operator A in the Schrodinger representation of the CCR, the function tr A is a globally de ned continuous function and therefore in R . We interpret the functions A() = tr A, with domains given by open subsets of state space, to be the numerical values of the physical quantity that is represented by the self-adjoint operator A. Not every Dedekind real number is of this form. Real numbers of this form are a proper subsheaf of the Dedekind real numbers in the spatial topos of sheaves on the state space ES . We will call them the quantum reals, they belong to the sheaf of locally linear functions.[5] De nition 3. If U is an open subset of ES then the function f from U to the standard reals is locally linear at in U and there is an open neighborhood U 0 of , with U 0 inside U , and a bounded self-adjoint operator A such that f U = A^ U . De nition 4. The sheaf of locally linear functions, A , is de ned by its sections over any open subset U of ES as the set of all locally linear functions on U with the requirement that if the open set V is contained in U then the sheaf of locally linear functions over V is obtained by restricting the locally linear functions over U . The global elements of the locally linear functions are given by the functions A^, where A is a self-adjoint operator, that are continuous on S . It suces to de ne algebraic relations between elements of A globally, because ES is locally connected and so we can treat functions which are de ned on disjoint connected components as if they were globally de ned. When U is an open neighbourhood of the state then the quantum real numbers belonging to the sections of A over U can be thought of as real numbers tangent to those Dedekind reals that have a tangent space at . 6. The Dedekind Reals R Stout[4] has shown that the usual order on the rational numbers Q can be extended to the following order on R . De nition 5. The order relation < on the Dedekind reals, R , is given by the de nition: , x < y if and only if 9 q 2 Q (q 2 x ) ^ (q 2 y,) where x is the upper cut of x and y, is the lower cut of y . The relation < is the subobject of R R consisting of such pairs (x; y). D
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Trichotomy does not hold universally for the order < on R . The order on R has the property that x y is not the same as (x < y) _ (x = y). De nition 6. The order relation x y is the subobject of R R consisting of the pair (x; y ) with x y and y, x,, where x is the upper cut of x and x, is the lower cut of x, and similarly for y. Stout[4] also showed that the statement (x y) is equivalent to the statement :(y < x). De nition 7. The open interval (x; y) for x < y is the subobject of R consisting of those z in R that satisfy x < z < y . The closed interval [x; y] for x < y is the subobject of R consisting of those z in R that satisfy x z y . The open intervals can be used to construct an interval topology T on R analogously to the interval topology on the standard real numbers that is generated from the open intervals by nite intersection and arbitrary union. The topology T on R is such that Q is dense in R with respect to T. If the max function is de ned using the order by the conditions: (i) x max(x; y) and y max(x; y), and (ii) if z x and z y then z max(x; y) and the norm function: j j : R ! R is de ned by jxj = max(x; ,x), then the norm j j satis es the usual conditions of non-negativity, that only 0 has norm zero and that the triangle inequality holds.
(R ; T ) is a metric space with the metric d(x; y) = x , y . It is both complete and separable[4]. R is a eld in the sense that for all a in R ,if a does not belong to the sheaf of germs of invertible functions, Unit(R ),then a = 0. 7. Properties of the Quantum Reals Theorem 3. A is a proper sub-sheaf of the sheaf of Dedekind numbers R and is dense in R in the metric topology T . The sheaf A inherits the orders and < from R . On the other hand A can be ordered as a consequence of the orders on the selfadjoint operators: 1. A is strictly positive, A > 0, if (Au; u) > 0 for u 6= 0, u 2 D(A). 2. A is non-negative, A 0, if (Au; u) 0 for all u 2 D(A). Lemma 8. The orders and < on A inherited from R are equivalent to those obtained from those on continuous self-adjoint operators. D
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JOHN V. CORBETT AND MURRAY ADELMAN
Proof. A is a non-negative self-adjoint operator i tr A 0 for all in ES , i.e. A^ 0 globally. When, as a Dedekind real number, a = A^ 0 then 0+ a+ and a, 0,. Globally, 0+ = fq 2 Q j q > 0g and 0, = fq 2 Q j q > 0g so that, if a = A^ 0 then A is a non-negative operator and if A is a non-negative operator then a = A^ 0. The positivity order for a continuous self-adjoint operator A is equivalent to A being bounded away from zero, i.e. there exists a rational number q > 0 such that (u; Au) > q for all u 6= 0. This gives the equivalence of the operator > with the > for Dedekind reals restricted to A , because for the latter a > 0 means globally that 9q 2 Q with q 2 0+ and q 2 a,. If a = A^ and b = B^ then the R D distance between a and b is given by the metric ja , bj on RD . There is another metric on the quantum numbers A ,1 it is given by the number jA\ , B j, where jC j is the operator jC j = (C 2) 2 , when C is self-adjoint. Proposition 9. The two metrics coincide on A , that is, jA\ , Bj = ^ ^ ja , bj, for all pairs of quantum numbers a = A and b = B . Proof. It is sucient to let B = 0. We will only consider the global sections. see for example section VI.2.7 of Kato[6], that It is well-known, (u; Au) (u; jAju) for all u in the domain D (A) = D (jAj), whence tr A tr jAj for all in ES , i.e. jA ^j jc Aj. ^ ,A^) and jc As on elements of R D , jA^j = max(A; Aj = jc Aj. The ^ ^ ^ lower cut of jAj is the union of the lower cuts of A ,andof ,A, that is jA^j, = (A^), [ (,A^),, which means that jA^j, jc Aj ,. The upper cut of jA^j is the intersection of the upper cuts of A^ and of ,A^, that is q 2 Q belongs to, jA^j+ if q is greater than or equal to both A^ and of , + + ^j+, therefore jA^j+ jc ,A^. Thus if q 2 jc Aj then q 2 j A A j . c c ^ ^ This shows that jAj A and therefore jAj = A . The metric is used to de ne Cauchy sequences in R D , this result means that we can de ne Cauchy Sequences in A in the same way. In order to ensure uniqueness of the limits of a Cauchy sequence we need a concept of apartness, ><, that is stronger than that of not equal to, 6=. De nition 10. The pair (a; b) of Dedekind reals are apart, a >< b, i (a > b) _ (a < b). Proposition 11. a >< b i ja , bj > 0.
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8. Calculus of Functions The concept of limit is available for functions: R ! R and di erential calculus for such functions can be developed. The de nition of the limit of a function G 2 R is modeled on the standard de nition; the limit of G as x tends to b is L i 8n > 0; 9 m > 0; 8x 2 R ; 0 < jx , bj < 1=m =) G(x) , L < 1=n . In this de nition both m and n are in Q . The uniqueness of the limit L holds in the sense that if L and M are two limits of G as x tends to b then L and M are apart, L , M = 0. For the requirements of this paper it is enough to consider only polynomial functions. For any b 2 R all powers bn of b are in R for n a natural number because products of continuous functions are continuous. Let both b and c 2 R , then bc = cb 2 R for the same reason. The product, however, is de ned only on the intersection of the domains of the continuous functions so that care has to be taken with the extent to which the product exists. Sums of numbers b + c exist to the extents given by the intersection of the extents of b and c. Hence we can construct polynomials D
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F (b) =
X
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am bm
where the sum is over nitely many terms and the coecients am 2 R . We can construct power series by de ning the convergence of the sequence of partial sums in the metric on R . The derivative of bm = mbm, which implies that the derivative of a monomial is de ned to the same extent as the monomial. This allows us to obtain the derivatives of polynomials and power series. We de ne continuity of a function F : R ! R at b in its domain by the requirement that D
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(1) 8n > 0; 9m > 0; 8x 2 R ; D
0 < jx , bj < m1 =) F (x) , F (b) < n1 :
In this de nition we have taken m and n to be in Q . A function is continuous on an interval I in R i it is continuous at each number in I . Interesting phenomena occur in this calculus and they deserve more study. However we have enough structure now to return to the dynamics of quantum mechanics. D
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9. Comparing the Dynamics Consider the example of a non-relativistic quantum particle of positive mass that moves in a central force eld which is derived from a potential function V . We assume that the quantum values X^ of the position of the particle satisfy Newton's equations of motion globally, that is, DQ^ = 1 P^ (2) and (3) DP^ = F (Q^ ) or (4) D (Q^ ) = F (Q^ ) where P^ represents the momentum of the particle and D denotes di erentiation with respect to time. F represents the force, it is the negative gradient of the potential function V as a function: R ! R . We will now prove that standard quantum mechanics is a local approximation to a global classical mechanics. More precisely, the theorem states that if the quantum values of the components of position and momentum of a particle are assumed to satisfy Newton's equations of motion globally then the self-adjoint operators corresponding to these values locally satisfy equations that well approximate Heisenberg's equations of motion. The theorem relates a set of operator equations (Heisenberg's equations) to a set of numerical equations (Newton's equations). The straightforward way to get a numerical equation from an operator equation is to multiply each side of the operator equation by a suitable trace class operator (a state) and then take the trace of each side. The original operator equation has become a family of numerical equations which is labelled by the states. Recall that Heisenberg's equations for an operator A are (5) DA = ,i[A; H ] where H is the Hamiltonian operator of the system and the square bracket denotes the operator commutator. Therefore, we get Heisenberg's equations of motion, when Q and P represent operators and the Hamiltonian operator is H = (1=2)P + V (Q), to be (6) DQ = 1 P 2
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and (7) DP = i[P; H ] = F (Q) : If we multiply each side of equations (6) and (7) by the operator and then take the trace of each side we get Heisenberg's numerical equations: (8) D(tr Q) = 1 tr P
and , , (9) D(tr P ) = tr (DP ) = tr i[P; H ] = tr F (Q) : Equation (8) is just Newton's equation (2) at the state . Equation (9) is similar to Newton's equation (3) at the state . The di erence between equations (3) and (9) is the same as that which restricts the validity of the result known as Ehrenfest's Theorem, namely, that, in general, (10) tr F (Q) 6= F (tr Q): To simplify the discussion we remove the explicit dependence of the equations (2), (3), (8), and (9) on the operators P and use equations of motion in the form of the second order di erential equations: Newton's equations to the extent W are , , (11) D Q^ W = F Q^ W : Heisenberg's numerical equations to the extent W are , (12) D Q^ W = F[ (Q) W : It is possible, however, that the di erence between the two sides of (10) is small at some state a and remains small for all states in an open neighborhood of a . In this case the equations (11) and (12) are approximately the same in that open set. We deduce that Heisenberg's equations approximate Newton's equation in that open set because if (11) holds in an open set W then it holds at each in W . We claim that for a suitable class of functions F , Heisenberg's equations approximate Newton's equations locally. The localness of the assertion is twofold, we mean that for every standard real number r we ^ can nd an open set,W , on which both the number Q W is arbitrarily , close to r, and F[ (Q) W is arbitrarily close to F Q^ W . The physical interpretation is that if an observer's measurement apparatus is located in a neighbourhood of the position r (in this world with one spatial dimension) then the di erence between the accelerations of the particle due to the two forces cannot be distinguished with this apparatus. 2
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The class of suitable functions is de ned through the concept of S continuity. De nition 12. We call functions G, S -continuous, if they are realvalued continuous functions on R such that,for the position operator Q, G(Q) : S ! S , is continuous in the standard countably normed topology on the Schwartz space S . Theorem 4. If the force F is S -continuous, then given " > 0, Heisenberg's equations of motion approximate Newton's equations of motion to within " on each member of a collection of open sets fW (r; ")g of state space ES , indexed by the standard real numbers r and " > 0. That is, on each fW (r; ")g, , [ ^ W < " : (13) F (Q) W , F Q Proof. The idea behind the proof is to nd states r on which F (tr r Q) closely approximates tr r F (Q), then F (Q^ ) will be close to F[ (Q) when , [ ^ ^ ^ ^ F (Q) is close to F Q(r ) 11 and F (Q) is close to tr r F (Q)11 as Dedekind numbers. The following pair of lemmas complete the proof. Lemma 13. If Q is a self-adjoint operator which has only absolutely continuous spectrum, then for any real number r in its spectrum we can construct a sequence of pure states fng such that for the given S -continuous function F , lim tr nF (Q) = F (r) : Proof. Weyl's criterion for the spectrum of self-adjoint operators[9] implies that, for any number r in the spectrum of Q, there exists a sequence of unit vectors fung, in the domain of Q, such that
lim (Q , r)un = 0 : Take n to be the projection onto the one dimensional subspace spanned by un. It is easy to check that lim tr nQ = r : The vectors fung can be chosen to be in S . For example, for any positive integer n, let ,
un(x) = n , exp , n (x , r) The sequence fung satis es the requirements of Weyl's lemma for the operator Q and the number r. 1 2
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Furthermore we can nd a sequence of vectors fung in S so that for n large enough the support of fung lies in a narrow interval centred on r. Then, by the spectral theorem for Q, the corresponding pure states n form a sequence so that both lim tr n F (Q) = F (r) and lim tr nQ = r : From Lemma 13, given a real number r in the spectrum of Q, the S continuous function F and a real number " > 0, there exists an integer N such that, for all j > N , both tr j F (Q) , F (r ) < " and (tr j Q) , r < where is such that F (r ) , F (x) < " (14) when jr , xj < : We choose r = j , for some j > N , and deduce that tr r F (Q) , F (tr r Q) < " (15) because tr r F (Q) , F (tr r Q) = tr r F (Q) , F (r) + F (r) , F (tr r Q) tr r F (Q) , F (r) + F (r) , F (tr r Q) : With this choice of r ,the open set W (r; ") can be de ned as W (r; ") = N (r ; Q; ) \ N (r ; F (Q); ") where satis es the requirements (14), and for any S -continuous function F and " > 0 we have that N (r ; F (Q); ") is given by 1 6
1 6
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n
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N r ; F (Q); " = ; tr F (Q) , tr r F (Q) < " :
Lemma 14. When r is chosen so that equation (15) holds, then for all in W (r; "), (16)
tr F (Q)
, F (tr Q) < " :
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JOHN V. CORBETT AND MURRAY ADELMAN
That is, for W = W (r; "), F[ (Q) W , F (Q^ W ) < " : In the de nition of the open neighborhood W, may depend upon r , as well as on F and ". Proof. For any pair of states, and r , we have tr F (Q) , F (tr Q) tr F (Q) , tr r F (Q) + tr r F (Q) , F (tr r Q) + F (tr r Q) , F (tr Q) : If is in N (r ; F (Q); ") the rst summand is < ", as is the second by choice of r . The nal summand is also because by assumption the function F : R ! R is continuous everywhere in the usual topology on R . Because given " > 0, there exists a ( x ) > 0, such that F ( x ) , a F (xa) < ", whenever jx , xa j < . Apply this to x = (tr Q) and xa = (tr r Q). Therefore, given r , for any in N (r ; Q; ) \ N (r ; F (Q); "), the inequality (16) holds. The question remains whether we can construct suciently many of these open sets. for a given smooth function F , the family In general, of open sets W (r; ") , does not form an open cover of state space ES . However for every standard real number r, and hence for every point in classical coordinate space, all the standard real numbers that lie within of r in the standard norm topology on R also lie in W (r; ") as Cauchy real numbers R . In this sense, the family of open sets W (r; ") covers the classical coordinate space of the physical system. 1 3
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10. Conclusion It is important to be clear about the meaning of this result. It involves the comparison of two di erent theories. Assume that Newton's equations of motion hold for the position and momentum variables of a non-relativistic massive quantum particle when they are expressed in terms of the real numbers R . Assume also that these real numbers R are given by sheaves of continuous functions over the state space ES and include quantum numbers like Q^ = tr Q and P^ = tr P for 2 fopen subsets of ES g, where P and Q are self-adjoint operators on an underlying Hilbert space. Then we can nd open subsets of ES such that on each open subset, the restrictions of the functions P^ and Q^ can be reinterpreted as the average values of the operators P and Q which almost satisfy the Heisenberg's equations of motion with the analogous Hamiltonian operator. This result only involves a comparison of the D
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dynamical equations of motion. Comparison of the trajectories requires that the initial data be compatible which leads to further constraints on the allowable trajectories. Nevertheless, there are trajectories that can be described with these real numbers. References
[1] J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, 1955), p. 212. [2] W. Lawvere, \Quanti ers as Sheaves", Actes Congres Intern. Math. 1, (1970). [3] S. MacLane and I. Moerdijk, Sheaves in Geometry and Logic (Springer{Verlag, New York, 1994). [4] L. N. Stout, Cahiers Top. et Geom. Di . XVII, 295 (1976); C. Mulvey, \Intuitionistic Algebra and Representation of Rings" in Memoirs of the AMS 148 (1974). [5] M. Adelman and J. V. Corbett, Applied Categorical Structures 3, 79 (1995). [6] T. Kato, Perturbation Theory for Linear Operators (SpringerVerlag, New York, 1966). [7] R.T. Powers, Commun. Math. Phys. 21, 85{124 (1971). [8] A. Inoue, Tomita{Takesaki Theory in Algebras of Unbounded Operators (Lecture Notes in Mathematics,1699, Springer, 1998) [9] W. Reed and B. Simon, Methods of Mathematical Physics I: Functional Analysis (Academic Press, New York, 1972). [10] A. Ja e, \Dynamics of a cut-o - eld theory", Ph.D. Thesis, Princeton University, 1965; see also B. Simon, J. Math. Phys. 12, 140 (1971). 4
E-mail address : E-mail address :
[email protected]
[email protected]
Department of Mathematics,, Macquarie University, Sydney, N.S.W. 2109, Australia
VILENKIN BASES IN NON-COMMUTATIVE Lp-SPACES P. G. DODDS AND F. A. SUKOCHEV Abstract. We study systems of eigenspaces arising from the rep-
resentation of a Vilenkin group on a semi nite von Neumann algebra. In particular, such systems form a Schauder decomposition in the re exive non-commutative Lp ,spaces of measurable operators aliated with the underlying von Neumann algebra. Our results extend classical results of Paley concerning the familiar WalshPaley system to the non-commutative setting.
1. Introduction It is a classical theorem of Paley [Pa] that the Walsh system, taken in the Walsh-Paley ordering, is a Schauder basis in each of the re exive Lp-spaces on the unit interval. Although the Walsh basis in not unconditional, except in the case p = 2, it was further proved by Paley that partitioning the Walsh system into dyadic blocks yields an unconditional Schauder decomposition. This paper provides an overview of recent work by the present authors (and collaborators) which develops the theory of orthogonal systems in the setting of re exive noncommutative Lp-spaces of measurable operators aliated with a semi nite von Neumann algebra. The work nds its roots in that of Paley and in the subsequent development of Paley's ideas in the more general setting of Vilenkin systems by Watari [Wa], Schipp [Sc], Simon [Si1,2] and Young [Y]. In the classical setting, the Walsh and Vilenkin systems arise as the system of characters of the familiar dyadic and Vilenkin groups. The present approach exhibits non-commutative orthogonal systems as eigenvectors corresponding to the action of an ergodic ow on the underlying non-commutative Lp-space. The classical results are recovered by specialisation to the case where the ow is given by the action of right translation on the space Lp(G), with G an arbitrary Vilenkin group. In this case, the eigenspaces are one-dimensional and are spanned by the characters of G. Research supported by the Australian Research Council (ARC). 30
VILENKIN BASES IN NON-COMMUTATIVE Lp -SPACES
31
2. Preliminaries There is an extensive literature concerning harmonic analysis on compact Vilenkin groups that is related to the theme of present article. In particular we refer the reader to [AVDR], [BaR], [Gos], [SWS], [Sc], [Si1,2], [Vi], [W], [Wa], [Y] and the references contained therein. We be concerned here with Vilenkin groups of the form Gm = Q1 shall Z m k , equipped with the product topology and normalized Haar k measure. Here m = fm(k) j k 2 N [ f0gg is a sequence of natural numbers greater than one and Zm k = f0; 1; 2 : : :; m(k ) , 1g is the discrete cyclic group of order m(k). The dual group Gdm of Gm can be identi ed with the sum 1 1 a a [ = Z Zm k mk =0
( )
( )
k=0
( )
k=0
( )
consisting of all sequences n = (n0; n1 ; : : : ) with nk 2 Zm k for all k and nk 6= 0 for at most nitely many k. See for example [SWS]. The pairing between Gm and Gdm is given by ht; ni = n (t); where 1 Y nk k (t); 8n = (n0 ; n1; : : : ) 2 Gdm n (t) = ( )
and
k=0
k (t) := exp(2i mt(kk) ); 8t = (t0 ; t1; : : : ) 2 Gm: The dual group Gdm is linearly ordered by the (reverse) lexicographical ordering: for n; p 2 Gdm we de ne n < p if and only if there exists k 2 N [ f0g such that nj = pj for all j > k and nk < pk . We shall always consider the system of characters f n : n 2 Gdmg with the enumeration induced by the reverse lexicographical ordering of Gdm. For k = 1; 2; : : : and 1 j m(k) , 1, de ne d0; dk ; d(k;j) 2 Gdm via d0 := 0; dk := fjk g1j ; d(k;j) := fjik g1i : dk ; d(k;j) correspond to k ; jk respectively. The system of characters f n : n 2 Gdmg forms a complete orthonormal system in L2 (Gm; dtm), called the Vilenkin system corresponding to m. Here dtm denotes normalised Haar measure on Gm. =1
=1
32
P. G. DODDS AND F. A. SUKOCHEV
We set M := 1 and Mk := m(k , 1)Mk, and to each n 2 Gdm we assign the natural number 0
1
n=
1 X k=0
nk Mk ; 0 nk < m(k):
This de nes an order preserving bijection between Gdm and N [ f0g. If we denote the character n by n, with n corresponding to n under this bijection, then we may write the Vilenkin system as f ng1 n . =0
There is a well-known and natural measure preserving identi cation between the Vilenkin group Gm and the closed interval [0; 1] given by the map t ! t(t) 2 [0; 1] where 1 X t(t) := Mtk : k+1 k In the special case that m(k) = 2 for all k 2 N [ f0g, the Vilenkin group Gm is the dyadic group D , and the characters f k g1 k may be identi ed with the usual Rademacher system frk g1 on [0 ; 1]. In this k 1 case, the Vilenkin system f n gn coincides with the familiar Walsh system fwk g1 k , taken in the Walsh-Paley ordering. =0
=0
=0
=0
=0
We begin with the following classical results concerning Vilenkin systems. If f 2 Lp (Gm; dtm), we set
ck(f ) :=
Z
Gm
f k dtm ; k 2 Gdm:
Theorem 1 Suppose 1 < p < 1. (i) 9Kp; 8f 2 Lp (Gm; dtm) ; 8n 2 Gdm
X
ck (f ) k
k Kp kf kp :
(1:1)
p
(ii) 9Kp; 8k = 1; 8f 2 Lp (Gm; dtm)
1 0
X @ X
k k=0
dk n
1
cn n A
Kpkf kp:
p
(1:2)
VILENKIN BASES IN NON-COMMUTATIVE Lp -SPACES
(iii) If supk m(k) < 1, 9Kp; 8k;j = 1; 8f 2 Lp (Gm; dtm)
1 m(k),1 0
X X k;j @
k=0 j =1
X dk;j n
1
cn n A
Kpkf kp:
33
(1:3)
p
Remarks In the terminology of Banach spaces (see, for example [LT1,2]), the assertion (1.1) is equivalent to the statement that the Vilenkin system f n ; n 2 Gdmg is a Schauder basis in Lp (Gm; dtm). Similarly, the estimate (1.2) asserts that the system
Xk = clmf n : dk n < dk+1 g; k = 0; 1; 2; : : : is an unconditional Schauder decomposition of Lp (Gm; dtm). The statement of (1.3) is that the system Xk;j = clmf n : dk;j n < dk;j+1g; 1 j m(k,1); k = 0; 1; 2; : : : is a ( ner) unconditional Schauder decomposition of Lp (Gm; dtm), provided that the Vilenkin group Gm is bounded in the sense that supk m(k) < 1. Here, clm(A) denotes the closed linear manifold generated by A. We remark further that if we set k :=
X
dk n
cn n ; k = 0; 1; 2; : : :;
then (1.2) equally asserts that the sequence f k ; k = 0; 1; 2; : : : g is an unconditional martingale di erence sequence in Lp (Gm; dtm). In the case of the classical Walsh-Paley system, that is, for m(k) = 2; k = 0; 1; 2; : : : , Theorem 1 is due to Paley [Pa, 1931], who showed further that statements (i), (ii) are equivalent. In the case of more general but bounded Vilenkin systems, Theorem 1 was established by Watari [Wa,1958]. In this case, (ii) is valid, but is not equivalent to (i). Watari showed that (i) is equivalent to (iii), but that (iii) is not equivalent to (i) if the Vilenkin group is not bounded. For general Vilenkin systems that are not necessarily bounded, Theorem 1(i) is due independently to Young, Schipp and Simon [Y,Sc,Si1, 1976].
34
P. G. DODDS AND F. A. SUKOCHEV
3. Non-commutative Vilenkin systems In order to establish a general framework, it will be convenient to recall some basic facts concerning representations of compact Abelian groups on Banach spaces. Let X be a Banach space, let G be a compact Abelian group with dual group G^ and normalized Haar measure dm. Let = f tgt2G be an action (strongly continuous representation) of G by automorphisms of X . We set c := sup k tkX!X < 1: (1:1) t2G
For each 2 G^ , we de ne the eigenspace X corresponding to and the representation by setting X := fx 2 X : t (x) = ht; ix; 8t 2 Gg: It should be noted that the eigenspace X may be f0g, that X \ X 0 = f0g if 6= 0, and that clmfX : 2 G^ g = X: It is important to note that if X = Lp(G; dm) and if is de ned by setting t (x)(s) := x(s , t); s; t 2 G then X = fc : c 2 C g: In other words, the eigenspaces corresponding to the action of G on Lp(G; dm) given by forward translation are precisely the one-dimensional eigenspaces spanned by the characters of G. The non-commutative framework that we require is provided by the well-known theory of non-commutative integration with respect to a semi- nite trace, introduced by Segal [Se] and Dixmier[Di]. We suppose that (M; ) is a semi- nite von Neumann algebra equipped with a faithful, normal semi- nite trace , and unit 1. For relevant de nitions see [Ta]. A closed densely de ned operator x is said to be aliated with M if ux = xu for any unitary u in the commutant M0 of M. An operator x aliated with M is said to be -measurable if (e(jxj)) < 1 for some > 0, where e(jxj) denotes the spectral projection ,1; (jxj). If 1 p < 1, we denote by Lp(M; ) the space of all -measurable operators x aliated with M for which kxkp := (jxjp) p < 1 (
1
]
VILENKIN BASES IN NON-COMMUTATIVE Lp -SPACES
p
35
where jxj = x x. The precise de nitions and relevant properties may be found in [FK],[GK1]. Let be an action of the Vilenkin group Gm on Lp(M; ). The system of eigenspaces n o Lp(M; )n : n 2 Gdm taken in the (reverse) lexicographical ordering will be called a Vilenkin decomposition of Lp(M; ). An important special case is given when (1) < 1 and the representation is given by an ergodic Gm- ow, that is is an ultraweakly continuous group of trace preserving automorphisms of M, and which is ergodic in the sense that the xedpoint algebra consists of scalar multiples of the identity (see [OPT]). In this case, extends to an isometric action on Lp(M; ) and each eigenspace Lp(M; )n for the exended action coincides the corresponding eigenspace Mn for the original action of on M. Further, each eigenspace Mn is the one-dimensional span of some unitary operator Wn 2 M and the sequence fWng is an orthonormal basis (in the natural sense) in the space L2 (M; ). We consider the special case obtained by taking M to be L1(Gm; dt) acting by multiplication on L2 (Gm; dt), with trace given by integration with respect to Lebesgue measure. If is given by right translation, then the Vilenkin decomposition fMng = f n : n 2 Gdmg is simply that given by the classical Vilenkin basis. This observation shows that Theorem 1 is actually a special case of the following Theorem.
Theorem 2 [DFPS] (i) If 1 < p < 1, then each Vilenkin decomposition fLp(M; )n; n 2 Gdmg is a Schauder decomposition that is, for all x 2 LpP(M; ) there exists a unique sequence xn 2 Lp(M; )n such that x = n2Gcm xn and there exists a constant Kp, depending only on p and the bound c of the representation such that
X
xk
Kpkxkp k
p
(2:1)
(ii) [SF1,2, FS] There exists a constant Kp depending only on p such that for all k = 1,
1 0
X @ X
k k=0
dk n
1
xn A
Kpkxkp:
p
(2:2)
36
P. G. DODDS AND F. A. SUKOCHEV
(iii)[DS] If sup m(k) < 1, there exists a constant Kp depending only on p such that for all k;j = 1,
1 m(k),1 0
X X k;j @
k=0 j =1
X
dk;j n
1
xn A
Kpkxkp:
(2:3)
p
We remark rst that the starting point for the proof of Theorem 2 may be taken to be the assertion of (ii). This is a consequence of the fact that re exive non-commutative Lp-spaces have the (so-called) UMDproperty combined with an application of the well- known transference principle. Details may be found in [SF1,2]. In the case of bounded Vilenkin groups and nite von Neumann algebras, Theorem 2 was established on [DS]. The case that the von Neumann algebra has in nite trace follows from [DS] and [CPSW]. The general theorem is proved in [DFPS]. As observed above, Theorem 2 contains Theorem 1 as a special case. However, the proof of Theorem 2 is new even in its commutative specialisation and is based on essentially non-commutative techniques. To illustrate some of the ideas on which the proof of Theorem 2 is based, we will consider some very special cases. We begin with the commutative example given by taking m =(n; 0; 0; : : :) for some natural number n 2 so that Gm may be identi ed with Zn. We take M to be L1(Zn) = ln1, where Zn is equipped with counting measure and acting on L2 (Zn) = ln2 by pointwise multiplication. We identify k 2 Zcn with the character k 2 L1(Zn) given by k (j ) = ("k )j ; " = e ni ; j = 0; 1; : : : ; n , 1 with " = e ni . The system f k gk2Zcn is a vector basis of lnp . The assertion that the system f k gk2Zcn is a Schauder basis of lnp with basis constant independent of n is the assertion that there exists a constant Kp such that 2
2
k
Xl
k=0
ck k kp Kpk
nX ,1 k=0
ck k kp; l = 0; 1; : : : ; n , 1;
(2:4)
for all scalars c0; c1; : : : cn,1. To see what is involved in the proof of (2.4), we consider the representations ; of Zcn; Zn respectively on ln2 given by
(k) =
nX ,1 i=0
("k )iei;i; (j ) =
nX ,1 i=0
ei;(i+j)mod n ; k 2 Zcn; j 2 Zn
VILENKIN BASES IN NON-COMMUTATIVE Lp -SPACES
37
Here fei;lg denotes the usual system of matrix units in Mn(C ). Note that the linear span of the system (k); k 2 Zcn is precisely the image of L1(Zn), identi ed as acting by multiplication on L2 (Zn). At the same time, the mapping
X
k2Zn
ck (k) !
X ck (k) c k2Zn
is an algebraic, trace-preserving -isomorphism from L1(Zn) to Mn(C ). This implies that
Xl
ck k
k=0
Lp (Zn)
Xl
Xl
=
ck (k)
=
ck (k)
k=0 k=0 Cp
Cp
for all 0 l n , 1. Here k kCp denotes the usual Schatten p-norm on Mn(C ) given by kxkpCp = Tr(jxjp); x 2 Mn(C ): Accordingly to establish (2.4), it suces to show the existence of a constant Kp, independent of n such that
Xl
,1
ck (k)
Kp
nX
k=0
k=0 ck (k)
; 0 l n , 1; Cp
Cp
(2:5)
A simple proof of the estimate (2.5) may be based on the well-known theorem of Macaev: if 1 < p < 1, then there exists a constant Kp, independent of n such that kT (x)kCp CpkxkCp where T is the operator of triangular truncation given by
T(
nX ,1
i;j =0
xij ei;j ) =
X i<j
xij ei;j ;
1 x e 2 M (C ) (see [GK1,2]). In the above argument, for all x = Pni;j,=0 ij i;j n the use of the operator of triangular truncation introduces ideas that are essentially non-commutative in order to prove an estimate in a setting that is commutative.
The general counterpart to the estimates (2.4), (2.5) given in the proof of Theorem 2(i) [DFPS] is based on the preceding ideas but implemented via the introduction of a generalised operator of triangular truncation. Boundedness of this generalised truncation operator is proved by applying a very general version of the Macaev theorem due to Zsido [Zs 80] (see also [DDPS]), which contains as a special case the
38
P. G. DODDS AND F. A. SUKOCHEV
classical Riesz projection theorem, and which further implies the fact that replexive non-commutative Lp-spaces have the UMD -property. A further essential ingredient in the proof of Theorem 2 is the application of general non-commutative Khintchine inequalities [LP 86], [LPP 91]. 4. Non-commutative Examples We will indicate here how the ideas of the preceding section yield an explicit Schauder basis in the non-commutative Lp-spaces associated with the hyper nite von Neumann II1 factor R. In a certain sense, this factor may be regarded in a natural manner as a non-commutative extension of the commutative von Neumann algebra L1([0; 1]). To set ideas, we consider a simple non-commutative example. We take m = (n; n; 0; 0; : : : ) so that Gm may be identi ed with Zn Zn. We let (M; ) be Mn (C ), equipped with normalised trace tr. If k 2 Zcn, then the equality (3:1) k (s , j ) = hj; ki k (s); s 2 Zn 1 identi es f k gk2Zcn as a Vilenkin system for the action of Zn on L (Zn) by right translation. If we now identify L1(Zn) as the commutative diagonal subalgebra in Mn(C ), then the relation (3.1) in matrix form becomes (j )(k)(j ) = hj; ki(k) (3:2) c for all j 2 Zn; k 2 Zn. These identities are of course easily checked by direct calculation and are a special case of the (so-called) WeylHeisenberg relations. The equalities (3.2) identify the matrices f(k) : k 2 Zcng as a system of eigenvectors for the action x ! (j )x(j ); x 2 Mn (C ): A corresponding Vilenkin decomposition for this action is easily seen to be (Mn(C )) k = span f(k)(m) : 0 m n , 1g : The equalities given in (3.2) may be written equivalently in the form (3:3) (k)(j )(k) = hj; ki(j ) The equalities (3.3) identify the matrices f(j ) : j 2 Zng as a system of eigenvectors for the action x ! (k)x(k); x 2 Mn(C ): This yields a corresponding Vilenkin decomposition (Mn (C )) j = span f(l)(j ) : 0 l n , 1g ; j 2 Zcn
VILENKIN BASES IN NON-COMMUTATIVE Lp -SPACES
39
If we now successively apply (3.2),(3.3), we obtain that (m)(l) ((k)(j )) (l)(m) = hl; kihm; j i(k)(j ) (3:4) for all (l; m) 2 Zn Zn and (k; j ) 2 Zcn Zcn = Z\ n Zn The equalities (3.4) identify the matrices f(k)(j ) : (k; j ) 2 Z\ n Zng as a Vilenkin basis in Lp(Mn(C ); tr) for the action x ! (m)(l)x(l)(m) ; x 2 Mn(C ): We remark that span f(k)(0) : 0 k n , 1g may be identi ed in the obvious way with Lp(Zn). We summarise as follows.
\
Corollary 1 For every n 2 N , the system f(k)(j )g(k;j)2ZnZn,
taken in the reverse lexicographical order, forms a Schauder basis of Lp(Mn(C ); tr) with basis constants independent of n. To proceed further, identify L1[0; 1] with the space L1(Gm; dtm) in the sense of measure spaces. We observe that the equality L1(Gm; dtm) = L1(Zm(0)) L1(Zm(1)) holds in the sense of in nite product of measure spaces. The hyper nite von Neumann II1 factor (R; ) is given by the tensor product (R; ) = (Mm(0) (C ); tr) (Mm(1) (C ); tr) with product trace given by (x0 x1 ) = tr(x0 )tr(x1) Details of this construction may be found in [Sa]. Observe that L1(Gm; dtm) may be identi ed with the commutative \diagonal "subalgebra (Mmd (0) (C ); tr) (Mmd (1) (C ); tr) where Mnd (C ) denotes the subalgebra of Mn (C ) consisting of all diagonal n n matrices. If l = (l0 ; l1; : : : ); n = (n0; n1 ; : : : ) 2 Gdm, we set (l) = (l0 ) (l1 ) : : : ; (n) = (n0 ) (n1 ) : : : It may be shown that the mapping x ! (l)(n)x(n)(l) extends to an ergodic GmGm- ow on R, with eigenspace (Lp(R; ))(k;j) given by the one-dimensional span of the unitary operator (k)(j) for each (k; j) 2 Gm\ Gm = Gdm Gdm. We obtain the following
40
P. G. DODDS AND F. A. SUKOCHEV
\ taken in the reverse lexicographic ordering, is a Schauder basis in Corollary 2 The system of unitary operators f(k)(j)g(k;j)2GmGm Lp(R; ); 1 < p < 1:
We note nally that the basis exhibited in the preceding Corollary contains the classical Vilenkin systems via the \diagonal"subalgebra (k)(0); k 2 Gdm: References [AVDR] G.N. Agaev, N.Ya. Vilenkin, G.M. Dzafarli and A.I. Rubinstein, Multiplicative systems and harmonic analysis on zero-dimensional groups, Elm, (1981) (Russian). [CPSW] Ph. Clement, B. de Pagter, F.A. Sukochev and H. Witvliet, Schauder decompositions and multiplier theorems, Studia Math. 138 (2000), 135163. [Di] J. Dixmier, Von Neumann algebras, North-Holland mathematical library 27, North-Holland publishing company, (1981). [DDPS] P.G. Dodds, T.K. Dodds, B. de Pagter and F.A. Sukochev, Lipschitz continuity of the absolute value and Riesz projections in symmetric operator spaces, J. Functional Analysis 148 (1997), 28-69. [DFPS] P.G. Dodds, S.V. Ferleger, B. de Pagter and F.A. Sukochev, Vilenkin systems and a generalised triangular truncation operator, Integral Equations Operator Theory (to appear). [DS] P.G. Dodds and F.A.Sukochev, Non-commutative bounded Vilenkin systems, Math. Scandinavica 87 (2000), 73-92. [FK] T. Fack and H. Kosaki, Generalized s-numbers of -measurable operators, Paci c J. Math. 123 (1986), 269-300. [FS] S.V. Ferleger and F.A. Sukochev,On the contractibility to a point of the linear groups of re exive non-commutative Lp -spaces, Math. Proc. Camb. Phil. Soc. 119 (1996), 545-560. [GK1] I.C. Gohberg and M.G. Krein, Introduction to the theory of linear nonselfadjoint operators, American Mathematical Society (1969). [GK2] I.C. Gohberg and M.G. Krein, Theory of Volterra operators in Hilbert spaces, American Mathematical Society (1970). [Gos] J.A. Gosselin, Almost everywhere convergence of Vilenkin-Fourier series, Trans. Amer. Math. Soc. 185 (1973), 345-370. [LP] F. Lust-Piquard, Khintchine inequalities in Cp (1 < p < 1), C.R.Acad.Sc.Paris 303 (1986), 289-292. [LPP] F. Lust-Piquard and G. Pisier, Non-commutative Khintchine and Paley inequalities, Ark. Mat. 29 (1991), 241-260. [LT1] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I. Sequence spaces, Springer-Verlag, (1977). [LT 2] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II. Function spaces, Springer-Verlag, (1979). [OPT] D. Olesen, G.K. Pedersen and M. Takesaki, Ergodic actions of compact abelian groups, J. Operator Theory 3 (2) (1980), 237-271.
VILENKIN BASES IN NON-COMMUTATIVE Lp -SPACES [Pa] [Sa] [Se] [Sc] [Si1] [Si2] [SF1] [SF2] [SWS] [Ta] [Vi] [W] [Wa] [Y] [Zs]
41
R.E.A.C. Paley, A remarkable system of orthogonal functions, Proc. Lond. Math. Soc. 34 (1932), 241-279. S.Sakai, C -algebras and W -algebras, Springer-Verlag, (1971). I.E. Segal, A non-commutative extension of abstract integration, Ann.Math. 57 (1953), 401-457. F. Schipp, On Lp -convergence of series with respect to product systems, Analysis Mathematica, 2 (1976), 49-64. P. Simon, Verallgemeinerte Walsh-Fourierreihen. II, Acta Math. Acad. Sci. Hungar. 27 (1976), 329-341. P. Simon, Remarks on Vilenkin bases, Annales Univ. Sci. Budapest., Sect. Comp. 16 (1996), 343-357. F.A. Sukochev and S.V. Ferleger, Harmonic analysis in symmetric spaces of measurable operators, Russian Acad. Sci. Dokl. Math 50 (1995), no. 3, 432-437. F.A. Sukochev and S.V. Ferleger, Harmonic analysis in UMD-spaces: application to the theory of bases, Mathematical Notes, 58 (1995), 13151326. F. Schipp, W.R. Wade and P. Simon, Walsh Series: an introduction to dyadic harmonic analysis, Adam Hilger, (1990). M. Takesaki, Theory of operator algebras I, Springer-Verlag, (1979). N. Ya. Vilenkin, A class of complete orthonormal systems, Izv. Akad. Nauk SSSR, Ser. Mat. 11 (1947), 363-400 (Russian); Amer. Math. Soc. Transl. (2) 28, 1-35, (1963). W.R. Wade, Harmonic analysis on Vilenkin groups, Fourier analysis, approximation theory and applications (Aligarh, 1993), New Age, New Delhi (1997), 339-369. C. Watari, On generalized Walsh-Fourier series, T^ohoku Math J. (2) 10 (1958), 211-241. W.-S. Young, Mean convergence of generalized Walsh -Fourier series, Trans. Amer. Math. Soc. 218 (1976), 311- 320. L. Zsido, On spectral subspaces associated to locally compact abelian groups of operators, Advances in Math. 36 (1980), 213-276.
P. G. Dodds and F. A. Sukochev, Department of Mathematics and Statistics, School of Informatics and Engineering, The Flinders University of South Australia, GPO Box 2100, Adelaide, SA 5001, Australia
ORBITAL CONVOLUTIONS, WRAPPING MAPS AND e-FUNCTIONS A.H. DOOLEY Abstract. We survey the theory of wrapping maps as applied
to compact groups and vector times compact semidirect products, and give an explicit description of the e-function for compact symmetric spaces. The latter is globally de ned.
1. Introduction Let G be a Lie group. The Kirillov orbit method gives a heuristic method which relates the Euclidean Fourier transforms of coadjoint orbits in g to the in nitesimal characters of the irreducible representations. At its simplest, it has the following form Z
j (X )tr (exp X ) = ei(X ) dO (); O where is an irreducible representation of G related to the coadjoint orbit O, O is G-invariant Liouville measure on O and j is the square root of the Jacobian of the exponential map. In the case of a compact Lie group, this formula is exact | for other groups, where is in nite dimensional and O need not have compact support, the formula needs more careful interpretation | in general, it should be seen as an equality of distributions. The reader should consult Kirillov's recent survey article [11] for a detailed discussion of the orbit method. In [4], the author and Norman Wildberger remarked that, for compact groups, the Kirillov formula follows in a simple way from the fact that the wrapping map is a homomorphism of Banach algebras between (say) M G (g) and M G (G). Hence M G is the set of G-invariant measures (Ad-invariant on g and central on G). For an Ad-invariant distribution of compact support on g, let be a distribution on G de ned for f 2 C 1(G), by h; f iG = h; j:f expig: The wrapping formula then states (1.2) G = ( g ): (1.1)
42
ORBITAL CONVOLUTIONS, WRAPPING MAPS AND e-FUNCTIONS
43
From this formula, it follows that the adjoint mapping 0 : M G (G)0 ! M G (g)0 is an injection of Gelfand spaces; thus, to each irreducible character of G, one obtains a character RofRg averaged over adjoint orbits, that is, a mapping of the form 7! O ei (x) dO ( )d (x). From this the Kirillov character (1.1) follows easily. The above theorem allows us to relate the central harmonic analysis of G to Euclidean harmonic analysis of g. This becomes particularly interesting when one realizes that the latter can be described explicitly. In fact, in [3], we gave the following formula for Ad-invariant convolution on g. Recall that each adjoint orbit O intersects the positive Weyl chamber t+ of the Cartan subalgebra t in a unique point | say. Then we have
= where
N (; ; ) =
Z
t+
N (; ; ) d;
X w2W
sgn w ew T ( );
T( ) being the projection on t of O , given by X Y sgn ww0ew Fw0 ; T = jW1 j + w;w02W 2 where F is the distribution on t given by Lebesgue measure on the ray through . In work currently in progress, these results are being extended in two directions | to some non-compact groups, and to compact symmetric spaces. I will describe these results in the next section and in section 4. 2. Semi-direct product groups In [6], we extend the wrapping map formula to G = V o K , V a vector group, K a compact group. Here already, there is a substantial technical hurdle to be overcome, in the sense that formula (1.2) requires the convolution of Ad-invariant distributions (or measures) to be de ned, and there are no non-trivial G-invariant distributions of compact support, as the G-orbits in g (and in g ) are not compact. This problem can be overcome by the following device. Notice that conjugacy classes, adjoint orbits and coadjoint orbits are all bred spaces over K -orbits. I will illustrate this for adjoint orbits only. For each A 2 k, split V into two subspaces, VA = fv 2 V : A v = vg and V A = VA? (the
44
A.H. DOOLEY
orthogonal complement with respect to a K -invariant inner product) | where A v is the derivative of the K -action. Then for v 2 V the orbit G A is bred over the compact orbit K v, the bre at k A being V kA. Now we replace the G-invariant distributions of compact support on G with a family of distributions on g = k + V which are K -invariant, compactly supported in the k-direction and for each A in this support, are given by an invariant mean (suitably normalized) on V A. It turns out that such distributions: (i) wrap to similarly de ned distributions on conjugacy classes, (ii) belong to the dual of a suitable G-stable family of functions on g | they are C 1 in the k-direction and for each A 2 k, are almost periodic in the V A direction, (iii) have a natural notion of convolution (using the above duality in a standard way), (iv) have as Fourier transforms, similar distributions on g. The formula (1.2) continues to hold for V o K with the above de nitions. This leads to a new proof of the Lipsman character theorem [13]. The full details are somewhat technical. We may interpret the above results as follows. Each of the conjugacy classes, adjoint orbits and coadjoint orbits possesses a natural convolution on hypergroup structures. Denote the associated hypergroups as Conj, Adj and Coadj respectively. Now provides a homomorphism : Adj ! Conj and we may identify Coadj as the dual hypergroup of Adj. The Lipsman character may thus be interpreted as the mapping 0 from Conj ! Coadj = (Adj): It is possible also to give very explicit formulae for the hypergroup structures of Adj and Coadj: they are no longer identical, in contrast to the compact case. The gist of this structure is that the compact orbits in V (or V ) are convolved as in section 1, and one forms the sums of the bres. Full details are in [6]. 3. Generalizations of the Duflo Isomorphism The wrapping map formula (1.2) can be considered as a global version of the Du o isomorphism. To see this, notice that the Ad-invariant distributions of support f0g in g wrap to central distributions of support feg in G. The latter may be identi ed with zu(g), the centre of the universal enveloping algebra; the former with SG (g), the centre of the symmetric algebra of g.
ORBITAL CONVOLUTIONS, WRAPPING MAPS AND e-FUNCTIONS
45
Recently, far reaching generalizations of the Du o isomorphism have been proved by Maxim Kontseivich [12]. He proves that for a Poisson manifold (X; ), there is a family r of star products which deform products (at r = 0) to convolutions (at r = 1). However, this series can only be shown to converge near f0g, and it is of considerable interest to ask how far they can be extended, and if global versions such as (1.2) are available in special cases. Andler, Dvorsky and Sahi [1] recently showed, using [12], that every bi-invariant di erential operator on a Lie group is locally solvable. Actually, in the case where X = G=H , G a Lie group and H the set of xed points of an involution , Kontseivich's construction coincides with that of Rouviere [14]. Rouviere's theory is as follows. We may write g = h + s, where h and s are the eigenspaces of (by which I denote also the di erential of the involution, by abuse of notation) of eigenvalues +1 and ,1 respectively. Then h is the Lie algebra of H , and p may be identi ed with the tangent space of G=H . There is, furthermore, an exponential map Exp : p ! G=H . We take an H -invariant neighbourhood s of o in p on which Exp is a di eomorphism. For X 2 s, let j (X ) be a suitable square root of J;0 (Exp)(X ). (We leave aside temporarily the existence of a smooth real-valued square root | in the case of interest below, it can be explicitly calculated.) Then AdjH : p ! p and j (Ad(h)X ) = j (X ) for all x 2 p, h 2 H . We may thus de ne a version of wrapping using the same ideas as above: for an H -invariant distribution (of compact support) on p, and for f 2 Cc1(G=H ), let h; f iG=H = h; j:f Expip: If now ; are supported in s, and are such that p is also supported in s, we can ask whether we have a formula such as \( p ) = ( ) G=H ()". It turns out that the formula requires some modi cation and that it should read (3.1) ( p;e ) = ( ) G=H (): In this formula, e(X; Y ) is a certain function of two variables on s s, and p;e is \twisted" convolution given, for , H -invariant and locally integrable, by Z (3.2) ( p;e )(X ) = (Y )(X , Y )e(X; Y )dY: p
(This formula needs an obvious adaptation in order for it to work for distributions | see [14].) It is instructive to understand where the e-function comes from, as we will be calculating it in some special cases in the next section. We
46
A.H. DOOLEY
write the right-hand side of (3.1), for u 2 C 1(G=H ), as
h( ) G=H (); ui = =
Z
Z
G=H G=H Z Z s
s
( )(x)()(y)u(xy)dxdy
(X )(Y )j (X )j (Y )u(Exp X Exp Y )dXdY:
We now claim that there exist h; k 2 H so that Exp X Exp Y = Exp(h:X + k:Y ). Accepting this for the moment, consider the change of variables (h:X; k:Y ) 7! (X; Y ). Denote the Jacobian of this change of variables by (X; Y ). Then the integral transforms to Z Z s
s
(h,1X )(k,1Y )j (h,1X )j (k,1Y ) (X; Y )u(Exp(X + Y ))dXdY:
Now j , and are all H -invariant, so we obtain Z Z )j (Y ) (X; Y )(j:u Exp)(X + Y )dXdY: (X )(Y ) jj ((X X +Y) s s Letting )j (Y ) (X; Y ); e(X; Y ) = jj ((X X +Y) we obtain the right-hand side. Rouviere is able to calculate e(X; Y ) as an in nite power series, which converges in a neighbourhood of o in s. To see why the elements h and k above exist, consider the CampbellBaker-Hausdor series for Exp. Exp X Exp Y =Exp(X + Y + 1 [X; Y ] + 1 ([X [X; Y ]]+[Y [Y; X ]])+ : : : ): 2 6 Now the hypothesis that is an involutive automorphism of g implies that [h; p] p, and [p; p] h. Thus we may rearrange the above series to get X + 16 [[X; Y ]; X ] + + Y , 61 [[X; Y ]; Y ] + + + H (X; Y ) = (I + 16 [X; Y ] + +)X + (I , 61 [X; Y ] + +)Y + H (X; Y ); = h:X + k:Y + H; where H 2 h. The result now follows. (For a fuller proof, and for computations of the series, see [14].)
ORBITAL CONVOLUTIONS, WRAPPING MAPS AND e-FUNCTIONS
47
4. Symmetric spaces of the compact type The question I would like to address in this section is: can we nd a global version of Rouviere's formula in the case of a symmetric space of the compact type? Let (G; K ) be a Riemannian symmetric pair of the compact type. Then K is the set of xed points of the Cartan involution , and g = k + p is the Cartan decomposition. Choose a maximal abelian subalgebra q of p and let A be the corresponding subgroup of G. We then have the Cartan decomposition of G, G = KA+K , where A+ = exp(a+). Here a+ is the positive Weyl chamber for a set of positive restricted roots +r . Let m denote the multiplicity of 2 +r . We may identify p as the tangent space at eK of X = G=K , and have the standard exponential map Exp : p ! X . The square root of the Jacobian of Exp is AdK invariant of p, and is given by Y sin (H ) m =2 j (H ) = ; (H ) 2 a+: ( H ) + 2 r
For X; Y 2 p, let X = Ad (k1)H1, Y = Ad(k2)H2 and X + Y = Ad(k3)H3. We de ne e(X; Y ) m Q Q cos 12 ( (H1 )+ w (H2 )+ w0 (H3 )) =2 j ( H ) j ( H ) 1 2 = j(H3) ; 2+r w;w02Wr (H1 )+ w (H2 )+ w0 (H3 )
where Wr denotes the restricted Weyl group and w (H ) is the image of the root by the Wr -action. The following theorem then holds. Theorem (i) e is de ned on all of p. (ii) Let and be K -invariant distributions of compact support on p. Then X = ( p;e ): Details of the proof of this theorem will appear in [7]. In order to prove it, we need to nd explicitly the hypergroup convolution of K orbits in p and of K -orbits in X . The e-function is then found by comparing the two structures. The gist of the calculation is already present in the case of the sphere X = SO(3)=SO(2), which is discussed in [15]. We brie y describe this calculation here. Let us discuss the convolution of two circles (radii r1 and r2) in the plane. (This corresponds to K -orbits in p.) One needs to write r1 + r2ei = rei
48
A.H. DOOLEY
and then compute \dr"in terms of \d". (It is obvious that the resulting measure is rotationally invariant.) A little rst year calculus yields 2r dr = d 2r1r2 sin and one identi es the denominator on the left-hand side as the area of a triangle in the complex plane with vertices at 0, r1 and r2 ei . By Heron's formula, this is given also by [(r2 , (r1 , r2 )2)((r1 + r2)2 , r2 )] 12 , which we may write as Y
12
(r r1 r2 ) ;
where the product is over all choices of signs. Thus, the convolution of two circles of radius r1 , r2 is a rotationally invariant density given by fr1 ;r2 (r) = Y 2r (r): 1 (r r1 r2) 2 [jr1,r2j;r1+r2]
If one now carries out the same calculation on the surface of the sphere S 2 | there is a convenient version of Heron's formula for spherical geodesic triangles | one obtains the density function: r)(cos r , cos(r1 + r2 ))] 21 gr1;r2 (r) = sin r [(cos(r1 , r2) , cos sin r1 sin r2 Y 12 1 sin r 1 = sin r1 sin r2 cos 2 (r r1 r2 ) using the half-angle formula for cosine. The e-function is now given by the ratio g=f 1 Y 1 cos 1 (r r1 r2 ) 2 sin r 2 2 e = sin r sin r : 1 (r r r ) 1 2 1 2 2 In essence, the proof for the symmetric space case is to reduce everything to two dimensions and to use these elementary ideas. For the n-dimensional sphere, one obtains
e(X; Y ) =
"
Y 2 cos(r r1 r2 )=2
# n,2 3
(jr1 ,r2j;r1+r2) (r): (r r1 r2 ) These formulae will have interesting consequences for harmonic analysis | for example in nding fundamental solutions of K -invariant di erential operators on X .
ORBITAL CONVOLUTIONS, WRAPPING MAPS AND e-FUNCTIONS
49
One can also prove a compact version of the Gindikin-Karpilevic formula for the Harish-Chandra cY -function. c() = c ( ); 2+r
where the right-hand side is the c-function for each of the symmetric spaces G =K , 2 +r . It seems most likely that the proof will also go through in the non-compact case and yield an elementary proof of this formula. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
M. Andler, A. Dvorsky, S. Sahi, Deformation quantization and invariant distributions, E-print Math 9.9./9905065, 12 May 1999 K.H. Chung, Compact group actions and harmonic analysis, thesis, UNSW 1999 A.H. Dooley, N.J. Wildberger and J. Repka, Sums of adjoint orbits, Lin. Multilin. Alg. 36 (1993), 79-101 A.H. Dooley and N.J. Wildberger, Harmonic analysis and the global exponential map for compact Lie groups, Funct. Anal. Appl. 27 (1993), 25-32 A.H. Dooley and N.J. Wildberger, Global character formulae for compact Lie groups, Trans. Amer. Math. Soc. 351 (1999), 477-495 A.H. Dooley and N.J. Wildberger, Orbital convolutions for semi-direct products, in preparation A.H. Dooley, Global versions of the e-function for compact symmetric spaces, in preparation. M. Du o, Caracteres des algebres de Lie resolubles, C.R. Acad. Sci. (Paris) 269 (1969) serie a, 437-438 S. Helgason, Groups and Geometric Analysis, Academic Press (1984) M. Kashiwara, M. Vergne, The Campbell-Hausdor formula and invariant hyperfunctions, Invent. Math. 47 (1978), 249-272 A. Kirillov, Merits and Demerits of the Orbit method, Bull. Amer. Math. Soc. 36 (1999), 433-488 M. Kontsevich, Deformation Quantization of Poisson manifolds I, EIMS qalg/9709040 (1997) R. Lipsman, Orbit theory and harmonic analysis on Lie groups with cocompact nilradical, J. Math. Pures et Appl. 59 (1980), 337-374. F. Rouviere, Invariant analysis and contractions of symmetric spaces, Compositio Math. 80 (1991), 11-136 N.J. Wildberger, Hypergroups, symmetric spaces and wrapping maps, in Probability Measures on Groups and related Structures, XI, World Scienti c Singapore, 1995, 406-425
School of Mathematics, UNSW Sydney NSW 2052, Australia
NORMS OF 0{1 MATRICES IN CP IAN DOUST To Derek Robinson on his 65th birthday
Abstract. We announce a new result (proved in collaboration with T.A. Gillespie) on the boundedness of a class of Schur multiplier projections on the von Neumann-Schatten ideals Cp . We also show that for 1 p 2 the average Cp norm of a 0{1 matrix grows just as quickly as the largest norm of such a matrix.
1. Introduction Calculating the norm of an operator which acts on one of the von Neumann Schatten ideals Cp is often rather dicult | even for rather simple operators | because of the scarcity of elements T 2 Cp for which one can easily calculate (or even approximate) kT kp = kT kCp . Even for the algebraically simple Schur projections which act by replacing certain xed entries of the matrices of elements of Cp by 0, proving boundedness results is typically quite hard. To x some notation, for 1 p < 1, let Cp denote the von Neumann Schatten ideal of compact operators on `2, with norm kT kp = trace((T T )p=2)1=p. We take C1 to be the set of all compact operators on `2 with the usual operator norm. We will let Cpn denote the n n matrices equipped with the corresponding norm. We will, as usual, think of elements of Cp as being in nite matrices. Let Z denote the set of all zero-one arrays [ai;j ]1 i;j =1. We are interested in the norms of projections de ned by Schur multiplication of such arrays. If A = [ai;j ] 2 Z , de ne the Schur projection corresponding to A to be the map PA : T 7! A T , where denotes Schur or elementwise multiplication of matrices. Let Bp = fA 2 Z : PA 2 B (Cp)g : The set of n n matrices with zero-one entries will be denoted by Z n. By requiring that each entry in an array is either zero or one with equal 2000 Mathematics Subject Classi cation. Primary 47B10. Secondary 43A22, 46B20, 47B49. Key words and phrases. Von Neumann-Schatten ideals of compact operators, Schur multiplier projections. 50
NORMS OF 0{1 MATRICES IN CP
51
probability we may regard Z and Z n as being probability spaces with the appropriate product measures. If 1 p < 1 and p1 + 1q = 1, then Cp = Cq , under the natural pairing hS; T i = trace(ST ). If A 2 Bp then trace(PA(S ) T ) = trace(S PA(T )) for all S 2 Cp, T 2 Cq and so it follows that kPAkp = kPAkq . In particular note that B2 = Z with kPAkp = 1 for all A 2 Z and that Bp = Bq . It is a trivial consequence of the ideal inequalities for Cp that if A 2 Z is a nonzero array which is constant on each row (or on each column) then kPAkp = 1 for all p. Proving boundedness for other types of arrays has been signi cantly more dicult. The rst major result was due to Macaev (see [7]) who showed that the upper triangular truncation map is bounded on Cp for 1 < p < 1 (but not for p = 1 or p = 1). In the 1980s Bourgain [3] showed boundedness results for a class of `Toeplitz' arrays which are analogous to multiplier results from Fourier analysis. In particular, for 1 < p < 1 there is a constant Kp such that if A 2 Z is any array which is constant on diadic blocks of (long) diagonals, then kPAkp Kp. It is still an open question as to whether if 2 < r < p < 1 there is always a Schur multiplier projection which is bounded on Cr , but not on Cp. An important special case when r and p are even integers has recently been solved (in the armative) by A. Harcharras [8]. Recently, Alastair Gillespie and I have proved boundedness for a new class of such projections. De nition 1.1. A zero-one array [aij ] will be said to be obtainable if for all indices i1 ; i2; j1; j2 , a a 1 0 i1 j1 i1 j2 ai2 j1 ai2 j2 6= 0 1 :
Theorem 1.2. For all p 2 (1; 1) there exists a constant Kp such that for every obtainable array A, kPAkp Kp. The proof, which depends on the fact that for 1 < p < 1, Cp is a
UMD space, uses a new result from spectral theory proved in [5]; the sum of two commuting real scalar-type spectral operators on a UMD space is a well-bounded operator. (Since completing this work we have been made aware of some closely related work by Clement, de Pagter, Sukochev and Witvliet [4] [11] [12].) As we show in [6], it is relatively easy to use this result to recover Bourgain's theorem (at least as it applies to 0{1 arrays). One can also prove Littlewood-Paley type decomposition results in Cp, of which we shall just give two simple examples here.
52
IAN DOUST
Split Z+ Z+ into rectangular subarrays as in either of the diagrams below.
1 0 B 1 0 B B B1 0 1 B 0 BB CC BB B2 B3 4 B7 CC B B C B CC BB 3 4 C B : : : : : : CC (ii) BB B2 CC (i) B BB B5 B6 B B C B CC 6 7 B C B B@ 5 B8 B9 B8 B9 A @ A
... ... ... ... The actual sizes of the subarrays is not important. Let Pk denote the projection given by Schur multiplication by the characteristic function of Bk . Theorem 1.3. Suppose that 1 < p < 1 and that ; 6= J N . Then P P converges in the strong operator topology and k2J k
X
Pk p 2Kp + 1 k2J
where Kp is the constant from Theorem 1.2. Actually, if all the `diagonal' subarrays B3k are just 1 1 subarrays, then this result is an easy consequence of Macaev's result. Even for the more general splittings of this form described above, it is probably not too hard to deduce this bound from known estimates (see, for example, Section 4 of [1]). Our techniques however cover a rather wider range of decompositions including some which are not at all related to triangular trunctions. Details will appear in [6]. 2. Norms of 0{1 matrices In proving the above results we were lead to considering what one can say about the bounds of Schur multiplier projections on Cpn. The following example is the standard way of showing that these projections can have bad norms. 1 1 Example 2.1. Let B1 = 1 ,1 . For m 2, de ne Bm = Bm,1
B1. For example 0 1 1 1 11 B 1 ,1 1 ,1CC : B2 = B @ 1 1 ,1 ,1A 1 ,1 ,1 1
NORMS OF 0{1 MATRICES IN CP
53
Thus Bm is a 2m 2m orthogonal matrix. Let n1 =1 2m. Then Bm B is just nI . Clearly then kBmkp = (n np=2)1=p = n 2 + p . Let Om be the n n matrix all of whose entries are 1. This is just n times a projection, so kOmkp = n for all p. Let Am be the n n array formed by replacing the ,1's in the matrix Bm by zeros. Thus Am = 21 (Bm + Om). Note that Am = PAm Om . If 1 p < 2 then 1 n( 12 + p1 ) , n kA k 1 n( 1p , 21 ) , 1 : kPAm kp kOm kp 2 = n 2 2 m p Using the earlier remarks about duality we see that there exists c > 0 such that for 1 p 1 and large n, kPAm kp cnj 12 , p1 j:
2
This order of growth is the worst that one can get from nn matrices. Theorem 6.2 of [2] shows that for any A 2 Zn, kPAk1 n1=2 . Using 1 1 interpolation and duality gives that kPAkp nj 2 , p j.This upper bound also follows from the following upper bound due to Ong [10]: kPAk1 min max `2 norm of a column; max `2 norm of a row : We shall say that A1 2 Z n is a subarray of A 2 Z if A1 is formed by deleting all but n of the rows and all but n of the columns of A. If A1 is a subarray of A then kPA1 kp kPAkp. With probability one, a randomly chosen in nite string of binary digits contains all nite strings as substrings. In other words, with probability one, an element of Z contains every array Am (from Example 2.1 above) as a subarray. It follows immediately that if p 6= 2 then Prob(kPAk < 1) = 0. A more delicate question concerns whether this `bad' array is typical of n n arrays. The answer is Yes. This is undoubtedly known to the experts, although as far as we are aware, this does not appear in the literature. (Indeed, throughout this area, much more seems to be known than is written down!) To simplify notation we shall write g(n) f (n) if there exist constants 0 < c1 < c2 < 1 such that c1 f (n) < g(n) < c2f (n) for all (suciently large) n. Throughout we shall use c as a generic absolute constant whose value may change from one line to the next. Theorem 2.2. E (k PAkp : A 2 Z n) nj 12 , p1 j. Proof. For each n let Xn denote the matrix-valued random variable where each entry is an independent P Gaussian variable with mean 0 and variance 1. That is, Xn = ni;j=1 gij Eij where each gij is an
54
IAN DOUST
independent N (0; 1) random variable and fEij gni;j=1 are the standard matrix units. Let frij gni;j=1 be a family of independent identically distributed random variables which take the values ,1 and 1 with equal probability. Suppose rst that 1 p 2. Some rather deep estimates of Szarek [14] give that 1 +1 E (k Xn kp ) n 2 p : Then, since Cp has cotype 2 for p in this range, [13, Theorem 3.9] implies that
cn
1 2
+ p1
n
X E gij Eij i;j =1 n
21=2
X E gij Eij
i;j =1 n
21=2
X c E rij Eij
i;j =1
But by Kahane's inequality [9, Theorem 1.e.13]
n n
21=2
X
X E rij Eij c E rij Eij
: i;j =1
i;j =1
Following the same argument as in Example 2.1, n
12 + p1
X 1 n E (k Akp : A 2 Z ) 2 E i;j=1 rij Eij , n c n ; (for large n) and so 1 ,1 E (k PA kp : A 2 Z n ) cn p 2 : 1 1 Since for any A 2 Z n, kPAkp n p , 2 the result is proved. The case when p > 2 follows from duality. 2 I would like to thank Alastair Gillespie and Quanhua Xu for several interesting discussions on these matters, and the referee for bringing my attention to some additional references. References [1] Arazy, J., Some remarks on interpolation theorems and the boundedness of the triangular projection in unitary matrix spaces, Integral Equations Operator Theory 1 (1978) 453{495. [2] Bennett, G., Schur multipliers, Duke Math. J. 44 (1977), 603{639.
NORMS OF 0{1 MATRICES IN CP
55
[3] Bourgain, J., Vector-valued singular integrals and the H 1 -BMO duality, in: Probability Theory and Harmonic Analysis (Mini-conference on Probability Theory and Harmonic Analysis, Cleveland, 1983), J.-A. Chao and W.A. Woyczynski (eds), Monographs and Textbooks in Pure and Appl. Math. 98, Marcel Dekker, New York, 1986, 1-19. [4] Clement, P., de Pagter, B., Sukochev, F.A. and Witvliet, H., Schauder decomposition and multiplier theorems, Studia Math. 138 (2000), 135{163. [5] Doust, I. and Gillespie, T. A., Well-boundedness of sums and products of operators, in preparation. [6] Doust, I. and Gillespie, T. A., Schur multipliers on Cp spaces, in preparation. [7] Gohberg, I. C. and Kren, M. G., Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I. 1969. [8] Harcharras, A., Fourier analysis, Schur multipliers on S p and non-commutative (p)-sets, Studia Math. 137 (1999), 203{260. [9] Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces II, Function spaces, Springer Verlag, Berlin{New York, 1979. [10] Ong, S.C., On the Schur multiplier norm of matrices, Lin. Algebra Appl. 56 (1984), 45{55. [11] de Pagter, B., Sukochev, F.A. and Witvliet, H., Unconditional decompositions and Schur-type multipliers, to appear in Proceedings of the 1998 IWOTA conference in Groningen, Birkhauser. [12] de Pagter, B. and Witvliet, H., Unconditional decompositions and UMDspaces, in: Semi-groupes d'operateurs et calcul fonctionnel, Ecole d'Ete (June 1998), Publications Mathematiques de l'UFR Sciences et Techniques de Besancon, Vol. 16, Besancon, 1998, 79{111. [13] Pisier, G., Factorization of linear operators and geometry of Banach spaces, CBMS Regional Conference Series in Mathematics, 60, American Mathematical Society, Providence, R.I., 1986 [14] Szarek, S.J., Condition numbers of random matrices. J. Complexity 7 (1991), 131{149.
School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
E-mail address:
[email protected]
SPECTRAL MULTIPLIERS FOR SELF-ADJOINT OPERATORS XUAN THINH DUONG, EL MAATI OUHABAZ AND ADAM SIKORA Abstract. In this article, we give a survey of spectral multipliers
and present (without proof) sharp Hormander-type multiplier theorems for a self adjoint operator A under the assumption that A has Gaussian heat kernel bounds and satis es appropriate estimates of the L2 norm of the kernels of spectral multipliers. Our theorems imply several important, previously known results on spectral multipliers and give new results for sharp estimates for the critical exponent for the Riesz means summability.
1. Introduction This paper contains discussion and survey of the topic of spectral multipliers and main results of [DOS] without giving proofs. Readers are referred to [DOS] for their proofs and more applications. Suppose that A is a positive de nite self-adjoint operator acting on L (X ), where X is a measure space. Such an operator admits a spectral decomposition EA() and for any bounded Borel function F : [0; 1) ! C, we de ne the operator F (A) by the formula 2
(1)
F (A) =
Z1 0
F () dEA():
By the spectral theorem the operator F (A) is continuous on L (X ). Spectral multiplier theorems investigate sucient conditions on function F which ensure that the operator F (A) extends to a bounded operator on Lq for some q, 1 q 1. Spectral multiplier has been a very active topic of Harmonic analysis. Roughly, it is sucient that F is di erentiable to some order with appropriate bounds on its derivatives. The more information we know about the underlying space X and the operator A, the sharper multiplier results can be obtained, i.e. less derivatives on F are needed for F (A) to be bounded. For example, Pd when X is the Euclidean space d R and A is the Laplacian d = k @k a sucient condition is that F possesses [d=2] + 1 derivatives which satisfy certain size estimates where [d=2] denotes the integral part of d=2. Recent results extend 2
=1
56
2
SPECTRAL MULTIPLIERS FOR SELF-ADJOINT OPERATORS
57
this to more general underlying spaces and more general self adjoint operators. For example, see, [He3, He2, DO, CS, Ale2] when A is an abstract positive self-adjoint operator which has heat kernel bounds (or nite propagation speed) and the underlying space X satis es doubling volume property. (See Assumption 2.1). Let us discuss two important examples of spectral multiplier theorems concerning group invariant Laplace operators acting on Lie groups of polynomial growth. Let G be a Lie group of polynomial growth and let X ; : : : ; Xk be a system of left-invariant vector elds on a G satisfying the Hormander condition. We de ne Laplace operator L acting on L (G) by the formula 1
2
L=
(2)
k X i=1
Xi : 2
If B (x; r) is a ball de ned by the distance associated with system X ; : : : ; Xk (see e.g. [VSC, xIII.4]), then there exist natural numbers d ; d1 0 such that (B (x; r)) rd0 for r 1 and (B (x; r)) rd1 for r > 1 (see e.g. [VSC, xVIII.2]). We call G a homogeneous group if there exists a family of dilations on G. A family of dilations on a Lie group G is a one-parameter group (~t )t> (~t ~s = ~ts ) of automorphisms of G determined by (3) ~t Yj = tdj Yj ; where Y ; : : : ; Yl is a linear basis of Lie algebra of G and dj 1 for 1 j l (see [FS]). We say that an operator L de ned by (2) is homogeneous if ~t Xi P = tXi for 1 i k. For a homogeneous Laplace operator d = d1 = lj dj (see [FS]). Spectral multiplier theorems for homogeneous Laplace operators acting on homogeneous groups were investigated by Hulanicki and Stein [HS] (see also [FS, Theorem 6.25]), De Michele and Mauceri [dMM]. The following theorem was obtained independently by Christ [Ch2] and Mauceri and Meda [MM]. See also [Si3]. Its proof relies on heat kernel bounds, L estimates from Plancherel theorems, translation and dilation invariant structures of homogeneous groups, Calderon Zygmund operator theory and interpolation theory. Theorem 1. Let L be a homogeneous operator de ned by the formula (2) acting on a homogeneous group G. Denote by d = d = d1 homogeneous dimension of the underlying group G. Next suppose that s > d=2 and that F : [0; 1) ! C is a bounded Borel function such that (4) sup k t F kWs2 < 1; 1
0
0
1
0
=1
2
0
t>0
58
XUAN THINH DUONG, EL MAATI OUHABAZ AND ADAM SIKORA
where tF () = F (t), kF kWsp = k(I , d2 = dx2 )s=2F kLp and 2 Cc1(R+) is a xed function, not identically zero. Then F (L) is of weak type (1; 1) and bounded on Lq when 1 < q < 1. Note that condition (4) is independent of the choice of 2 Cc1(R+). The Hormander multiplier theorem describes the Fourier multiplier on Rd (see [Ho1]). If we apply Theorem 1 to Rd we obtain a result equivalent to the Hormander multiplier theorem restricted to radial Fourier multipliers. Therefore we call Theorem 1 the Hormander-type multiplier theorem and condition (4) the Hormander-type condition. Theorem 1.1 is optimal for a general homogeneous group, see estimate 1.6. However, for speci c groups such as Heisenberg and related groups, it is possible to obtain multiplier results where the number of derivatives needed is roughly half of the topological dimension n. Often the homogeneous dimension d is strictly greater than the topological dimension n, hence the number of derivatives needed could be less than d=2. See for example, [MS, He4, Du, CS]. In the setting of general Lie groups of polynomial growth spectral multipliers were investigated by Alexopoulos. Note that in this setting, the local dimension d0 and dimension at in nity d1 are di erent in general and the group G does not have dilation invariants as in the case of a homogeneous group. Using nite propagation speed property, estimates on upper bounds of heat kernels and their space gradients, Alexopoulos proved the following (see [Ale1]). Theorem 2. Let L be a group invariant operator acting on a Lie group of polynomial growth de ned by (2). Suppose that s > d=2 = max(d0; d1)=2 and that F : [0; 1) ! C is a bounded Borel function such that (5) sup k t F kWs1 < 1; t>0
where t F () = F (t) and kF kWsp = k(I , d2 = dx2 )s=2 F kLp . Then F (L) is of weak type (1; 1) and bounded on Lq when 1 < q < 1. Condition (5) is also independent of the choice of . We note that Theorem 1.2 does not appear exactly the same but is essentially equivalent to the results of [Ale1]. In [He3] Hebisch extended Theorem 2 to a class of abstract operators acting on spaces satisfying \doubling condition" (see also [Ale2]). The order of di erentiability in the Alexopoulos-Hebisch multiplier theorem is optimal. This means that for any s < d=2 we can nd a function F such that F satis es condition (5) but F (A) is not of weak type (1; 1). Indeed, let A be a uniformly elliptic, self-adjoint second-order
SPECTRAL MULTIPLIERS FOR SELF-ADJOINT OPERATORS
59
di erential operator on Rd, e.g. A = d , where d is the standard Laplace operator. One can prove that (6) C (1 + j j)d= kAi kL1 !L1;1 C (1 + j j)d= (see [SW]). (See also [St1, pp. 52] and Christ [Ch2]). However, if we put F () = jji , then C 0 (1 + j j)s= sup ktF kWs1 C 0 (1 + j j)s= : 2
1
2
2
1
t>0
2
2
2
Therefore for any s < d=2 Theorem 2 does not hold. If X is a manifold with exponential volume growth, i.e. V (x; r) cekr and L is the Laplace-Beltrami operator, spectral multipliers was investigate by M. Taylor in [Tay] where it was shown that a sucient p condition is that F is holomorphic on a strip of width k for F ( L) to be bounded on Lp for 1 < p < 1. For more speci c spaces such as certain Iwasawa AN groups, see [He5, CGHM] where it was shown that only a nite number of derivatives are required for F (L) to be bounded on L or to be of weak type (1,1). Note that the exponential volume growth is like the dimension d = 1. The theory of spectral multipliers is related to and motivated by the study of convergence of the Riesz means or convergence of other eigenfunction expansions of self-adjoint operators. To de ne the Riesz means of the operator A we put ) for R (7) R () = (1 , =R 0 for > R: We then de ne the operator R (A) using (1). We call R (A) the Riesz or the Bochner-Riesz means of order . The basic question in the theory of Riesz means is to establish the critical exponent for the continuity and convergence of the Riesz means. More precisely we want to study the optimal range of for which the Riesz means R (A) are uniformly bounded on L (X ) (or other Lq (X ) spaces). Since the publication of Riesz paper [Ri] the summability of the Riesz means has been one of the most fundamental problems in Harmonic Analysis (see e.g. [St2, IX.2 and xIX.6B]). Despite the fact that the Riesz means have been extensively studied we do not have the full description of the optimal range of even if we study only the space LP(X ). On one hand we know that for the Laplace operator d = d @ acting on Rd and the Laplace-Beltrami operator acting on k k compact d-dimensional Riemannian manifolds the critical exponent is equal (d , 1)=2 (see [So1]). This means that Riesz means are uniformly continuous on L (X ) if and only if > (d , 1)=2 (see also [ChS, Ta]). On the other hand, if we consider more general operators like e.g. 1
1
1
=1
2
1
60
XUAN THINH DUONG, EL MAATI OUHABAZ AND ADAM SIKORA
uniformly elliptic operators on Rd it is only known that Riesz means are uniformly continuous on L (X ) if > d=2 (see [He1]). One of the main points of our work is to investigate the summability of Riesz means for d=2 > (d , 1)=2. The Alexopoulos-Hebisch multiplier theorem discussed above gives optimal value for the exponent d=2 of the number of derivatives needed in spectral multipliers, but it does not give the optimal range of the exponent for the Riesz summability. Indeed, if k kWs1 < 1, then s. However, kR kWs2 < 1 if and only if > s , 1=2. This means that in virtue of Theorem 2 one obtains uniform continuity of Riesz means on Lq for any > d=2 and for all q 2 (1; 1), whereas Theorem 1 shows Riesz summability for > (d , 1)=2 (see also [Ch2, pp. 74]). As we mentioned earlier (d , 1)=2 is a critical index for Riesz summability for standard Laplace operator on Rd and Laplace-Beltrami operator on compact manifolds. To conclude we see that the optimal number of derivatives in multiplier theorems is d=2. However, in condition (5) we required d=2 derivatives in L1. In the Hormander-type condition (4) we required d=2 derivatives in L . Note that functions t F are compactly supported so condition (5) is strictly stronger than (4). We would like to investigate when it is possible to replace condition (5) in the Alexopoulos-Hebisch multiplier theorem by condition (4) from Theorem 1. As we investigate spectral multipliers in a general setting of abstract operators rather than in a speci c setting of group invariant operators acting on Lie groups, we do not have certain estimates which are consequences of invariant structures of Lie groups. Also we only assume suitable bounds on heat kernels but not pointwise bounds on their space derivatives. The subject of Bochner-Riesz means and spectral multipliers is so broad that it is impossible to provide comprehensive bibliography of it here. Hence we quote only papers directly related to our investigation and refer reader to [Ale1, Ch2, Ch1, ChS, CS, dMM, Du, He1, He3, Ho1, Ho3, HS, MM, SeSo, So1, So2, Si2, St1, St2, Ta] and their references. 1
1
1
2
2. Main results In this section we rst introduce some notation and describe the hypotheses of our operators and underlying spaces. We then state our main results. Assumption 1. Let X be an open subset of Xe , where Xe is a topological space equipped with a Borel measure and a distance : Let e (x; y ) < rg be the open ball (of Xe ) with centre at B (x; r) = fy 2 X; x and radius r. We suppose throughout that Xe satis es the doubling
SPECTRAL MULTIPLIERS FOR SELF-ADJOINT OPERATORS
61
property, i.e., there exists a constant C such that e 8r > 0: (8) (B (x; 2r)) C(B (x; r)) 8x 2 X; Note that (8) implies that there exist positive constants C and d such that e r > 0: (9) (B (x; r)) C (1 + )d(B (x; r)) 8 > 0; x 2 X; In a sequel we always assume that (9) holds. We state our results in terms of the value d in (9). Of course for any d0 d (9) also holds. However, the smaller d the stronger multiplier theorem we will be able to obtain. Therefore we want to take d as small as possible. Note that in the case of the group of polynomial growth the smallest possible d in (9) is equal to max(d ; d1). Hence our notation is consistent with statements of Theorems 1 and 2. Note that we do not assume that X satis es doubling property. This poses certain diculties which we overcome by using results of singular integral operators of [DM]. An example of such a space X is a domain of Euclidean space Rd. If we do not assume any smoothness on its boundary, then doubling property fails in general. Now we describe the notion of the kernel of the operator. Suppose that T : L (X; ) ! Lq (X; ) for q > 1. Then by KT (x; y) we denote the kernel of the operator T de ned by the formula 0
1
(10) hTf ; f i = 1
Z
2
X
Tf f d =
Z
1 2
X
KT (x; y)f (y)f (x) d(x) d(y): 1
2
for all f ; f 2 Cc(X ). Note that kT kL1 X; !Lq X; = sup kKT ( ; y)kLq X; : 1
2
(
)
(
)
(
y2X
)
Hence if kT kL1 X; !Lq X; < 1, then its kernel KT is a well de ned measurable function. Vice versa, if supy2X kKT ( ; y)kLq X; < 1, then KT is a kernel of the bounded operator T : L (X; ) ! Lq (X; ), even if q = 1. Next we denote the weak type (1; 1) norm of an operator T on a measure space (X; ) by kT kL1 X; !L1;1 X; = sup (fx 2 X : jTf (x)j > g), where the supremum is taken over > 0 and functions f with L (X; ) norm less than one. Assumption 2. Let A be a self-adjoint positive de nite operator. We suppose that the semigroup generated by A on L has kernel (
)
(
)
(
1
(
)
(
)
1
2
)
62
XUAN THINH DUONG, EL MAATI OUHABAZ AND ADAM SIKORA
pt(x; y) = K
exp(,tA) (x; y ) which for all t > 0 satis es the following Gaussian upper bound m=(m,1) (11) jpt(x; y)j C(B (y; t)),1=m exp , b (x;t1y=)(m,1) where C; b and m are positive constants and m 2. Such estimates are typical Gaussian estimates for elliptic or subelliptic di erential operators of order m (see e.g. [Da1, Ro, VSC]). We will call pt (x; y) the heat kernel associated with A. When order m = 2, Gaussian estimates (2.4) is equivalent to nite propagation speed, see [Si1]. When m 6= 2, we can have (2.4) but nite propagation speed property does not hold. In our following main results, we suppose that Assumptions 1 and 2 hold. The values d and m always refer to (9) and (11). Theorem 3. Suppose that s > d=2 and assume that for any R > 0 and all Borel functions F such that supp F [0; R]
Z
jKF mpA (x; y)j d(x) C(B (y; R, )), kR F kLp X for some p 2 [2; 1]. Then for any Borel bounded function F such that sup ktF kWsp < 1 the operator F (A) is of weak type (1; 1) and is t> bounded on Lq (X ) for all 1 < q < 1. In addition p (13) kF (A)kL X; !L ;1 X; Cs sup kt F kWs + jF (0)j : t> Note that if (12) holds for p < 1, then the pointwise spectrum of A is empty. Indeed, for all p < 1 and all y 2 X (12)
(
2
)
1
1
2
0
1(
1
)
(
)
0
(14) 0 = C k f = g kLp = C k a fag kLp (B (y; 21a )kK fag mpA ( ; y)kL2 X; 1 2
p
2
(
)
(
)
so fag ( m A) = 0. Hence for elliptic operators on compact manifolds, (12) cannot be true for any p < 1. To be able to study these operators as well we introduce some variation of assumption (12). Following [CS] for a Borel function F such that supp F [,1; 2] we de ne the norm kF kN;p by the formula X =p N 1 p kF kN;p = N sup jF ()j ; l ,N 2 l,N1 ; Nl where p 2 [1; 1) and N 2 Z . For p = 1 we put kF kN;1 = kF kL1 . It is obvious that kF kN;p increases monotonically in p. The next theorem is a variation of Theorem 3. This variation can be used in the 2
=1
+
1
[
)
SPECTRAL MULTIPLIERS FOR SELF-ADJOINT OPERATORS
63
case of operators with nonempty pointwise spectrum (compare [CS, Theorem 3.6]). Theorem 4. Suppose that is a xed natural number, s > d=2 and that for any N 2 Z and for all Borel functions F such that supp F [,1; N + 1] +
Z
(15)
X
jKF
(
mpA) (
; y)j d(x) C(B (y; 1=N )), kN F kN ;p 2
1
2
for some p 2. In addition we assume that for any " > 0 there exists a constant C" such that for all N 2 Z+ and all Borel functions F such that supp F [,1; N + 1]
p
(16) kF ( m A)kL1 X; !L1 X; C"N d "kN F kN ;p: Then for any Borel bounded function F such that supt> ktF kWsp < 1 the operator F (A) is of weak type (1; 1) and is bounded on Lq (X ) for all q 2 (1; 1). In addition 2
(
)
(
+
)
2
1
kF (A)kL
(17)
X;)!L1;1 (X;)
1(
Cs sup ktF k + kF k t>1
Wsp
L1
:
Remarks 1. It is straightforward that (12) always holds with p = 1
as a consequence of spectral theory. This means that Alexopoulos' multiplier theorem i.e. Theorem 2 follows from Theorem 3. Theorem 1 also follows from Theorem 3. Indeed, it is easy to check that for homogeneous operators (12) holds for p = 2 (see Section 6 [DOS] or [Ch2, Proposition 3]). 2. The main point of our theorems is that if one can obtain (12) or (15) then one can prove stronger multiplier results. If one shows (12) or (15) for p = 2, then this implies the sharp Hormander-type multiplier result. Actually we believe that to obtain any sharp spectral multiplier theorem one has to investigate conditions of the same type as (12) or (15), i.e. conditions which allow us to estimate the norm kKF mpA ( ; y)kL2 X; in terms of some kind of Lp norm of the function F. 3. We call hypotheses (12) or (15) the Plancherel estimates or the Plancherel conditions. In the proof of Theorems 3 and 4 one does not have to assume that p 2 in estimates (12) or (15). However (12) or (15) for p < 2 would imply Riesz summability for < (d , 1)=2 and we do not expect such a situation. Note that (12) is weaker than (15) and we need additional hypothesis (16) in this case. However, in practice once (15) is proved, (16) is usually easy to check and we can often put " = 0. (
)
(
)
64
XUAN THINH DUONG, EL MAATI OUHABAZ AND ADAM SIKORA
4. We conclude this paper with a theorem on Riesz summability for d=2 > (d , 1)=2. Theorem 3 with p = 2 implies Riesz summability for all > (d , 1)=2 and that in addition it seems that Theorem 3 with p = 2 is essentially stronger than sharp Riesz summability. However, one can obtain only weak type (1; 1) estimates in virtue of Theorem 3 and formally Theorem 3 does not imply continuity and convergence of Riesz means on L (X; ). However, Theorem 3 and 4 can be modi ed to prove that uniform continuity of Riesz means of order greater than (d=2 , 1=p) on all spaces Lq (X; ) for q 2 [1; 1]. We claim the following Theorem. Theorem 5. Suppose that operator A satis es condition (12), or (15) and (16) for some p 2 [2; 1]. Then for any > d=2 , 1=p and q 2 [1; 1] sup kR (A)kLq X; !Lq X; C < 1: R> Hence for any q 2 [1; 1) and f 2 Lq (X; ) lim k (A)f , f kLq X; !Lq X; = 0; R!1 R where R is de ned by (7). For the proofs of our Theorems, we refer reader to [DOS]. Here let us only mention that the proofs of Theorems 5 and 3 are less complicated than most of earlier spectral multiplier results. Our strategy is to use the complex time heat kernel bounds (see [Da1, DO]) to show W d = functional calculus for the considered operator A. Then we use Mauceri-Meda interpolation trick (see [MM]) and our Plancherel type assumption (12) to obtain Wd=p " functional calculus. This is enough to show Riesz summability (i.e. Theorem 5. To prove Theorem 3 we need also some Calderon-Zygmund singular integral techniques. However in contrast to the standard Calderon-Zygmund singular integral estimates we do not use estimates for the gradient of the kernel of singular integral operators. Instead of that we follow the ideas of [DM, He3, He2]. 1
0
(
)
(
(
)
)
(
)
2 ( +1) 2
2+
References
[Ale1] G. Alexopoulos. Spectral multipliers on Lie groups of polynomial growth. Proc. Amer. Math. Soc., 120(3):973{979, 1994. [Ale2] G. Alexopoulos. Lp bounds for spectral multipliers from Gaussian estimates of the heat kernel. preprint, 2000. [ChS] F. M. Christ and C. D. Sogge. The weak type L1 convergence of eigenfunction expansions for pseudodi erential operators. Invent. Math., 94(2):421{453, 1988. [Ch1] Michael Christ. Weak type (1; 1) bounds for rough operators. Ann. of Math. (2), 128(1):19{42, 1988.
SPECTRAL MULTIPLIERS FOR SELF-ADJOINT OPERATORS
65
[Ch2] Michael Christ. Lp bounds for spectral multipliers on nilpotent groups. Trans. Amer. Math. Soc., 328(1):73{81, 1991. [CS] Michael Cowling and Adam Sikora. A spectral multiplier theorem on SU(2). to appear in Math. Z. [CGHM] Michael Cowling, Saverio Giulini, Andrzej Hulanicki, and Giancarlo Mauceri. Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth. Studia Mathematica, 111(2):103{121, 1994. [Da1] E. B. Davies. Heat kernels and spectral theory. Cambridge University Press, Cambridge, 1989. [Da2] E. B. Davies. Uniformly elliptic operators with measurable coecients. J. Funct. Anal., 132(1):141{169, 1995. [dMM] Leonede De-Michele and Giancarlo Mauceri. H p multipliers on strati ed groups. Ann. Mat. Pura Appl. (4), 148:353{366, 1987. [DO] Xuan Thinh Duong and El-Maati Ouhabaz. Heat kernel bounds and spectral multipliers on spaces of polynomial growth and irregular domains. preprint, 1998. [Du] Xuan Thinh Duong. From the L1 norms of the complex heat kernels to a Hormander multiplier theorem for sub-Laplacians on nilpotent Lie groups. Paci c J. Math., 173(2):413{424, 1996. [DM] Xuan Thinh Duong and Alan McIntosh. Singular integral operators with nonsmooth kernels on irregular domains. Rev. Mat. Iberoamericana, 15(2):233{265, 1999. [DOS] Xuan Thinh Duong, El Maati Ouhabaz and Adam Sikora. Plancherel type estimates and sharp spectral multipliers preprint, 2000. [FS] G. B. Folland and Elias M. Stein. Hardy spaces on homogeneous groups. Princeton University Press, Princeton, N.J., 1982. [He1] Waldemar Hebisch. Almost everywhere summability of eigenfunction expansions associated to elliptic operators. Studia Math., 96(3):263{275, 1990. [He2] Waldemar Hebisch. A multiplier theorem for Schrodinger operators. Colloq. Math., 60/61(2):659{664, 1990. [He3] Waldemar Hebisch. Functional calculus for slowly decaying kernels. preprint, 1995. [He4] Waldemar Hebisch. Multiplier theorem on generalised Heisenberg groups. Colloq. Math., 65(2):231{239, 1993. [He5] Waldemar Hebisch. The subalgebra of L1(AN ) generated by the Laplacean. Proc. Amer. Mat. Soc., 117(2):547{549, 1993. [Ho1] Lars Hormander. Estimates for translation invariant operators in Lp spaces. Acta Math., 104:93{140, 1960. [Ho3] Lars Hormander. On the Riesz means of spectral functions and eigenfunction expansions for elliptic di erential operators. In Some Recent Advances in the Basic Sciences, Vol. 2 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1965{1966), pages 155{202. Belfer Graduate School of Science, Yeshiva Univ., New York, 1969. [HS] Andrzej Hulanicki and Elias M. Stein. Marcinkiewicz multiplier theorem for strati ed groups. manuscript. [MM] Giancarlo Mauceri and Stefano Meda. Vector-valued multipliers on strati ed groups. Rev. Mat. Iberoamericana, 6(3-4):141{154, 1990. [MS] D. Muller and E. Stein. On spectral multipliers for Heisenberg and related groups J. Math. Pures Appl., 73: 413{440, 1994.
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XUAN THINH DUONG, EL MAATI OUHABAZ AND ADAM SIKORA
[Ri] Marcel Riesz. Sur la sommation des series de Fourier,. A.S., 1:104{113, 1923. [Ro] Derek W. Robinson. Elliptic operators and Lie groups. The Clarendon Press Oxford University Press, New York, 1991. Oxford Science Publications. [SeSo] A. Seeger and C. D. Sogge. On the boundedness of functions of (pseudo) di erential operators on compact manifolds. Duke Math. J., 59(3):709{736, 1989. [Si1] Adam Sikora. Sharp pointwise estimates on heat kernels. Quart. J. Math. Oxford Ser. (2), 47(187):371{382, 1996. [Si2] Adam Sikora. On the L2 ! L1 norms of spectral multipliers of \quasihomogeneous" operators on homogeneous groups. Trans. Amer. Math. Soc., 351(9):3743{3755, 1999. [Si3] Adam Sikora. Multiplicateurs associes aux souslaplaciens sur les groupes homogenes. C.R. Acad. Sci. Paris series 1, 315:417{419, 1992. [SW] Adam Sikora and James Wright. Imaginary powers of Laplace operators. to appear in Proc. Amer. Math. Soc. [So1] Christopher D. Sogge. On the convergence of Riesz means on compact manifolds. Ann. of Math. (2), 126(2):439{447, 1987. [So2] Christopher D. Sogge. Fourier integrals in classical analysis. Cambridge University Press, Cambridge, 1993. [St1] Elias M. Stein. Singular integrals and di erentiability properties of functions. Princeton University Press, Princeton, N.J., 1970. Princeton Mathematical Series, No. 30. [St2] Elias M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. [Ta] Terence Tao. Weak-type endpoint bounds for Riesz means. Proc. Amer. Math. Soc., 124(9):2797{2805, 1996. [Tay] Michael Taylor Lp estimates on functions of the Laplace operator. Duke Math. J., 58: 773{793, 1989. [VSC] N. Th. Varopoulos, L. Salo -Coste, and T. Coulhon. Analysis and geometry on groups. Cambridge University Press, Cambridge, 1992. Xuan Thinh Duong, School of Mathematics, Physics, Computing and Electronics. Macquarie University, N.S.W. 2109 Australia
E-mail address :
[email protected]
El Maati Ouhabaz, Institut de Mathematiques Universite de Bordeaux 1 351, Cours de la Liberation 33405 Talence cedex. France
E-mail address :
[email protected]
Adam Sikora, Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra ACT 0200, Australia
E-mail address :
[email protected]
FROM XY TO ADE DAVID E. EVANS Abstract. We survey the role of non-commutative operator al-
gebras in statistical mechanics and the relation between the classi cation of modular invariant partition functions in conformal eld theories and braided subfactors.
As exposed in the treatises of Bratteli and Robinson [21] non-commutative operator algebras have a long tradition of providing a framework for understanding quantum statistical mechanics. For example, the one-dimensional -model is studied in the Pauli or Fermion algebra N M with the XY Hamiltonian Z 2
H=,
X
f(1 + )xj xj+1 + (1 , )yj yj+1 + 2zj g:
j 2Z
Here j ; = x; y; z are the usual Pauli matrices placed at the j th site of the tensor product. Typically one studies time evolution on the Pauli algebra via the one-parameter group of *-automorphisms t = eiHt ()e,iHt suitably de ned. In such lattice models one is interested in determining the set of equilibrium states, using the Gibbs conditions, KMS condition or a variational principle, minimizing the thermodynamic quantity (energy - temperature.entropy), as well as the return to equilibrium of locally perturbed models. Robinson played a seminal role in this theory, which is described in detail in [21]. Amongst other things, this led to the development of the theory of derivations on operator algebras, the in nitesimal generators of time evolution, which is still relevant today with the Powers-Sakai conjecture [58] a particularly challenging open problem. This led Robinson to working on the in nitesimal generators of (completely) positive semigroups on operator algebras and subsequently his most recent work on heat kernel methods. The Powers-Sakai conjecture asks whether every one-parameter dynamics on a UHF algebra (an in nite tensor product of matrix algebras) or more generally on a simple AF algebra (an inductive limit of nite dimensional algebras) is approximately inner, obtained as a limit of inner 85
86
DAVID E. EVANS
one-parameter groups as in the above XY -example. Kishimoto [49] has recently shown that a stronger form of the conjecture is false, namely the core problem of whether the in nitesimal generator has an AF-core in a suitable sense. If is a strongly continuous one-parameter group of *-automorphisms of a simple AF algebra, where the domain of the generator is AF, then is approximately inner. However, [49] constructs on any non type I simple AF C -algebra examples of approximately inner one-parameter groups of *-automorphisms where the domain of the generator is not AF. These can be regarded as one-parameter continuous analogues of the exotic examples of compact group actions on AF algebras whose xed points are not AF ( rst shown by Blackadar [5] for Z=2 on the Pauli algebra, and latter by Bratteli et al [18] for nite groups and Evans and Kishimoto [33] for compact groups). Returning to our original starting point of this paper, the XY -model, notice that it degenerates at certain values of (; ), namely at (0; 1), to the Ising nearest neighbour model. This is a classical Hamiltonian, and it would therefore appear to be arti cial to study it via a noncommutative framework, the Pauli algebra. Nevertheless, there is a natural role for non-commutative operator algebras in the study of such classical statistical mechanical models which is the point of this present survey. This connection begins with the transfer matrix method. Let us take a two dimensional nearest neighbour Ising model on a square lattice Z2 with Hamiltonian
H =,
X
; nn
J
with the summation over the vertices or sites ; on Z2 which are nearest neighbours (nn). We switch from one to two dimensions because the one dimensional version does not have a phase transition at a non zero temperature. At each site or vertex point of the lattice we have a variable, a spin or magnetization with either a positive or negative orientation or value represented by +1 or ,1. Then a state = ( ) of the Ising model is a distribution of pluses and minuses over the square lattice L = Z2, so any con guration is represented by a point in con guration space the compact Hausdor space P = f1gL. Thus the N natural home to study this Ising model is the space C (P ) = Z2 C 2
FROM XY TO ADE
87
the commutative C -algebra of all continuous functions on the compact con guration space. At each inverse temperature we may be interested in the simplex K of equilibrium states, given say by solutions to the equations of Dobrushin, Lanford and Ruelle [30][50] or the variational principle: minimize(energy - temperature.entropy). In the algebraic approach, one uses the transfer matrix formalism to transform the setting to that of a one dimensional quantum model, represented by a non commutative "one dimensional" C -algebra and time evolution t . The transfer matrix T is obtained for the partition function of a strip of nite length M and width length one. With boundary conditions ; along the two lengths the corresponding partition function T de nes us the transfer matrix T . The partition function Z of a nite rectangular lattice of length M and width N is then obtained by multiplying the strip partition functions, namely transfer matrix entries and summing over internal edges. For periodic boundary conditions we obtain (1)
Z=
X
exp(, H ()) =
X
T 12 T 23 : : : TN 1 = trace T N :
N
In this way we move from the commutative algebra C (P ) = Z2 C 2 N to the non-commutative Pauli algebra A = Z M2 where the local transfer matrices T generate the even part A+. Time evolution can be formally written as t = T it()T ,it, i.e. we consider T = e,H where H is now a quantum Hamiltonian which is no longer a (one dimensional) N Ising Hamiltonian. Spatial translation by Z2 in the N classical model Z2 C 2 corresponds to spatial translation in AP = Z M2 together with an evolution fT n()T ,n : n 2 Zg in the orthogonal transfer direction. For each inverse temperature one looks for a map F ! F from (local) classical observables in C (P ) to the quantum algebra A, and a map ! ' from states on C (P ) (or measures on P ) to linear functionals on the local observables in AP such that one can recover the classical expectation values or correlation functions from a knowledge of the quantum ones alone: (F ) = ' (F ). Fixing some boundary conditions, then for each inverse temperature , let ' denote the corresponding state on A. [In general positivity of ' is not automatic but follows from re ection positivity of ]. Then if c denotes the inverse critical temperature of Onsager, there exist automorphisms
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f : 6= cg of A [34] which do not depend on boundary conditions, and real analytic in = 6 c such that ' =
N
(
'1 > c '0 < c:
, p
and ,for Here '0 = Z ! , where = 11 N2 is the disordered state, , 1 + , + or - boundary conditions, '1 = Z ! where = 0 ; = 01 respectively and '1 = ('+1 + ',1) =2 for free or periodic boundary conditions. Thus with free or periodic boundary conditions, we conclude that ' is pure for 0 < c (also for = c by di erent methods [2]) and is a mixture of two inequivalent pure states for ' for > c. If h i denote the classical states corresponding to + and - boundary conditions respectively, then we can deduce that hF i is real analytic in > c when F is a local classical observable, as hF i = '1 (F ) using analyticity of . A dynamical system t on AP is formally given as t = T it()T ,it which has a unique ground state for < c and two extremal ground states ' for > c. The Ising model can be generalized to the possibility of having more than two values or spins possible at any lattice site, and moreover some constraints or rules to determine allowable con gurations. A particular value at one site may force only restricted choices at nearest neighbours. This would be achieved by distributing values of a xed graph G at sites of the lattice L in such a way that if and are nearest neighbours in L, then the corresponding values and are joined in the graph. The state space P can be de ned for any graph, but if G contains some multiple edges, we consider distributions of edges of G on edges of L. For the Dynkin diagram A3 with vertices labelled by f; g and square lattice L one obtains two copies of the Ising model as in Fig. 1 by placing the frozen spin on the even or odd sublattices of L. This graph may be generalised to the Dynkin diagrams of Fig. 1 for the models of Andrews, Baxter and Forrester [1]. These in turn can be generalised by considering the Weyl alcove A(n;k) of the level k integrable representations of the Kac-Moody algebra SU (n)^ : Boltzmann weights associated to a local con guration around minimal squares of the lattice can be chosen to satisfy the integrable YangBaxter equation [25].
FROM XY TO ADE
r
,,A@3 + r
r
r
r
r
r
r
r
89
r
r
Ising model: Dynkin diagram A3 and con guration space Figure 1.
The centre of SU (n), the abelian Z=n, acts on A(n;k), e.g. Z=2 on the Dynkin diagram Ak+1 by a ip i ! k , i which may or may not have a xed point depending on the parity of k. The interesting case is when there is a xed point. In any case, the Boltzmann weights are preserved under the symmetry, and yield new Boltzmann weights on the orbifold graphs A(n;k)=(Z=p), whenever p divides n, satisfying the integrable Yang Baxter equation [28] [35]. For example, when n = 2; k = 2m we blow up the xed point m to a pair (a copy of Z=2) and replace each distinct pair i; k , i (i 6= m) interchanged by the symmetry with a singleton yielding the graph Dm+2. The case A3 is self dual in that A3=(Z=2) = A3 . Nevertheless, the situation here is not entirely trivial. This is Kramers-Wannier high temperature{low temperature duality. This duality replaces the Boltzmann weights at a temperature t with ones at dual temperature t. Again the xed point of the symmetry t ! t is what provides the interesting structure | at the critical temperature tc of Onsager. We have mentioned the phenomena of AF algebras with non-AF xed point algebras under symmetries. Such examples were rst found using similar orbifold constructions. As a continuous version of the ip on a Dynkin diagram which yields symmetries on AF algebras, consider instead the ip on the interval around its midpoint or a ip on a circle around an axis in its plane through its centre. The orbifold space is best described by taking the cross product. For a pair of points interchanged by the symmetry, the local crossed product is simply a two by two matrix algebra. The diagonal elements represent the continuous functions on the pair, and the o -diagonal elements come from the transition between the two points. Each xed point is replaced by a pair arising from the transitions only generating a copy of C 2 as the continuous functions on the group (dual). Thus gluing together, we
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represent the action of the ip on the interval [0; 1] by the C -algebra of continuous functions on a half-interval [0; 1=2], which on the half-open half-interval [0; 1=2) (the part of the interval which has a non-trivial orbit) are two by two matrix valued but at other end point 1=2 are diagonal. So the spectrum of this algebra is a continuous version of the D-Dynkin diagram. Topologically it is an interval with two non Hausdor points at one end. The analogous action of the ip on the two torus which conjugates each variable has four xed points, yielding a sphere with four singular points. The corresponding cross product is the space of two by two matrix valued functions on the sphere which are diagonal at four distinguished points. Replacing the two torus by a non-commutative torus generated by two unitaries U and V satisfying the commutation relation V U = qUV where q = exp2i we obtain a non commutative toroidal orbifold when taking the symmetry which inverts the generators U and V . It is Morita equivalent to the algebra of a singular ow on a sphere obtained as the quotient of the Kronecker ow on the torus as illustrated in [36], page 137 or http://www.cf.ac.uk/maths/opalg/ncto/. Remarkably these algebras are AF (when is irrational)([20][19] or [32],[65]) although the corresponding irrational rotation algebras and algebras of the Kronecker ow are not. The non-commutative torus has a representation on L2 (T2 ) where U and V are represented as multiplication and translation operators. In this representation, or at least if one takes the Fourier transform, the Hamiltonian H = U + U ,1 + (V + V ,1 ) are the Mathieu or discrete Schrodinger operators with almost periodic potentials. The natural home to study these operators is the xed point algebra under our ip because when is irrational then U + U ,1 and V + V ,1 generate the xed point algebra. It is still a tantalizing mystery as to whether there is a relation between the AF property of the xed point algebras, a strong form of non-commutative disconnectedness, and the Cantor spectra of such almost Mathieu operators - which are at least known to be Cantor for generic coupling constant and rotation number . Symmetries on such algebras, where there are underlying xed points can produce algebras with totally di erent properties. Similarly, symmetries on subfactors, statistical mechanical models, conformal eld theories can produce totally di erent subfactors etc from what one started with.
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91
From a lattice model one may obtain a eld theory by taking a continuum or scaling limit, letting the lattice spacing go to zero whilst simultaneously approaching the critical temperature. As the scale or correlation length becomes in nite, one obtains a scale invariant or conformal invariant theory. Belavin et al [4] suggested that the scale invariance at a critical point is enhanced to conformal invariance. One of the invariants of the conformal eld theory is the central charge a multiplier in projective representations Lm of the vector elds ,zm+1 d=dz on the circle, the Virasoro algebra. However the central charge can already raise its head in the statistical mechanical model. Going back to the partition function Z of Eq. (1) the free energy f = ,logZ=NM is independent of boundary conditions as N; M ! 1. However the asymptotics depend on boundary conditions; if 1 << N << M , then Z exp(,NMf + Mc=N 6), where c is the central charge. (See e.g. [26] (or [36], Chapter 8) for explicit computation in the case of the Ising model). Let us however proceed to the conformal eld theory at criticality. It is argued on physical grounds that the partition function Z ( ) in a conformal eld theory on the torus should be invariant under reparametrization of the torus by SL(2; Z): Z ( ) = Z ((a + b)=(c + d)) [23]. In the string theory formulation, modular invariance is essentially built into the de nition of the partition function (although Nahm [53] has argued the case for modular invariance in terms of the chiral algebra and its representations rather than a functional integral setting). In the transfer matrix picture of the statistical mechanical picture, we wrote the partition function as an average over e, H , where H is the Hamiltonian. The Hamiltonian is now L0 + L 0 , c=12 where L0 ; L 0 are commuting generators of rotation groups, c the central charge and the shift by c=24 comes from mapping the Virasoro algebra on the plane to a cylinder. We also have a momentum P = L0 , L 0 describing evolution along the closed string, so taking both evolutions into account we rst compute
Z ( ) = tr e, H eiP = tr e2i (L0 ,c=24) e,2i (L 0 ,c=24) : Here 2i = , + i parametrizes the metric of the torus, and we then have to average over . If we choose one from each orbit under the action of SL(2; Z) and integrate we implicitly assume that Z ( )
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is modular invariant. In our SU (n) setting the Hilbert space decomposes according to the associated loop group representations. The loop group LG is the group of smooth maps from S 1 into a compact Lie group G under pointwise multiplication. We are interested in projective representations of LG o Rot(S 1) where Rot(S 1 ) is the rotation group, which are highest weight representations in that the generator L0 of the rotation group is bounded below. Such representations are called positive energy representations and are classi ed by irreducible representations of G (by restriction to the constant loops) and a level k describing the multiplier in the projective representation. For unitary irreducible positive energy representations, the possibilities are severely restricted. Indeed k must be integral and for a given value of the level, there are only a nite number of admissible (vacuum vector) irreducible representations of G: For G = SU (n) the admissible ones are precisely the vertices of A(n;k), the same labeling set as used to construct our statistical mechanical model. The partition function then decomposes as
Z ( ) =
X
Z; ( ) ( )
where (2)
Z = 0; 1; 2; :::; Z00 = 1
and characters ( ) = tre2i (L0 ,c=24) , Im > 0. Here the label 0 refers to the vacuum representation, and the condition Z00 = 1 refers to the physical concept of uniqueness of the vacuum state. The matrix Z arising in this way is called a modular invariant mass matrix. (More precisely, for current algebras the characters depend also on variables other than , corresponding to Cartan subalgebra generators which are omitted here for simplicity. These extra variables mean we are dealing with SL(2; Z) rather than PSL(2; Z)). From the canonical generators 0 , 1 1 1 S= 1 0 , and T = 0 1 of SL(2; Z) we obtain the unitary Kac-Petersen matrices S = [S ]; T = [T ] transforming characters, where S is symmetric as well as S0
FROM XY TO ADE
S00 > 0 and T is diagonal:
X
(,1= ) =
S ( ); ( + 1) =
93
X
T ( ):
Then the classi cation of modular invariant partition functions can be reformulated as a matrix problem. Find all matrices Z subject to Eq. (2) commuting with S and T . This is a rather concrete problem. For SU (2) at level k, SU (2)k , the admissible weights are the spins = 0; 1; ::; k and the Kac-Peterson matrices are given explicitly as r 2 + 1) S = k + 2 sin ( +k1)( +2
( + 1)2 i T = exp i 2k + 4 , 4
with ; = 0; 1; :::; k, and the characters as (+1)2 =4(k+2) X q (q) = (q)3 (2n(k + 2) + + 1)qn(n(k + 2) + + 1) n2Z
Q
n if q = e2i , and the Dedekind function (q) = q1=24 1 n=1 (1 , q ): For the Ising model, the characters are (in the notation of Fig. 1),
= [#2 =2]3=2 ; = ([#3 =]3=2 [#4 =n]3=2 )=2 with the #-functions:
p
#3 = = q,1=48
p
#4 = = q,1=48
p
#2 = =
1 , Y
1 + qn+1=2
n=0
1 , Y
1 , qn+1=2
n=0
q1=24
1 Y
(1 + qn):
n=1
Here the Kac-Petersen matrices are simply 0 1 p2 1 1 01 0 0 p p S = 12 @ 2 p 0 , 2 A ; T = ei=24 @ 0 ei3=8 0 1 , 2 1 0 0 ei
1 A
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so that there is only one modular invariant the diagonal mass matrix or: , Z = j#2j3 + j#3 j3 + j#4 j3 =2jj3: Whilst the mass matrix is trivial, the partition function itself has some structure. The following is also a modular invariant particular function Z = (j#2 j + j#3j + j#4 j)=2jj L this time for the coset model su(2)1 su(2)1=su(2)2, which also exhibits Ising fusion rules (as does (E8)1 (E8 )1=(E8 )2) and so(5)1). At rst sight, it might appear that generally there may be an in nite number of solutions to this modular invariant problem. However, there is a following estimate [12]: P Z dd which is a strengthening of the inequality of Gannon [42]: Z 1=S002 if d = S0=S00 : Thus since Z is positive and integral there are at most nitely many solutions, for a xed representation of SL(2; Z). In the case of SU (2), there are at most three solutions for a xed level k. This is the ADE classi cation of Capelli, Itzykson and Zuber [22]. A Dynkin diagram is associated to each invariant through the identi cation of diagonal terms of the invariant f : Z 6= 0g = I with eigenvalues fSf=S00 : 2 Ig of the corresponding Dynkin diagram if f = 1 the fundamental representation of SU (2). The A refers to the diagonal invariant, D to orbifold invariants and E to the three E6; E7 ; E8 exceptional invariants. For SU (3) there is an anologous ADE classi cation due to Gannon [43]; di Francesco and Zuber [28] sought to show systematically the existence of graphs with spectra matching the modular invariant, give a meaning to these graphs themselves and compute them in a number of examples. As we have said there are at most nitely many solutions to the modular invariant conditions. There is always one solution the trivial diagonal invariant: X 2 j j 2A
where the corresponding mass matrix is diagonal Z = . In some sense, [52] [29] [11] 'every' modular invariant is diagonal if looked at properly. If we can extend the A system to a B system so that the characters decompose X = b ; 2 B 2A
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95
according to some branching rules, then the diagonal B-modular invariant will give an A-modular invariant X 2 XX j j = j b j2: 2B
2B 2A
In some sense, every modular invariant should look like this or with a possible twist w( ) ; for a permutation w of the extended fusion rules, preserving the vacuum. The problem in general is then to nd such extensions. When there is no twist present we have what are sometimes called type I invariants: X Z = b ; b :
These are automatically symmetric: Z = Z . In the presence of a non-trivial twist, we have the type II invariants X Z = b bw( ) :
These are not necessarily symmetric, but at least there is symmetric vacuum coupling Z0 = Z0. Not every modular invariant is even symmetric in this sense, (e.g. for SO(16n)1) but every known SU (n) modular invariant is even symmetric in the usual sense. Our aim is to study or even construct modular invariants from subfactors. The framework can be summarised as follows. We have a hyper nite III1 factor N on which there is a system of endomorphisms f 2 Ag labelled by our positive energy representations or our original states in the original statistical mechanical setting. We induce these endomorphisms to endomorphisms on a larger ambient factor M | there will be two natural ways to do this labelled . The modular invariant will then be constructed or recovered as Z = h +; ,i where the right hand side will be computed as dimensions of intertwiner spaces or the number of common sectors when we decompose into irreducibles. The original endomorphism 2 A will be irreducible but may not be. The factor N will carry the modular data for S and T matrices, varying the inclusion may change the modular invariant but somehow the inclusion will have to be related to the original A-system. The system A on the factor N can be constructed via the method of Jones-Wassermann. First for any positive energy representation
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2 A, the objects LI SU (n) and LI 0 SU (n), if LI ; LI 0 denote loops on SU (n) concentrated on complementary non-trivial intervals I and I0 in the circle. We can thus form the inclusion (3) LI SU (n)00 LI 0 SU (n)0 : For = 0, the vacuum representation, there is Haag duality and this inclusion is not proper but gives us a single hyper nite III1 factor N (more precisely a net N (I ) of such factors). The inclusion Eq. (3) then determines a system of endomorphisms 2 A, so that the inclusion Eq. (3) is isomorphic to N N , with index [N , N ] = d2. Wassermann [66] has shown that the fusion rules of such endomorphisms , are precisely the same as that of SU (n) at a root of unity q = e2i=(k+n) . Moreover, rotation through 1800 on the circle, interchanges the role of I and I0 . This has the e ect that the system A is naturally braided, i.e. not only is the system commutative = as sectors if ; 2 A End(N ), but there is a choice "(; ) of unitaries taking to satisfying the Yang Baxter equation, braiding fusion equation etc. ]; ; 2 A, deterThus we have commutative matrices N = [N mining the fusion X = N
with composition of endomorphisms, or rather sectors, their unitary equivalence classes and a natural notion of addition. Fusion by the endomorphism of the conjugate of is given by N = Ntr, the transpose. Thus fN : 2 Ag is a family of commuting normal matrices and so simultaneously diagonalisable. By the Verlinde theory the unitary matrix which performs this diagonalisation is the S matrix itself. Inverting the consistency condition or the regular representation X (4) N N = N N we obtain X S = N S S S
or
0
X S
jSihSj: S 0 The modular invariants will provide representations other than the regular representation Eq. (4), and pick out subsets fS=S0 : 2 Ig N =
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97
where I = f : Z 6= 0g; the diagonal part of the modular invariant. At the same time these representations will replace N by other families G of graphs | to be associated with the modular invariant or at least the subfactor N M which yields that particular invariant. As we have said the inclusion, which is meant to duplicate the modular invariant, should be related to the original A-system. This is achieved as follows. There is a conjugation on endomorphisms of N , (extending for groups the notion of inverting automorphisms or conjugating a representation in the dual) compatible with the conjugation on A. Similarly one can conjugate endomorphisms or sectors of M , or those between N and M , M and N . In particular, we can take the inclusion = N ! M , its conjugate = M ! N and compose to get endomorphisms = on M and = jN = on N called the canonical and dual canonical endomorphisms respectively. What we need is lies in the system generated by A, i.e. decomposes as a sum of sectors from A. Note that we do not need to specify M when we ask whether a particular endomorphism of N is a dual canonical endomorphism. It may not be particularly clear in a given situation whether a certain endomorphism is a dual canonical endomorphism or what M may be. When Z is a modular invariant typical candidates for P dual canonical P P N opp endomorphisms will be Z ; Z on N and Z 0 0 N on N N opp where N opp is the opposite algebra, etc. The rst non trivial (i.e. exceptional) invariant for SU (2) occurs at level 10: (5)
ZE6 = j 0 + 6j2 + j 4 + 10 j2 + j 3 + 7 j2:
The diagonal part of the invariant I = f : Z 6= 0g matches the spectrum of the Dynkin diagram E6 , namely fS1=S0 = 2 cos ( +1)=12 : = 0; 6; 4; 10; 3; 7g: For this reason Capelli, Itzykson and Zuber labelled the invariant by the graph E6 . In the subfactor setting we can derive this graph as follows. First, we turn to the conformal embedding description of this invariant due to Bouwknegt and Nahm [17] which provides the extended system B which diagonalises the invariant. The embedding SU (2)10 SO(5)1 means there is a mapping of SU (2) in SO(5) such that the three level 1 representations B of SO(5) decompose into level 10 representations of SU (2) with nite multiplicity. The
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DAVID E. EVANS
system SO(5)1 has three inequivalent representations, b,v,s basic, vector and spinor which reproduce the Ising fusion rules. They decompose (cf Eq. (5)) as (6) b = 0 + 6 ; v = 4 + 10 ; s = 3 + 7 so that the E6 modular invariant for SU (2)10 arises from the diagonal invariant for SO(5)1: ZE6 = j bj2 + j v j2 + j sj2: Moving now to the loop group factors the conformal embedding gives us an inclusion of factors: L SU (2) LSO(5) using the vacuum representation on LSO(5), a net of subfactors N (I ) M (I ): Fixing I, we have subfactor N M on which there are systems A = SU (2)10 and B = SO(5)1 of endomorphisms acting respectively. These two systems can be related by a form of Mackey inductionrestriction which in the subfactor setting goes back to Longo-Rehren [51]. Using the braiding "+ or its opposite braiding ",, we can lift endomorphisms in A to those of M: = ,1Ad"(; ) . The maps [] ! [ ] preserve all the operations of conjugation, addition and multiplication of sectors [67][8][9][10]. However, they are not injective, and may be reducible. We nd that f + : 2 Ag decomposes into six irreducible sectors such that the graph E6 is multiplication by 1+ [67] [9]. In fact [ 1+] = G; is part of a system of matrices with non-negative entries fG : 2 Ag which represents the original A-fusion rules. This had been noticed empirically in e.g. [28] [55] which now gets a subfactor explanation. To bring the B system into the game we use -restriction, = to take M -sectors to N -sectors. This map is not multiplicative, but in the type I situation there is a reciprocity: h ; i = h; i (with inequality on the type II setting) as long as say is a subsector of the induced system E6 respectively. Since restriction takes the E6 systems into the A-system by Eq. (6), namely b = = 0 + 6 ; v = 4 + 10 ; s = 3 + 7 the reciprocity means that the B system, coming from the three level 1 representations of SO(5) must lie in the induced systems E6, i.e.
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99
A3 = B E6+ \ E6,. In fact we can identify as sectors b = 0; v = 10; s = 3 , 9; and E6+ \ E6, = A3 precisely. The dual canonical endomorphism lies in A, but a priori we do not have much information about its Fourier transform . In fact as sectors: = idM + 1+ 1, so that 2 E6+ _ E6,, the full induced system. Indeed A = E6+ _ E6, , the system generated by E6 is precisely all subsectors of f : 2 Ag P 2 in latter has global dimension w = A d, whilst if w = P fact dthe 2 denotes the global dimension of the induced system then [10]: 2A w=w =
X
Z0d
with the sum over only the degenerate sectors in A | which have trivial monodromy with all other sectors. In this case the A-system, as far SU (n)k is non-degenerate, the vacuum is the only degenerate sector. Moreover we can recover the modular invariant as
Z = h +; ,i; ; 2 A: In this case the E6 systems are commutative (but not braided) as is the E6+ _ E6, system | but this is not always the case. The neutral system A0 = A+ \ A,, if A are the induced chiral systems, is braided, with the braiding non-degenerate if that of A is. Complexifying the nite dimensional algebras A we can decompose them in the non-degenerate case as [15]: (7)
A =
MM
2A0 2A
Mat(b ):
Here b ; are the chiral branching coecients h ; i; 2 A; 2 A. (In the case of chiral locality where the extended net M (I ), is local, i.e. observables associated with disjoint intervals commute, then b ; = h ; i = h ; i; 2 A0; 2 A:) In particular the extended systems are commutative only when b ; 1; 2 A0; 2 A Thus the informal inclusions SU (n)n SU (n2 ,1)1 give non-commutative chiral systems when n 4; and it explains the computations of Feng Xu [67] who found non-commutativity in case n = 4 by a direct computation. Thus we can decompose the modular invariant as
Z = h +; ,i =
X
2A0
b ; b ;
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DAVID E. EVANS
and from Eq. (7) by counting dimension we see that
jAj =
X
(b ; )2 = trbt b :
In the case of chiral locality where b+ = b, , so that the invariant is type I, we see that jAj = trZ t Z , more generally jAj only sees the type I parent of a type II invariant. Thus with
ZD10 = j 0 + 16j2 + j 4 + 12 j2 + j 6 + 10 j2 + 2j 8j2 + j 2 + 14 j2 ZE7 = j 0 + 16j2+j 4+ 12 j2+j 6+ 10 j2+j 2 j2+( 2+ 14 ) ,8 + 8 ( 2+ 14 ), then in either case trb b = 10 so that multiplication by [ 1] gives the graph D10 in either case so we do not get the graph E7 for ZE7 (where we can use the dual canonical endomorphisms 0 + 16 , 0 + 16 + 8 respectively). In general (and this will work when either chiral locality holds or fails) we look at the action of A on the M -N sectors M AN which are the irreducible sectors of = (which can be identi ed with A L when chiral locality holds). This action decomposes as: Mat(Z ), with M = SS 1Z :
0
Thus we get the desired representation with spectrum matching the diagonal part of the modular invariant, and counting dimension then jM AN j = trZ; e.g. trZE7 = 7 so that we do indeed now recover the correct graph. The subfactor framework is rich enough to produce a Moore-Seiberg type decomposition of modular invariants as well as handle possibly non-symmetric modular invariants. As we have already observed, in the case of chiral locality, b+ ; = b, ;(= h; i) for 2 A; 2 A0: So the question arises as to how far we can identify b+ and b, , say b, ; = b+w( ); for a permutation w of the extended neutral system B or if we need di erent labellings B+ or B, to handle possibly non-symmetric modular P P Z : invariants. Now locality holds if and only if = Z = 0 0 P P In general we de ne + = Z0; , = Z0 : Using the theory of intermediate subfactors of [45], we can show [16] that both are dual canonical endomorphisms for inclusions N M which satisfy chiral locality and M M . This means we can use -induction on both
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101
inclusions N M, to obtain type I modular invariants Z , such that Z+0 = Z0+ = Z0; Z,0 = Z0, = Z0; where we can identify both neutral systems M A0M with M A0M . If M+ = M, ; then we can write = Xb b Z 2A0
in particular (Z + = Z ,) and using the identi cations M A0M with M A0M to produce an automorphism ! of the neutral elements M A0M we have: X Z = b bw( ) : In the case of E7 invariant we have N M M where M+ = M, and the dual canonical endomorphism for N M ; N M are 0 + 16 , 0 + 16 + 8 as we have said before. It may happen that M+ 6= M, and this does occur for SO(16n)1 where there are non-symmetric modular invariants where we must use di erent labelling M+ A0M+ ;M, A0M, on the left and right to decompose ,1 Z ext + ; , as , ;#( + ) , where # = #, #+ is the identi cation. The situation is summarised [10] using recent work of Rehren [62] on canonical tensor product subfactors as a pair of inclusions: O O N N opp M+ M,opp B N N opp B and where the dual canonical endomorphisms for N N P N P N M+ M,opp B as Z opp; 2A0 #+ ( ) #, ( )opp respectively. There is a connection between the two chiral inductions and the picture of left- and right-chiral algebras in conformal eld theory. Suppose that our factor N is obtained as a local factor N = N (I ) of a quantum eld theoretical net of factors fN (I )g indexed by proper intervals I R on the real line, and that the system N XN is obtained as restrictions of DHR-morphisms (cf. [44]) to N . This is in fact the case in our examples arising from conformal eld theory where the net is de ned in terms of local loop groups in the vacuum representation. Taking two copies of such a net and placing the real axes on the light cone, then this de nes a local net fA(O)g, indexed by double cones O on two-dimensional Minkowski space (cf. [61] for such constructions). Given a subfactor N M , determining in turn two subfactors N M obeying chiral
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DAVID E. EVANS
locality, will provide two local nets of subfactors fN (I ) M(I )g as a local subfactor basically encodes the entire information about the net of subfactors [51]. Arranging M+ (I ) and M,(J ) on the two light cone axes de nes a local net of subfactors fA(O) Aext (O)g in Minkowski space. The embedding M+ M,opp B gives rise to another net of subfactors fAext (O) B (O)g, where the net fB (O)g obeys local commutation relations. The existence of the local net was already proven in [62], and now the decomposition of [ext ] tells us that the chiral extensions N (I ) M+(I ) and N (I ) M, (I ) for left and right chiral nets are indeed maximal (in the sense of [61]), following from the fact that the coupling matrix for fAext (O) B (O)g is a bijection. This shows that the inclusions N M should in fact be regarded as the subfactor version of left- and right maximal extensions of the chiral algebra. References
[1] Andrews, G. E., Baxter, R. J., and Forrester, P. J.: Eight vertex SOS model and generalized Rogers{Ramanujan type identities. J. Stat. Phys., 35, 193{266 (1984). [2] Araki, H., Evans, D.E.: On a C -algebra approach to phase transition in the two dimensional Ising model. Commun. Math. Phys. 91, 489-503 (1983). [3] Behrend, R.E., Pearce, P.A., Petkova, V.B., Zuber, J.-B.: Boundary conditions in rational conformal eld theories Nucl. Phys. B570, 525-589 (2000). [4] Belavin, A. A., Polyakov, A. M., and Zamolodchikov, A. B. In nite conformal symmetry in two-dimensional quantum eld theory. Nucl. Phys. B, 241, 333{ 380 (1980). [5] Blackadar, B.: Symmetries of the CAR algebra. Annals of Math., 131, 589{623 (1990). [6] Bockenhauer, J.: Localized endomorphisms of the chiral Ising model. Commun. Math. Phys. 177, 265-304 (1996). [7] Bockenhauer, J.: An algebraic formulation of level one Wess-Zumino-Witten models. Rev. Math. Phys. 8, 925-947 (1996). [8] Bockenhauer, J., Evans, D.E.: Modular invariants, graphs and -induction for nets of subfactors. I. Commun. Math. Phys. 197, 361-386 (1998). [9] Bockenhauer, J., Evans, D.E.: Modular invariants, graphs and -induction for nets of subfactors. II. Commun. Math. Phys. 200, 57-103 (1999). [10] Bockenhauer, J., Evans, D.E.: Modular invariants, graphs and -induction for nets of subfactors. III. Commun. Math. Phys. 205, 183-228 (1999). [11] Bockenhauer, J., Evans, D.E.: Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors. Commun. Math. Phys. 213, 267-289 (2000).
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[12] Bockenhauer, J., Evans, D. E.: Subfactors from modular invariants. math.OA/0006114. [13] Bockenhauer, J., Evans, D. E.: Subfactors and modular invariants. math. OA/0008056. [14] Bockenhauer, J., Evans, D.E., Kawahigashi, Y.: On -induction, chiral generators and modular invariants for subfactors. Commun. Math. Phys. 208, 429-487 (1999). [15] Bockenhauer, J., Evans, D.E., Kawahigashi, Y.: Chiral structure of modular invariants for subfactors. Commun. Math. Phys. 210, 733-784 (2000). [16] Bockenhauer, J., Evans, D.E., Kawahigashi, Y.: Longo-Rehren subfactors arising from -induction. Publ. RIMS, Kyoto Univ. to appear, math.OA/0002154. [17] Bouwknegt, P., Nahm,W.: Realizations of the exceptional modular invariant A(1) partition functions. Phys. Lett. B, 184, 359-362 (1987). 1 [18] Bratteli, O., Elliott, G. A., Evans, D. E., and Kishimoto, A.: Finite group actions on AF algebras obtained by folding the interval. K -theory, 8, 443{464 (1994). [19] Bratteli, O., Evans, D. E. and Kishimoto, A.: Crossed products of totally disconnected spaces by Z2 Z2. Ergodic Theory and Dynamical Systems, 13, 445{484 (1994). [20] Bratteli, O. and Kishimoto, A.: Non-commutative spheres, III, Irrational rotations. Commun. Math. Phys., 147, 605{624 (1992). [21] Bratteli,O., Robinson,D.W.: Operator algebras and quantum statistical mechanics. I and II 2nd editions Springer, Berlin (1987 and 1997) [22] Cappelli, A., Itzykson, C., Zuber, J.-B.: The A-D-E classi cation of minimal and A(1) 1 conformal invariant theories. Commun. Math. Phys. 113, 1-26 (1987). [23] Cardy, J. L., Operator content of two-dimensional conformally invariant theories. Nucl. Phys. B270 186-204 (1986). [24] Carey, A. L., Evans, D. E.: The operator algebras of the two dimensional Ising model. In: Birman, J. et al (eds.): Braids. Contemp. Math. 78, 117-165 (1988). [25] Date, E., Jimbo, M., Miwa, T., Okado, M.: Solvable lattice models. In: Theta functions { Bowdoin 1987, Part 1. Proc. Symp. Pure Math. 49, Providence, R.I.: American Math. Soc., pp. 295-332 (1987) [26] Davies, E.B., Pearce, P.A.: Conformal invariance and critical spectrum of corner transfer matrices. J. Phys. A23, 1295-1312 (1990). [27] Di Francesco, P., Mathieu, P., Senechal, D.: Conformal eld theory. New York: Springer-Verlag 1996. [28] Di Francesco, P., Zuber, J.-B.: SU (N ) lattice integrable models associated with graphs. Nucl. Phys. B338, 602-646 (1990). [29] Dijkgraaf, R., Verlinde, E.: Modular invariance and the fusion algebras. Nucl. Phys. (Proc. Suppl.) 5B, 87-97 (1988). [30] Dobrushin,R. L.: Gibbsian random elds for lattice systems with pair-wise interactions. Functional Analysis and its applications, 2, 292-301 (1968).
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[31] Doplicher, S., Haag, R., Roberts, J.E.: Fields, observables and gauge transformations. II. Commun. Math. Phys. 15, 173-200 (1969). [32] Elliott, G. A. and Evans, D. E.: The structure of the irrational rotation C algebra. Annals of Mathematics, 138, 477{501 (1993). [33] Evans, D. E. and Kishimoto, A.: Compact group actions on UHF algebras obtained by folding the interval. J. Funct. Anal., 98, 346{360 (1991). [34] Evans, D. E., Lewis, J.T.: On a C -algebra approach to phase transition in the two dimensional Ising model. II. Commun. Math. Phys. 102, 521-535 (1986). [35] Evans, D. E., Kawahigashi, Y.: Orbifold subfactors from Hecke algebras. Commun. Math. Phys. 165, 445-484 (1994). [36] Evans, D. E., Kawahigashi, Y.: Quantum symmetries on operator algebras. Oxford: Oxford University Press 1998. [37] Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras. I. Commun. Math. Phys. 125, 201-226 (1989). [38] Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras. II. Rev. Math. Phys. Special issue, 113-157 (1992). [39] Frohlich, J., Gabbiani, F.: Braid statistics in local quantum theory. Rev. Math. Phys. 2, 251-353 (1990). [40] Frohlich, J., Gabbiani, F.: Operator algebras and conformal eld theory. Commun. Math. Phys. 155, 569-640 (1993). [41] Fuchs, J., Schellekens, A.N., Schweigert, C.: Galois modular invariants of WZW models. Nucl. Phys. B437, 667-694 (1995). [42] Gannon, T.: WZW commutants, lattices and level{one partition functions. Nucl. Phys. B396, 708-736 (1993). [43] Gannon, T.: The classi cation of ane SU (3) modular invariants. Commun. Math. Phys. 161, 233-264 (1994). [44] Haag, R.: Local Quantum Physics. Berlin: Springer-Verlag 1992. [45] Izumi, M., Longo, R., Popa, S.: A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras. J. Funct. Anal. 155, 25-63 (1998). [46] Jimbo, M.: A q-analogue of U (N + 1), Hecke algebra and the Yang-Baxter equation. Lett. Math. Phys. 11, 247-252 (1986). [47] Jones, V.F.R.: Index for subfactors. Invent. Math. 72, 1-25 (1983). [48] Kawahigashi, Y.: On atness of Ocneanu's connections on the Dynkin diagrams and classi cation of subfactors. J. Funct. Anal. 127, 63-107 (1995). [49] Kishimoto, A.: Examples of one-parameter automorphism groups of UHF algebras. math.OA/0008003. [50] Lanford, O.E., Ruelle, D.: Observables at in nity and states with short range correlations in statistical mechanics. Commun. Math. Phys., 13, 194215 (1969).
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[51] Longo, R., Rehren, K.-H.: Nets of subfactors. Rev. Math. Phys. 7, 567-597 (1995). [52] Moore, G., Seiberg, N.: Naturality in conformal eld theory. Nucl. Phys. B313, 16-40 (1989). [53] Nahm,W.: Lie group exponents and SU (2) current algebras. Commun. Math. Phys.,118, 171-176,(1988). [54] Ocneanu, A.: Paths on Coxeter diagrams: From Platonic solids and singularities to minimal models and subfactors. (Notes recorded by S. Goto) In: Rajarama Bhat, B.V. et al. (eds.), Lectures on operator theory, The Fields Institute Monographs, Providence, Rhode Island: AMS publications 2000, pp. 243-323. [55] Petkova, V.B., Zuber, J.-B: From CFT to graphs. Nucl. Phys. B463, 161-193 (1996). [56] Petkova, V.B., Zuber, J.-B.: Conformal eld theory and graphs. In: Proceedings Goslar 1996 \Group 21", hep-th/9701103. [57] Popa, S.: Classi cation of subfactors of nite depth of the hyper nite type III1 factor. Comptes Rendus de l'Academie des Sciences, Serie I, Math., 318, 10031008 (1994). [58] Powers, R.T., Sakai, S.: Existence of ground states and KMS states for approximately inner dynamics. Commun. Math. Phys., 39, 273-288 (1975). [59] Rehren, K.-H.: Braid group statistics and their superselection rules. In: Kastler, D. (ed.): The algebraic theory of superselection sectors. Palermo 1989, Singapore: World Scienti c 1990, pp. 333-355. [60] Rehren, K.-H.: Space-time elds and exchange elds. Commun. Math. Phys. 132, 461-483 (1990). [61] Rehren, K.-H.: Chiral observables and modular invariants. Commun. Math. Phys. 208, 689-712 (2000). [62] Rehren, K.-H.: Canonical tensor product subfactors. Commun. Math. Phys. 211, 395-406 (2000). [63] Rehren, K.-H., Stanev, Y.S., Todorov, I.T.: Characterizing invariants for local extensions of current algebras. Commun. Math. Phys. 174, 605-633 (1996). [64] Verstegen, D.: New exceptional modular invariant partition functions for simple Kac-Moody algebras. Nucl. Phys. B346, 349-386 (1990). [65] Walters, S. G.: Inductive limit automorphisms of the irrational rotation C algebra. Commun. Math. Phys., 171, 365{381 (1995). [66] Wassermann, A.: Operator algebras and conformal eld theory III: Fusion of positive energy representations of LSU (N ) using bounded operators. Invent. Math. 133, 467-538 (1998). [67] Xu, F.: New braided endomorphisms from conformal inclusions. Commun. Math. Phys. 192, 347-403 (1998). School of Mathematics, University of Wales Cardiff, PO Box 926, Senghennydd Road, Cardiff CF24 4YH, Wales, U.K., EvansDE@cardi .ac.uk
THE HEAT-FLOW METHOD IN CONTACT GEOMETRY ROBERT GULLIVER
Contact geometry treats such questions as the existence and classi cation of contact structures on manifolds of odd dimension and speci ed topological structure. See inequality (1) below. The geometric/analytic approach treated in this report introduces parabolic systems of partial di erential equations (PDEs) in a way which complements the more algebraic methods, which until now are better known in contact geometry. This is a report on joint work in progress with Hansjorg Geiges of the University of Leiden, Netherlands and Matthias Schwarz of the University of Leipzig, Germany. Many of the speci c results reported on here appeared rst in a paper [1] by Steve Altschuler, which introduced the heat- ow method to study contact structures, and in a recent preprint [2] of Altschuler and Lani Wu. 1. Introduction to Contact Geometry Many of the participants in this conference apply analytical methods to geometrically motivated problems, or use geometric methods to strengthen their analysis. However, it cannot be assumed that everyone is familiar with all of the most modern concepts and techniques of di erential geometry. For that reason, this section will be devoted to an introduction to contact geometry appropriate for analysts, among others, and may be skipped by those with a good knowledge of the area. I was until rather recently a complete novice in this area of geometry, and the reader should not expect a polished nor absolutely concise presentation. See [4], [5] and [6] for more complete references to the literature. I expect that analysts will be interested to see this novel application of parabolic operators. Date : September 29, 2000; November 24, 2000. This work was initiated during my stay at the Max Planck Institute for Mathematics in the Sciences, Leipzig, and completed while I was visiting Monash University and the University of Melbourne. I would like to thank MPI, Monash and Melbourne for their generous hospitality. 106
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A hyperplane distribution in an open set M of R 2n+1 ; or in a smooth (2n + 1)-dimensional manifold M; speci es a subspace x of dimension 2n in R 2n+1 ; or rather in the tangent space to M at each point x 2 M; which depends smoothly on the point x: 1.1. Example: a foliation. A familiar example of a hyperplane distribution would be the two-dimensional distribution in R 3 spanned by the vector elds e1 (x) = (1; 0; 2x1) and e2 (x) = (0; 1; 2x2): Here we have written x = (x1; x2 ; x3 ): This distribution is especially easy to visualize, since e1 (x) and e2 (x) are a basis for tangent vectors to the family of paraboloids of revolution x3 , x12 , x22 = C; for various real constants C: This family of surfaces is a foliation of R 3 ; which means that every point of R3 lies on one of the surfaces, the surfaces and the family are smooth, and in some neighborhood of any point, the family looks like the family of coordinate planes x3 = const :; up to a local di eomorphism. In this situation, we say that the distribution is integrable, meaning in this case, where the rst and second components of e1 and e2 are (1; 0) and (0; 1); that their third components 2x1 and 2x2 are simultaneously the partial derivatives of a scalar function, locally. (The scalar function is x12 + x22 + C; of course.) Integrability is equivalent to saying that for any two vector elds V; W in ; the Lie bracket [V; W ] also lies in : Alternatively, we may describe a hyperplane distribution as the kernel of a nowhere vanishing di erential 1-form : (A 1-form is the dual of a vector eld, so that for any vector eld V; (V ) de nes a scalar function and depends linearly and pointwise on V:) Given ; the 1-form is determined up to a nonvanishing scalar factor by the requirement that (e1) = (e2) = 0; where e1; e2 form a local basis for the distribution : (Computationally, has the same components as the cross product of e1 and e2 .) The integrability condition for the distribution may be written in terms of the 1-form as an identity between 3-forms: ^ d = 0: (The exterior derivative d of a 1-form is the 2-form de ned by the alternating part of the matrix of rst partial derivatives; the wedge product of di erential forms is the alternating part of their tensor product.) A contact structure is a hyperplane distribution which is maximally non-integrable. In terms of Lie brackets, we may write ! (V; W ) for the transversal component ([V; W ]) of the Lie bracket of two vector elds V; W in : This makes ! a 2-form. The integrability condition requires that ! 0; for to be a contact structure, we require not merely that ! 6= 0 but far more: that the 2n-form !n = ! ^ ! ^ ^ ! be nowhere zero on . Via the appropriate Riemannian metric, this is equivalent to saying that ! de nes an almost-complex structure on the hyperplane
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distribution : Restricted to = ker ; ! is the same as ,d : Thus, the contact criterion may be written entirely in terms of the 1-form : (1) ^ d n 6= 0: Note that inequality (1) depends on but is independent of the choice of 1-form ; since if e = f for some nonvanishing scalar function f; then e ^ d en f n+1 ^ d n: Note also that since d is a two-form, the left-hand side of (1) is a di erential form of degree 2n + 1; so on R 2n+1 or M 2n+1 ; it has only one real component. In this sense, contact structures and contact forms only have their full meaning in domains and manifolds of odd dimension. A 1-form on a (2n + 1)-manifold which satis es inequality (1) is called a contact form. Inequality (1) is unusual, in the context of geometric analysis, for two reasons: it is a strict partial di erential inequality, and it is an underdetermined \system" consisting of one real, rst-order, fully nonlinear partial di erential inequality for the 2n +1 real components ai(x) of the 1-form : Speci cally, in the 5-dimensional case n = 2, we may write in local coordinates (x0 ; : : : ; x4) as
=
4 X i=0
ai(x) dxi:
Then the contact inequality (1) is equivalent to the inequality X @a @a sgn() a(0) @x(1) @x(2) 6= 0; (3) (4) where the sum is over all permutations of f0; 1; 2; 3; 4g: Systems of partial di erential equations of this general form are rather poorly understood at present. In the case of contact geometry, however, we shall see that there is a parabolic method available to attack inequality (1); see Section 2 below. 1.2. Example: the standard contact structure. A familiar example of a contact structure would be the two-plane distribution in R 3 with the subspace x at the point x = (x1; x2 ; x3) having basis vector elds e1(x) = (x1 ; x2; 0) and e2 (x) = (,x2 ; x1; r2); where we have written r2 = x12 + x22: In order to visualize ; we note that e1 is the horizontal vector eld pointing away from the x3 -axis, and that e2 is a vector orthogonal to e1 and with slope r; as measured from the (x1; x2 )plane. Then the distribution is not a foliation, which may be seen as follows. Suppose (x1 (t); x2 (t)); 0 t T; describes a closed curve in the (x1 ; x2 )-plane. Since x is never vertical, there is a unique way to lift this curve to a curve x(t) = (x1 (t); x2(t); x3 (t)) in R 3 ; so that
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the tangent vector x0 (t) is always in the distribution x(t) : If were integrable, then the space curve would stay on the same surface of the foliation, and would therefore be a closed curve. However, to be speci c, suppose that the plane curve (x1(t); x2 (t)) describes the boundary @ ; in the positive sense, of = one-fourth of an annulus: in polar coordinates (r; ); is given by a < r < b; 0 < < =2: Then along each of the two straight sides 0; =2; the tangent vector lifts to a multiple of e1 = (x1 ; x2; 0), so x3 (t) remains constant. But along the quarter-circle r b; 0 < < =2; the tangent vector lifts to a multiple of e2 = (,x2 ; x1 ; r2), so x3 (t) increases by the slope times the length in the plane = b2 =2: Returning along the quarter-circle r a; as decreases from =2 to 0; x3(t) decreases by a2 =2: Thus, the change in x3(t) as t increases from 0 to T is (b2 , a2)=2; which is exactly twice the area of the quarter-annulus : In fact, for any domain in the (x1 ; x2 )-plane, the change in x3 (t) as (x1(t); x2 (t)) describes @ equals twice the area of : This may be seen by computing a form 0 so that = ker 0 : 0 = x2 dx1 , x1 dx2 + dx3 : Since x0(t) is in the distribution x(t) ; we get 0(x0 (t)) = 0; which means that x03 (t) = ,x2 (t)x01(t)+ x1 (t)x02 (t); hence the change x3 (T ) , x3 (0) in the height as (x1(t); x2 (t)) goes around @ equals the integral around @ of ,x2 dx1 + x1 dx2; which is twice the area of : The 1-form 0 is the standard contact form on R 3 ; and is the standard contact structure. More precisely, this is the rotationally symmetric version; the contact form x2 dx1 + dx3 is translationally invariant in two coordinate directions, and is also known as \the" standard contact form. In higher dimensions, the standard contact form on R 2n+1 is (2)
0 = dx0 +
n X k=1
(x2k dx2k,1 , x2k,1 dx2k );
which is invariant under the (n + 1)-dimensional group generated by rotation in the (x2k,1; x2k )-plane, 1 k n; plus translation along the x0-axis. A natural question is: when are two contact forms equivalent? The local version of this question has a surprisingly simple answer: Theorem 1.1. (Darboux): Let be a contact form on a neighborhood of x in R 2n+1 : Then on a smaller neighborhood of x, there is a di eomorphism into R 2n+1 such that is mapped to the standard contact form 0 :
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Darboux' Theorem may be interpreted as saying that the contact condition (1) is a very \soft" condition, as compared to the familiar partial di erential equations traditionally treated by geometric analysts. This softness is apparent from the recent work of Gromov, Eliashberg and others on noncompact manifolds, which showed, for example, that any noncompact, odd-dimensional manifold which has a hyperplane distribution with an almost-complex structure also carries a contact structure (see [8] and references therein.) 1.3. Global non-uniqueness: the Lutz Twist. Since contact structures are locally unique, it might seem reasonable to think that a topologically simple space like R 3 has only one contact structure up to a change of coordinates. However, there are subtle criteria which distinguish other contact structures on R3 from the standard 0: Recall the description in subsection 1.2 of basis vector elds e1 ; e2 for the standard contact structure on R 3 : e2 is orthogonal to the radial vector e1 ; and has slope r; which means that it makes an angle ' = arctan r with the (x1 ; x2 )-plane. As r ! 1; e2 becomes vertical, so ' ! =2: Instead, suppose that ' = '(r) increases beyond =2 to make one or more revolutions before slowly approaching arctan r +2m (m 2 Z) as r ! R < 1: Outside the cylinder r < R; the contact structure may be continued smoothly, to join up with the standard contact structure. This construction is known as the Lutz twist (see [9].) In terms of the contact form, in cylindrical coordinates (r; ; x3); 0 = dx3 , r2 d is replaced by = h0 (r)dx3 , h1(r) d for some functions h0 ; h1 : [0; 1) ! R with h01 h0 , h1 h00 > 0; and with h0(r) = 1; h1(r) = r2 for all r R: Then h1 and h0 are related to the angle ' by rh0(r) tan '(r) = h1(r): This new contact structure is overtwisted, that is, there is a topological disk D R 3 with jD nowhere zero along @D and j@D 0: In fact, let r0 be the rst value of r with '(r0) = : Then we may choose D = f(r; ; x3) : x3 = r02 , r2 ; 0 2; 0 r r0g: It may be shown that no such disk exists in R 3 with the standard contact structure. 1.4. Compact Manifolds. What about compact manifolds? The only known obstruction to the existence of an orientable contact structure on an oriented, odd-dimensional manifold M 2n+1 is the requirement that some hyperplane distribution on M should have an almost-complex structure; this can be written as a topological condition on M , that the even-dimensional Stiefel-Whitney classes w2i (certain natural cohomology classes with Z=2Z coecients) are in the image of cohomology with integer coecients. However, there are many manifolds which satisfy
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this condition but have not been shown to carry a contact structure. Speci cally, one would like to know whether there is a contact structure on the odd-dimensional torus T 2n+1. We shall assume for the rest of this paper that manifolds are compact, oriented and have no boundary. A readily visualized example is the interesting case of the torus T 2n+1, which is just the cube [,; ]2n+1 R 2n+1 after opposite faces have been identi ed. A contact form may be found on the three-torus T 3 as the rst case of a classical construction. Begin on the two-dimensional torus T 2; introduce local coordinates (q0 ; q1) on T 2 and then extend these coordinates to the 4-dimensional phase space, or cotangent bundle, T (T 2): Then a cotangent vector x at the point x = (q0 ; q1) has components (p0; p1), meaning that x = p0 dq0 + p1 dq1: (In certain applications, (q0; q1 ) are coordinates of position and (p0; p1 ) are components of the momentum vector.) Then ! = dp0 ^ dq0 + dp1 ^ dq1 is the natural symplectic form on phase space T (T 2): One notes that ! is the exterior derivative d ; where is the canonical 1-form p0 dq0 + p1 dq1 on phase space. When is restricted to the unit-sphere bundle M 3 := f(x; p) : x 2 T 2; p 2 Tx(T 2); jpj2 = 1g, it satis es the contact condition (1). Here jpj2 = p20 + p21: The veri cation of inequality (1) reduces to showing that p0 @ jpj2 =@p0 + p1 @ jpj2 =@p1 6= 0 on M . Meanwhile, on T 2; there is a global basis of tangent vector elds, which implies that the unit sphere bundle M 3 of T 2 is T 2 S 1 = T 3: In coordinates (q0; q1 ; ) for T 3 = (R =2Z)3 ; we have = cos dq0 + sin dq1 : This is the most natural construction for a contact structure on T 3: The construction above generalizes to higher dimensions. Let N be an oriented (n +1)-dimensional manifold, equipped with a Riemannian metric, and introduce local coordinates (q0 ; : : : ; qn; p0; : : : ; pn) for the cotangent bundle T N of N; where (q0 ; : : : ; qnP ) are local coordinates on n N and a cotangentPnvector is represented as i=0 pi dqi: Let be the canonical 1-form i=0 pidqi: When is restricted to the unit-sphere bundle M 2n+1 ; de ned as (q; p) : q 2 N; p 2 TqN; jpj2 = 1 ; it satis es the contact condition (1). That is, the unit sphere of the cotangent bundle of any manifold carries a natural contact structure. This is how contact structures arise naturally, on suitable energy surfaces in Hamiltonian systems. When one applies the same construction to N = T 3; n = 2; one nds a contact 1-form 1 on the 5-dimensional unit sphere bundle of T N: But the unit sphere bundle M 5 is now T 3 S 2; not T 5 : However, T 5 can still be given a contact structure, as was rst shown by Lutz [9]. Another way to nd a contact structure on T 5 is to apply the following result of Gromov (see [8] and [5], p. 456):
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Theorem 1.2. : If M2 is a branched covering of M1 ; branched along
a codimension-2 submanifold of M1 ; and if M1 has a contact form 1 whose restriction to also makes into a contact manifold, then M2 has a contact form close to the pullback of 1: In our case, T 2 may be written as a branched double cover F : T 2 ! S 2 of the sphere, branched simply over four points of S 2 ; which we may assume are the four equidistant points (1; 0; 0); (0; 1; 0) along the equator fp2 = 0g of S 2 R 3 : We construct a branched covering Fe : M2 ! M1 from M2 = T 5 = 3 T T 2 to M1 = T 3 S 2; by twisting F; as follows. Let q = (q0 ; q1; q2 ) be coordinates for T 3 = (R =2Z)3 ; and let p = (p0; p1 ; p2) be coordinates for S 2; where p02 +p12 +p22 = 1: For each q2 2 R =2Z; let (q2 ) : S 2 ! S 2 be the rotation in the (p0; p1)-plane by angle q2 ; leaving p2 xed. Then Fe : M2 ! M1 is de ned by Fe(q0 ; q1; q2 ; z) := (q0 ; q1; q2 ; (q2 )(F (z))) : Fe : M2 ! M1 is a branched covering, with branch locus = f(q; p) 2 T 3 S 2 : p0 = cos(q2 + k=2); p1 = sin(q2 + k=2); p2 = 0; k 2 Zg: has four connected components k ; k = 0; 1; 2; 3; each of which projects di eomorphically onto the the T 3 factor P2 of M1 : Write 1 for the canonical contact 1-form i=0 pi dqi on M1 ; viewed as the unit cotangent bundle of T 3: On each component k of ; we have 1 jk = cos(q2 + k=2) dq0 + sin(q2 + k=2) dq1: We compute ( 1 ^ d 1) jk = ,dq0 ^ dq1 ^ dq2 ; k = 0; 1; 2; 3; which shows that is a (disconnected) contact 3-manifold with contact form 1 j : We may now apply Theorem 1.2 to nd a contact form on M2 = T 5 which is close to the pullback of 1 : Thus, the 5-torus T 5 has a contact structure. The existence of a contact structure on the 7-torus, and on numerous higher-dimensional manifolds, was unclear until now.
2. The Heat Flow Recall that we are assuming that manifolds are compact, connected, oriented and have no boundary. In addition, we will assume that a Riemannian metric has been chosen. A property of parabolic PDEs familiar to analysts is the strong maximum principle: if the solution f (t; x) satis es at the initial time f (0; x) 0 but f (0; x) 6 0; then at time t > 0; f (t; x) will be positive everywhere. That is, heat ows instantaneously to warm a connected domain. This property makes parabolic methods ideal for the study of strict inequalities such as the contact inequality (1). The idea is to
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use a hands-on construction to make f (0; x) > 0 for x in an appropriate, possibly quite small, open set, while f (0; ) 0 everywhere, and then to replace f (0; x) with the strictly positive solution f (t; x) at some small positive time t: Since f (t; x) is close to f (0; x) in certain strong norms, other relevant conditions will be maintained. Altschuler in [1] considers any orientable compact 3-manifold M : he combines the Lutz twist with the strong maximum principle to construct a contact form on M: (The result was proved using entirely di erent methods in [10]; see also [4].) Altschuler's technique is to start with a foliation, or equivalently with a 1-form 1 satisfying 1 ^ d 1 0; and then to use the Lutz twist to construct a 1-form 2 satisfying 2 ^ d 2 > 0 on a certain open set U; with 2 = 1 near @U: The resulting 1-form 2 on all of M satis es 2 ^ d 2 0; such 1-forms have been called confoliations by Eliashberg and Thurston [4]. Altschuler then de nes a degenerate parabolic system of equations for a 1-form (t; x); 0 < t < "; x 2 M; and uses 2 as the initial condition at time t = 0: The system of PDEs is chosen so that the scalar quantity f (t; x) := ( ^ d ); which is initially nonnegative everywhere and strictly positive on U; becomes everywhere positive for small t > 0: One diculty is that the system of PDEs is degenerate parabolic, so that \heat" will ow reliably only in certain directions. Altschuler de nes the system of equations so that heat ows in directions tangent to ker 2 ; which is the original foliation ker 1 on the more troublesome set M nU; and ensures that the Lutz twist was carried out so that the open set U meets each leaf of ker 1: A nonlinear version of the system of equations Altschuler uses on a 3-manifold is @ = ( ^ df ) ; where f (t; x) = ( ^ d ): (3) @t Here, for a p-form on an oriented Riemannian (2n +1)-manifold, is a (2n+1,p)-form, the Hodge star of ; which depends linearly on and is de ned at each point so that for any oriented orthonormal coframe 0; : : : ; 2n of 1-forms, (p ^ : : : ^ 2n ) = 0 ^ : : : ^ p,1 : The system (3) appears quite complicated, but it may be dealt with successfully by the following trick. The real-valued function f (t; x) satis es a single degenerate parabolic PDE: @f = ( ^ d ( ^ df )) + h ^ df; d i: (4) @t Thus, the system (3) uncouples weakly, in the sense that appears in the PDE (4) only as a coecient. Once f (t; x) is determined, the equation (3) for (t; x) becomes a parameterized system of ODEs. Of
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course, the unknown 1-form also appears in the coecients of (4), so this version of Altschuler's method succeeds by requiring t to remain small, implying that (t; ) is close to the initial 1-form 2 : More generally, on a (2n + 1)-manifold, choose a (2n , 1)-form , and consider the system of equations @ = ( ^ df ) ; where f (t; x) := ( ^ d ): (5) @t The PDE satis ed by f is now @f @
(6) @t = ( ^ d ( ^ df )) + @t ^ d : Again, the system (5) uncouples weakly. Further, we have Proposition 2.1. : Equation (6) is a weakly parabolic PDE, a degenerate heat equation, where the right-hand side de nes a second-order partial di erential operator, which is strongly elliptic when restricted to the distribution H TM given by
H = (ker( ))? : Recall that for a 2-form on M; ker x := f v 2 TxM j (v; ) = 0 on TxM g. The coecients of the principal part of the PDE (6) at (t; x) are AT A; where the skew-symmetric matrix A represents x in coordinates which are orthonormal at x; the subspace Hx is spanned by the columns of AT A: In the nonlinear version of Altschuler's heat ow on a 3-manifold, as we have seen, one chooses = , the evolving 1-form itself. We would like to apply Proposition 2.1 in this case. At a given small time t > 0; the 1-form (t; ) never vanishes, so we may complete to a local orthonormal basis (1 ; 2 ; 3) with a nonvanishing scalar multiple of 1: We compute ker( ) = ker(1 ) = ker(2 ^ 3 ) = R e1 ; and therefore H = (ker( ))? = R e2 + R e3 = ker : In particular, for small positive t; the distribution H is close to ker 2: Thus, if 2 is a contact form on an open set U which is a neighborhood of some point on each leaf of the original foliation ker 1, then heat will ow out of U to warm each point of M: In general, one may show that Lemma 2.2. : If is locally decomposable as a product of 1-forms, then
H := (ker( ))? = ker :
THE HEAT-FLOW METHOD IN CONTACT GEOMETRY
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3. The Higher Lutz Twist For higher dimensions 2n + 1, in the recent paper of Altschuler and Wu [2], the degenerate parabolic system (5) is studied with the choice
= ^ (d )n,1: They show the existence of a smooth solution for all positive time, via a parabolic regularization. (More precisely, in [2] Altschuler and Wu consider a partly linearized degenerate parabolic system which is easier to analyze than equation (5), and slightly more complicated, but has consequences equivalent to those of equation (5).) The PDE (6) now becomes @f = n ( ^ d ( ^ df )) + h ^ df; (d )ni: (7) @t Thus, the system of equations uncouples in the same sense as in the 3-dimensional case n = 1 (compare equations (3) and (4).) Another part of their paper carries out a higher analogue of the Lutz twist for the ve-dimensional product case M 5 = N 3 F 2; using a contact structure on the 3-dimensional manifold N and its parallelizability. They are thereby able to prove that every product 5-manifold of this form carries a contact structure. Incidentally, this gives another construction of a contact form on the 5-torus T 5: Let us proceed in an analogous, but in applications rather di erent, fashion. Consider a (2n +1)-manifold M 2n+1 = N 2n,1 F 2 which is the product of a contact (2n , 1)-manifold (N; N ) and an oriented surface F . We shall write 1 for the 1-form on M = N F pulled back from N : For simplicity, assume that (N; N ) has a closed Reeb orbit . This means that 0 (s) 6= 0 and that d ( 0(s); v) = 0 for all parameter values s along the curve and for all vectors v 2 T (s) M: Then, according to an extension of Darboux' Theorem 1.1, in some neighborhood W of in N , there are multipolar coordinates (z; r1 ; 1 ; : : : ; rn,1; n,1); r12 + : : : + rn2,1 < R2 ; k 2 R mod 2; so that N is the standard contact form (2), which in these coordinates means that
N = dz +
n,1 X r 2 dk : k=1
k
In a small ball B F 2; let polar coordinates (rn; n ) be chosen, 0 rn < R; n 2 R mod 2: For some choice of real-valued functions hk (r1 ; : : : ; rn), 0 k n; de ne (8)
2 = h0 (r1; : : : ; rn) dz +
n X k=1
hk (r1; : : : ; rn) dk :
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Then 2 will satisfy the contact inequality (1) in U := W B provided h0 h1 : : : hn @1 h0 @1 h1 : : : @1 hn 1 (9) ... ... > 0; for ri 0 : r1 : : : rn ... @n h0 @n h1 : : : @n hn Here the operator @k denotes @=@rk : Inequality (9) is equivalent to the orientation preserving local di eomorphism property for the central projection of (h0 ; : : : ; hn) 2 R n+1 to the sphere S n: Observe that inequality (9) continues to hold when 2 is multiplied by a positive scalar function. Recall that we wish to carry out this higher Lutz twist on the open set U , but we need to construct 2 on all of M . Therefore it will be necessary for the coecients hk (r1; : : : ; rn) to satisfy boundary conditions on @U , so that the extension of 2 to all of M by de ning 2 = 1 on M nU will be smooth. However, only the oriented contact structure is important to us, which means that 2 only needs to be de ned modulo a (nonconstant) positive multiple. Speci cally, there needs to hold on the boundary hk (r1 ; : : : ; rn) = rk2 h0 (r1; : : : ; rn); 1 k n , 1; and hn(r1; : : : ; rn) = 0; as well as inequality (9) in the interior of U: This requires us to nd a mapping from the sector V := f(r1; : : : ; rn) 2 (0; 1)n : r12 + : : : + rn2 < R2g to the sphere S n which is a di eomorphism of V with an open subset of S n; having the following boundary values on @V . On the curved part of the boundary fr12 + : : : + rn2 = R2g; we require hk = rk2 h0 ; 1 k n , 1; and hn = 0: For 1 k n; on the face frk = 0g; we require hk = 0; 1 k n: In the ve-dimensional case n = 2; this may be done using a conformal mapping from the quarter-disk V to the hemisphere of S 2 with a slit from an interior point to the equator removed. The boundary of the quarter-disk covers the slit twice and the equator once. For the general case n 2, another more hands-on construction of the map from V into the hemisphere of S n may be carried out. Closed Reeb orbits may be rare for a given contact manifold (N; N ), but the above procedure may be modi ed appropriately. A covering argument may then be used to arrange disjoint open sets of M of the above form so that their projections from M = N F to N cover all of N: For small time t > 0; the solution (t; ) will be close to the initial value 2: On the complement of the union of the sets U where the higher Lutz twist has been carried out, we have 2 = 1: Write 1 = 1 ^ (d 1)n,1: Since 1 is the pullback of the contact form N ; we see that 1 is the pullback of a volume form on N; and thus is decomposable as a product of 1-forms. It follows from
THE HEAT-FLOW METHOD IN CONTACT GEOMETRY
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Lemma 2.2 that on the complement of the sets U; the distribution H1 of elliptic directions for the system (5), with replaced by 1 ; equals ker 1 ; which is the tangent plane TF to fyg F 2 in TM = TN TF: Therefore, for small time t > 0; the distribution H is close to the foliation TF: It follows by Proposition 2.1, for small time t > 0; that heat ows out of the union of open sets U along directions arbitrarily close to TF to warm all of M = N F: Numerous points omitted here, in part rather technical, will be treated in [7] to prove Theorem 3.1. : If N 2n,1 is a compact contact manifold and F 2 is a compact oriented surface, then M 2n+1 = N F has a contact 1-form, which is C 2 -close to a 1-form 2 obtained from the contact form of N by means of the higher Lutz twist. By induction on n = 2; 3; : : : ; with N = T 2n,1 and F = T 2; we deduce Corollary 3.2. : Any odd-dimensional torus T 2n+1 carries a contact structure. References [1] S.J. Altschuler, A geometric heat ow for one-forms on three dimensional manifolds, Illinois J. Math. 39 (1995), 98{118. [2] S.J. Altschuler and L.F. Wu, On deforming confoliations, preprint 1999. [3] S.-S. Chern, Pseudo-groupes continus in nis in Geometrie Di erentielle, Colloques Internat. CNRS, Strasbourg 1953. [4] Y. Eliashberg and W. Thurston, Confoliations, Univ. Lecture Ser. 13, Amer. Math. Soc., Providence, 1998. [5] H. Geiges, Constructions of contact manifolds, Proc. Camb. Phil. Soc. 121 (1997), 455{464. [6] H. Geiges, Contact topology in dimension greater than three, to appear in Proc. Eur. Congr. Math. (Barcelona, 2000). Progress in Math., Birkhauser, Basel. [7] H. Geiges, R. Gulliver and M. Schwarz, Heat ow and contact structures on product manifolds, in preparation. [8] M. Gromov, Partial Di erential Relations, Ergebnisse vol. 9, Springer, Berlin 1986. [9] R. Lutz, Sur la geometrie des structures de contact invariants, Ann. Inst. Fourier (Grenoble) 29 (1979), 283{306. [10] J. Martinet, Formes de contact sur les varietes de dimension 3, Lecture Notes in Math. 209 (1971), 142{163: Springer, Berlin. School of Mathematics, University of Minnesota
E-mail address :
[email protected]
MANIPULATING THE ELECTRON CURRENT THROUGH A SPLITTING M. HARMER, A. MIKHAILOVA AND B. S. PAVLOV Abstract. The description of electron current through a splitting
is a mathematical problem of electron transport in quantum networks [5, 1]. For quantum networks constructed on the interface of narrow-gap semiconductors [29, 2] the relevant scattering problem for the multi-dimensional Schodinger equation may be substituted by the corresponding problem on a one-dimensional linear graph with proper selfadjoint boundary conditions at the nodes [11, 10, 25, 24, 16, 19, 4, 28, 20, 18, 6, 5, 1]. However, realistic boundary conditions for splittings have not yet been derived. Here we consider some compact domain attached to a few semiin nite lines as a model for a quantum network. An asymptotic formula for the scattering matrix for this object is derived in terms of the properties of the compact domain. This allows us to propose designs for devices for manipulating quantum current through a splitting [3, 15, 22, 9, 21].
Introduction: current manipulation in the resonance case
In this paper we discuss the scattering problem on a compact domain with a few semi-in nite wires attached. This is motivated by the design of quantum electronic devices for triadic logic. In the papers [3, 15] a special design of the one-dimensional graph which permits manipulation of the current through an elementary ring-like splitting is suggested. This permits, in principle, manipulation of quantum current in the resonance case to form a quantum switch. Another device for manipulating quantum current through splittings is discussed in [22, 9]. In [21] the special design of the splitting formed as a circular domain with four one-dimensional wires attached is used to produce a triadic relay. Date : 20 September 2000 Revised 20 January 2001.
118
MANIPULATING THE ELECTRON CURRENT THROUGH A SPLITTING 119
In order to illustrate the basic principle of operation consider the selfadjoint Schrodinger operator L , + q(x) @ = 0: @n @
on some compact domain . In this paper we will only consider the case R 3 [22] (for other cases see also [3, 15, 21]). Roughly speaking the solution of the Cauchy problem @ = L i@t (1) (x; 0) = 0(x) is given in terms of eigenfunctions 'n X (x; t) = neint 'n(x): n
Picking a speci c mode '0 with energy 0 we suppose that '0 disappears on some subset l0 . Connecting `thin channels' at various
λ0
λ0
Ω
l0
gure 1. Resonance switch points on the boundary of and introducing an excitation of energy 0 along the channels we can hope to create a switching e ect. Essentialy this is achieved by varying q(x) so that l0 \ @ coincides with the connection point of a `thin channel'. Implicit in our construction is the assumption that the energy of the electrons in the device is equal to some resonance eigenvalue of the Schrodinger operator on . We refer to this as the resonance case1. This has interesting implications when we consider the e ect of decreasing the length scale|or equivalently scaling up the energy|viz. the e ect of non-zero temperature becomes negligable for suciently small length scales, see [14]. 1
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M. HARMER, A. MIKHAILOVA AND B. S. PAVLOV
Another assumption which we have made above is that 0 is a simple eigenvalue of L. We will show that the case of multiple eigenvalues is a simple generalisation of the case for simple eigenvalues, see [3, 15]. In the rst section we give a brief description of the connection of the thin channels (here they are modelled by one-dimensional semi-lines) to the compact domain, for more details see [22]. In the second section we derive an asymptotic formula for the scattering matrix in terms of the eigenfunctions on the compact domain. In the last section we brie y discuss some simple models of a quantum switch constructed on the basis of this asymptotic formula. 1. Connection of compact domain to thin channels As we mentioned above the thin channels are modelled by onedimensional semi-lines. This is justi ed by an appropriate choice of materials (narrow-band semiconductors) and energies, see [29, 2, 21]. We assume that these channels are attached at the points fa1 ; a2 ; :::; aN g @ (perturbation of the operator L at inner points faN +1; aN +2 ; : : : ; aN +M g may be considered using the same techniques as for fa1; a2; :::; aN g [2, 3, 21] although we do not consider this here). We refer to L, de ned above, as the unperturbed Schrodinger operator. L is restricted to the symmetric operator L0 de ned on the class D0 of smooth functions with Neumann boundary conditions which vanish near the points a1 ; a2; : : : ; aN . The de ciency subspaces, Ni, of the restricted symmetric operator L0 , [L?0 i] esi = 0 for complex values of the spectral parameter coincide with Greens functions G(x; as) of L which are elements of L2 ( ) but do not belong to the Sobolev class W21 ( ). In the case when is a compact one-dimensional manifold (a compact graph) these Greens functions are continuous and can be written in terms of a convergent spectral series [3]. However, when R 2 ; R 3 the de ciency elements will have singularities and we must use an iterated Hilbert identity to regularise the values of the Greens function at the poles. It is well known, for R 3 , that the Greens function admitts the representation inside
(2)
G0 (x; y) =
p
ei jx,yj + g(x; y; ) 4jx , yj
MANIPULATING THE ELECTRON CURRENT THROUGH A SPLITTING 121
where g(x; y; ) is continuous. The potential-theory approach gives the asymptotics of the Green function near the boundary point as 2 @
(3) G0 (x; as) 2jx1, a j + Ls(x) + Bs(x; ); s where Ls is a logarithmic term depending only on @ and Bs is a bounded term containing spectral information [8]. In order to choose regularised boundary values we use the following lemma [22] (here we assume L > ,1 is semi-bounded from below): Lemma 1. For any regular point N+from the complement of the specM trum (L) of L and any a 2 fas gs=1 the following representation is true: G(x; a) = G,1(x; a) + ( + 1)G,1 G (x; a); where the second addend is a continuous function of x and the spectral series of it in terms of eigenfunctions 'l of the nonperturbed operator
L
( + 1)G,1 G(x; a) = ( + 1)
X 'l(x)'l(a) l
(l + 1)(l , )
is absolutely and uniformly convergent in . The proof of this lemma is based on the classical Mercer theorem along with the Hilbert identity [22]. It is well known that the domain of L?0 can be written as the direct sum (4) D0? = D0 + Ni + N,i so for any u 2 D0? X X u = u0 + A+s Gi(x; as) + A,s G,i(x; as):
We de ne
u 2 D?
s
s
in terms of the coordinates As A+s +"A,s ; 0
Bs xlim u(x) , !as
X t
#
At
the singular and regular amplitudes respectively since it is clear from the above lemma that As is the coecient of the singular part and Bs the coecient of the regular part of u 2 D0?. The boundary form of L?0 may be written in terms of As Bs as a hermitian symplectic form X (5) hL?0u; vi , hu; L?0vi = BsuAvs , AusBsv :
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M. HARMER, A. MIKHAILOVA AND B. S. PAVLOV
1.1. Self-adjoint extensions. We recall that to each boundary point as, s = 1; : : : ; N , there is attached a semi-in nite ray. On the s-th ray we de ne the symmetric operator d2 + q (x ); ls;0 = , dx s s s on functions which vanish at xs = 0 (which is identi ed with as 2 ). Let us consider the symmetric operator L0 l1;0 l2;0 ::: lN;0. The connection between the compact domain and the rays is given by (a particular) self-adjoint extension of this operator. The boundary form ? is easily seen to be of the adjoint L?0 l1?;0 l2?;0 ::: lN; 0 (6)
N , X s
N , X u0s(0)vs(0) , us(0)vs0 (0) :
BsuAvs , AusBsv +
s
It is well known that the self-adjoint extensions of L0 l1;0 l2;0 ::: lN;0 correspond to Lagrange planes in the Hermitian symplectic space of boundary values equipped with the above boundary form [26]. In general, if A, B are (vectors of) boundary values for some symmetric operator then any self-adjoint extension can be described by i (U , I)A + 1 (U + I)B = 0 2 2 for some unitary matrix U [14, 13]. We choose the particular family of self-adjoint extensions which correspond to the following boundary conditions at the N points of contact of the rays A 0 B s s (7) us(0) = 0 u0s(0) ; for s = 1; : : : ; N and > 0. The resulting self-adjoint extension we denote by L . The parameter is a measure of the strength of the connection between the rays and the compact domain|in the limit ! 0 the resolvent of L converges uniformly to the resolvent of L on each compact subset of the resolvent set of L [22]. 2. Asymptotics of the scattering matrix For the remainder we assume that the potential on the rays qs(xs) 0 is zero. We use the ansatz (8) us = fs(xs; ,k)s1 + fs(xs; k)Ss1;
MANIPULATING THE ELECTRON CURRENT THROUGH A SPLITTING 123
for the scattered wave generated by the incoming wave from the ray attached to the point a1. Here fs(xs; k) are the Jost solutions [27], in this case (qs(xs) 0) just the exponentials fs(xs; k) = eikxs ; and = k2 is the spectral parameter. From the boundary conditions (7) we get N equations As = fs0 (0; ,ik)s1 + fs0 (0; ik)Ss1 (9) Bs = fs(0; ,ik)s1 + fs(0; ik)Ss1: Inside the eigenfunction u(x; k) may be written as a sum of Greens functions at the spectral parameter = k2
u(x; k) =
N X s
CsG (x; as):
Using the Cayley transform between the spectral points i and one gets a relationship between these Greens functions and the de ciency elements (as de ned above) so that [22] I + L lim [G(x; as; ) ,
It follows that for the scattering wave the symplectic variables are related by As = Cs X Bs = gs()Cs + Ct G(as; at ); t6=s
that is B = QA where (11)
0 g1() G(a1; a2) G(a1; aN ) 1 B G(a2 ; a1) g2() G(a2 ; aN ) C B CA : Q() = @ .. .. .. .
.
.
G(aN ; a1) Putting this into (9) we can solve for the scattering matrix to get + ik 2Q : (12) S = , II , ik 2Q gN ()
124
M. HARMER, A. MIKHAILOVA AND B. S. PAVLOV
Let us choose an eigenvalue 0 of the unperturbed operator L on . We suppose that 0 has a p-dimensional eigenspace, which we denote R0, with orthonormal basis f'0;igp. The following important technical statement close to Lemma 1 above is true [22]: Theorem 1. The elements of the Q-matrix have the following asymptotics at the spectral point 0 :
Qst ()
p X '0;i(as)'0;i(at )
0 ,
i=1
+ Q0(as; at ; );
where Q0 (as; at; ) is a continuous function at the point = 0.
We will use this result to prove an asymptotic formula for the scattering matrix in the limit of weak connection between the compact domain and the rays. Consider the mapping P : L2( ) ! C N which gives the vector of values of a function in L2 ( ) at the nodes of each of the N rays. To distinguish between functions and elements of C N we use the notation
P( ) = j i 2 C N ; and we denote
R0 P (R0 ):
Proposition 1. It is possible to choose an orthonormal basis f 0;igp for R0 which forms an orthogonal, but not necessarily normalised, basis for R0 under P . Proof: Given some orthonormal basis f'0;i gp for R0 we see that 0;i =
p X j =1
Uij '0;j
is also an orthonormal basis where U 2 U(p). The inner product of the image under P
h 0;ij 0;j i =
p X
r;s=1
Uir h'0;r j'0;siUjs
shows that nding an orthogonal basis for R0 amounts to nding the unitary matrix U which diagonalises Ars = h'0;r j'0;si. 2
MANIPULATING THE ELECTRON CURRENT THROUGH A SPLITTING 125
This allows us to write Q in `diagonal' form Q = 1, [j 0;1ih 0;1j + + j 0;mih 0;mj] 0 + Q0 () (13) = Dl + Q0 () 0 , where m p is the dimension of R0 . Theorem 2. If 0 is an eigenvalue of L then for vanishing coupling 0 the scattering matrix of L has the form X S (0) = ,I + 2P0 , 2 (ik0 2P0?Q0 P0?)s s=1
(14) = ,I + 2P0 + O( 2) where P0 is the orthogonal projection onto R0 . Proof: Using equation (13), ik 2D ,1 ik 2D 0 0 2 2 S () = , I + , + ik Q0 I , , , ik Q0 : 0 0 2 D0 Since D0 = D0?, the matrix E0 = I , ik 0 , is invertable. Consequently the denominator can be written ik 2D ,1 ,1 0 2 , ik Q = [ I , ik 2 Q0 E0,1 ]E0 I, 0 0 , = E0,1 I , ik 2Q0 E0,1 ,1 : Again the matrix I , ik 2Q0 E0,1 has an inverse for 0 since Q0 = Q?0. This gives the following expression for the scattering matrix S () = , E0?E0,1 + ik 2Q0 E0,1 I , ik 2Q0 E0,1 ,1 X(ik 2Q E ,1)s: = , E0?E0,1 + ik 2Q0 E0,1 0 0 s=0 p Denoting i h 0;ij 0;ii and diagonalising we can write, ik 22 ,1 ik 22
E0,1 = diag 1 , , 1 ; : : : ; 1 , , m ; 1; : : : ; 1 0 0 , , 0 0 = diag , , ik 22 ; : : : ; , , ik 2 2 ; 1; : : : ; 1 :
Therefore (15)
0
1
0
,1 = P ? = I , P : 0 0
lim E !0 0
m
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M. HARMER, A. MIKHAILOVA AND B. S. PAVLOV
Furthermore 22 22 , + ik , + ik 0 0 m 1 ? ,1 E0 E0 = diag , , ik 22 ; : : : ; , , ik 22 ; 1; : : : ; 1 0
1
0
m
which gives us the limit (16) lim E ?E ,1 = P0? , P0 = I , 2P0: !0 0 0 From these limits we get X S (0) = , I , 2P0 + ik0 2Q0 P0? (ik0 2Q0 P0?)s = ,I + 2P0 , 2
X s=1
s=0
(ik0 2P0?Q0P0?)s
= ,I + 2P0 + O( 2):
2
This formula appears to imply that there may be non-zero transmission in the case of zero connection between the rays. Actually the transmission coecients are not continuous with respect to uniformly in [3, 22]. The physically signi cant parameters of the system are obtained by averaging as functions of over the Fermi distribution so that there is no transmission for = 0. Corollary 1. If 0 is an eigenvalue of L such that P0 = I then the above formula is independent of , ie. S (0) = I Consequently, when we have pure re ection at an eigenvalue of the unperturbed operator, we have pure re ection regardless of the strength of the interaction between the rays and the compact domain. 3. Simple models In [21] the authors discuss the case where is the unit disc in R2 and there are four one-dimensional wires attached at the points ' = 0; ; =3. The dynamics on is given, using polar coordinates (r; ), by the dimensionless Schrodinger equation (17) , + [V0 + "rcos()] = on the domain with Neumann boundary conditions at the boundary: @ = 0: @n r=1 The dimensionless magnitude " of the governing eld is choosen so that the eigenfunction corresponding to the second smallest eigenvalue has only two zeroes on the boundary of the unit circle which divide the
MANIPULATING THE ELECTRON CURRENT THROUGH A SPLITTING 127
circumference in the ratio 2 : 1. It is then easy to see that by rotating the potential V one may redirect the quantum current from the wire attached to the point ' = 0 to any other wire with all of the other wires blocked [21]. The analysis in this case is similar to the analysis given above except there is now only a logarithmic singularity in the Greens function and the Krein formula for in nite de ciency indices [17, 23] and in nitedimensional Rouchet theorem [12] play a central r^ole. A large amount of the calculation was done using Mathematica. In [15], using the above asymptotic formula for the scattering matrix to choose appropriate parameters, the author discusses the case where
is simply a one-dimensional ring and there is an angle of =2 between the rays|see gure 2. a). By applying a uniform eld to the ring, q = 0 for the open state and q = ,3 for the closed state, it is easy to see that a switching e ect is produced where the Fermi energy is assumed to correspond to the smallest eigenvalue of the unperturbed operator on the ring, ie. 0 = 1. See also [7] where a similar construction is considered. For the purposes of comparison we also consider a device|see gure
qo= 0 q c =-3
a) Interference switch
qo= 0 q c= 3
gure 2.
b) Potential barrier switch
2.b), the angle between the rays is now |with similar parameters where now we switch the current by raising a potential barrier, q = 0 for the open state and q = 3 for the closed state, instead of using interference e ects. It is easy to see that, unlike the rst case, the eciency of such a switch will be limited by tunneling. Using Maple we numericaly integrate over the Fermi distribution to
128
M. HARMER, A. MIKHAILOVA AND B. S. PAVLOV
produce plots of the averaged conductance ^c ^o in the closed and open states respectively|see gure 3. a) and b) which show the plots for the `interference switch' and the `potential barrier switch' respectively. Here is a temperature parameter in the model. 4 3.8 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2
= 0:0225 = 0:045 = 0:09 = 0:18 0
0.2
0.4
|β|
0.6
0.8
1
0.8
1
gure 3. a) log10(^c=^o ) versus
6.5 6 5.5
= 0:0225
5 4.5
= 0:18
4 3.5 0
0.2
0.4
|β|
0.6
gure 3. b) log10 (^c=^o ) versus
MANIPULATING THE ELECTRON CURRENT THROUGH A SPLITTING 129
It appears that the open state|possibly due to tunneling e ects| is more dicult to achieve. This also appears to explain why, in the limit of small , the properties of the switches improve: weak coupling between the ring and rays improves the open state of the switches. On the other hand, in the limit ! 1, the ratio ^c=^o for the second example rapidly decreases to a bound due to tunneling which may be calculated from the transmission coecient ^ c lim 4:57 103; !0 ^o =1 see gure 3 b). The rst switch does not have this bound and consequently for suciently low temperature or small radius (see rst footnote) we conjecture that it will have better properties. References [1] V. M. Adamyan. Oper. Theory: Adv. Appl., 59:1{10, 1992. [2] I. Antoniou, B. S. Pavlov, and A. M. Yafyasov. Quantum electronic devices based on metal-dielectric transition in low-dimensional quantum structures. In D. S. Bridges, C. Calude, J. Gibbons, S. Reeves, and I. Witten, editors, Combinatorics, Complexity, Logic (Proceedings of DMTCS '96), Singapore, 1996. Springer. [3] V. Bogevolnov, A. Mikhailova, B. S. Pavlov, and A. Yafyasov. About scattering on the ring. Department of Maths Report series 413 ISSN 1173-0889, University of Auckland, New Zealand, 1999. To be published in `Operator Theory: Advances and Applications', (IWOTA 98, dedicated to I. Gohberg). [4] R. Carlson. Inverse eigenvalue problems on directed graphs. Trans. Amer. Math. Soc., 351(10):4069{4088, 1999. [5] P. Exner. Contact interactions on graph superlattices. J. Phys. A: Math. Gen., 29:87{102, 1996. [6] P. Exner and P. Seba. Free quantum motion on a branching graph. Rep. Math. Phys, 28:7{26, 1989. [7] P. Exner, P. Seba, and P. Stovicek. Quantum interference on graphs controlled by an external electrical eld. J. Phys. A: Math. Gen., 21:4009{4019, 1988. [8] M. D. Faddeev. Asymptotic behaviour of the green's function for the neumann problem at the boundary point. J. Sov. Math., 30:2336{2340, 1985. [9] B. Gejler, B. S. Pavlov, and I. Popov. Possible construction of a quantum triple logic device. New technologies for narrow-gap semiconductors, progress report (july 1, 1998{june 30 1999), ESPRIT NTCONGS, 1999. ESPRIT project No. 28890. [10] N. I. Gerasimenko. The inverse scattering problem on a noncompact graph. Teoret. Mat. Fiz., 75:187{200, 1988. [11] N. I. Gerasimenko and B. S. Pavlov. Scattering problems on compact graphs. Teoret. Mat. Fiz., 74:230{240, 1988. [12] I. S. Gohberg and E. I. Sigal. Operator extension of the theorem about logarithmic residue and rouchet theorem. Mat. Sbornik., 84:607, 1971.
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[13] M. Harmer. Hermitian symplectic geometry and the Schrodinger operator on the graph. Department of Maths Report series 444, University of Auckland, New Zealand, 2000. [14] M. Harmer. The Matrix Schrodinger Operator and Schrodinger Operator on Graphs. PhD thesis, University of Auckland, 2000. [15] M. Harmer. Scattering on the annulus. Department of Maths Report series 445, University of Auckland, New Zealand, 2000. [16] V. Kostrykin and R. Schrader. Kirchho 's rule for quantum wires. J. Phys A: Math. Gen., 32:595{630, 1999. [17] M. Krein. Comptes Rendue (Doklady) Acad. Sci. USSR (N.S.), 52:651{654, 1946. [18] P. B. Kurasov. J. Math. Anal. Appl., 201:297, 1996. [19] Yu. B. Melnikov. Scattering on graphs as a quantum few-body problem. preprint IPRT #09-93, 1993. [20] Yu. B. Melnikov and B. S. Pavlov. Two-body scattering on a graph and application to simple nanoelectronic devices. J. Math. Phys, 36:2813{2825, 1995. [21] A. Mikhailova and B. S. Pavlov. Quantum domain as a triadic relay. Department of Maths Report series 439, University of Auckland, New Zealand, 2000. [22] A. Mikhailova, B. S. Pavlov, I. Popov, T. Rudakova, and A. M. Yafyasov. Scattering on a compact domain with few semi-in nite wires attached: resonance case. Department of Maths Report series 420, University of Auckland, New Zealand, 1999. submitted to Mathematishe Nachrichten. [23] M. Naimark. Bull. Acad. Sci. USSR Ser. Math., 4:53{104, 1940. [24] S. P. Novikov. Schrodinger operators on graphs and symplectic geometry. In E. Bierstone, B. Khesin, A. Khovanskii, and J. Marsden, editors, The Arnol'dfest, 1998. Proceedings of the Fields Institute Conference in Honour of the 60-th Birthday of Vladimir I. Arnol'd, to appear in the Fields Institute Communications Series. [25] S. P. Novikov and I. A. Dynnikov. Discrete spectral symmetries of lowdimensional di erential operators and di erence operators on regular lattices and two-dimensional manifolds. Uspekhi Math. Nauk-Russian. Math. Surveys, 52(5):1057{1116, 1997. [26] B. S. Pavlov. The theory of extensions and explicitly solvable models. Uspekhi Math. Nauk-Russian. Math. Surveys, 42(6):127{168, 1987. [27] M. Reed and B. Simon. Methods of Modern Mathematical Physics. Academic Press, New York, 1972. [28] P. M. Soardi. Potential theory of in nite networks. Lecture Notes in Mathematics, Vol. 1590. Springer-Verlag, Berlin, 1994. [29] A. M. Yafyasov, V. B. Bogevolnov, and T. V. Rudakova. Quantum interferentional electronic transistor (qiet). theoretical analysis of electronic properties for low-dimensional systems. (Progress Report 1995), Preprint IPRT N 99-95.
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MANIPULATING THE ELECTRON CURRENT THROUGH A SPLITTING 131 Laboratory of the Theory of Complex Systems-Institute of Physics, St Petersburg University, Ulyanovskaja 1, Petrodvorets 198504, St Petersburg, Russia
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WHAT'S NEW FOR THE BELTRAMI EQUATION ? TADEUSZ IWANIEC AND GAVEN MARTIN Abstract. The existence theorem for quasiconformal mappings
has found a central r^ole in a diverse variety of areas such as holomorphic dynamics, Teichmuller theory, low dimensional topology and geometry, and the planar theory of PDEs. Anticipating the needs of future researchers we give an account of the \state of the art" as it pertains to this theorem, that is to the existence and uniqueness theory of the planar Beltrami equation, and various properties of the solutions to this equation.
This paper surveys the recent work of the authors' paper [14] and parts of our monograph [15]. Readers interested in more details, and in particular rather greater discussion of related work by others, should consult those works. In what follows we use fairly standard notation, in particular B denotes a disk, usually the unit disk, in the complex plane C and C = C [ f1g is the Riemann sphere. The Beltrami equation has a long history. Gauss rst studied the equation, with smooth coecients, in the 1820's while investigating the problem of existence of isothermal coordinates on a given surface. The complex Beltrami equation was intensively studied by Morrey in the late 1930's, and he established the existence of homeomorphic solutions for measurable [18]. It took another 20 years before Bers recognised that homeomorphic solutions are quasiconformal mappings. Since then these ideas have found diverse applications in a variety of areas such as holomorphic dynamics, Teichmuller theory, low dimensional topology and geometry, and the planar theory of PDEs. 1. PDEs There is a strong interaction between linear and non-linear elliptic systems in the plane and quasiconformal mappings. The most general rst order linear (over R) elliptic system takes the form @ f = 1 @f + 2 @f Research supported in parts by grants from the U.S. National Science Foundation and the N.Z. Marsden Fund. 132
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where 1 and 2 are complex valued measurable functions such that , 1 < 1 a:e:
j1(z)j + j2(z)j K K +1 The complex Beltrami equation is simply that equation which is linear over the complex numbers. (1) @f (z) = (z) @f (z) Classically one assumes ellipticity bounds: ,1 <1 (2) kk1 = K K +1 When = 0 we have the Cauchy{Riemann system. These sets of equations are particular cases of the genuine non-linear rst ord er system (3) @f = H (z; @f )M where H : C ! C is Lipschitz in the second variable, , 1 j , j; H (z; 0) 0 jH (z; ) , H (z; )j K K +1 A feature of (3) is that the di erence of two solutions need not solve the same equation but it is K {quasiregular (the term used to describe non-injective quasiconformal functions). Thus quasiconformal maps are a central tool used to establish a priori estimates needed for the existence and uniqueness. 2. Classical Regularity Theory Typically one seeks solutions to the Beltrami equation in the Sobolev 1;2 ( ). However, the solutions to this equation have the folspace Wloc lowing striking regularity result nally established in complete form by K. Astala (the Area Distortion Theorem) [2]. Theorem 2.1. Let be a measurable function de ned in with k1;qk1 = k < 1. Let f be any solution to the Beltrami equation with f 2 Wloc ( ), 1;p( ) for all p < 1 + 1 . q > 1 + k. Then f 2 Wloc k 1;1+k ( ) not in any higher Moreover, there may be solutions in Wloc 1;2 ( ) not in W 1+1=k ( ). Sobolev space, and there may be solutions in Wloc loc Notice the indices p and q in the above result form a Holder conjugate pair. The Neumann iteration procedure based on invertibility of the Beltrami operator I , S , (S the complex Hilbert transform) yields a representation formula ( rst found by Bojarski [5]) and existence.
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Theorem 2.2. Let be a measurable function in C and suppose kk1 < 1. Then there is a homeomorphic solution g : ! C to the Beltrami equation. 1;2 ( ; C ) solution f is of the form Moreover every Wloc f (z) = (g(z)) where : g ( ) ! C is a holomorphic function. This last fact is known as \factorisation".
3. A Fundamental Example This example re ects what is possible in the degenerate elliptic setting, and shows why it is necessary to use the Orlicz-Sobolev spaces in order to discuss the ne properties of solutions. Theorem 3.1. Let A : [1; 1) ! [1; 1) be a smooth increasing function with A(1) = 1 and such that Z 1 A(s) ds < 1: (4) s2 1 Then there is a Beltrami coecient compactly supported in the unit disk, j(z)j < 1, with the following properties: j(z)j 1. The ellipticity bound K (z) = 1+ 1,j(z)j satis es (5)
Z
B
eA(K (z)) dz < 1
W 1;1(B){solution
2. Every loc to the Beltrami equation fz = fz a:e: B (6) continuous at the origin is constant. 3. There is a bounded solution Tw = f (z) to the Beltrami equation in the space weak-W 1;2(B) 1q<2 W 1;q (B) which homeomorphically maps the punctured disk onto the annulus 1 < jwj < R. A few remarks. 1;1 (B) is really the weakest space in which one can begin to First, Wloc discuss what it means to be a solution. Secondly, the integrability condition (4) implies that A is sublinear. As examples the function (7) A( ) = (log )1+
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satis es (4) for all > 0, but not for = 0. More generally, if we put log1 = log ; logn+1 = log(logn ) we have the iterated logarithm functions. Then for each n > 0 the function e log1(1 + ) log2(e + ) log3(ee + ) (logn(ee + ))1+ again satis es (4) for all > 0, but not for = 0. Finally, (3) is no accident. We can prove that if eA(K ) 2 Lploc(B) for some p 0 and certain types of A (and in particular the log log : : : examples), with Z
B
eA(K (z)) dz = 1
then there is a homeomorphic (and hence continuous) solution in W 1;p(B) for all p < 2. In fact the solution lies in an Orlicz{Sobolev class just 1;2 below Wloc 4. Mappings of Finite Distortion We next give a general de nition of the mappings which mostly occur. Roughly, solutions to a Beltrami equation with ellipticity bounds which are pointwise nite will be mappings of nite distortion as soon as they are ACL{absolutely continuous on lines.
De nition A mapping f : ! C is said to have nite distortion if: 1;1 ( ), 1. f 2 Wloc
2. The Jacobian determinant, J (z; f ) = det Df (z), of f is locally integrable and does not change sign in
3. There is a measurable function K = K(z) 1, nite almost everywhere, such that f satis es the distortion inequality (8) jDf (z)j2 K(z) jJ (z; f )j a:e:
Notice that the hypotheses are not sucient to guarantee that f 2 1;2 ( ) unless the distortion function K is bounded. Nor do they Wloc imply that the Jacobian does not vanish on a set of positive measure. The motivational philosophy behind the condition that the distortion function is exponentially integrable is now clear. We wish to exploit the BMO , H1 duality (and even more re ned versions of this) to achieve uniform estimates on approximating sequences of solutions.
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5. Maximum Principle and Continuity One of the rst tasks is to establish a maximum type principle and modulus of continuity estimates. A continuous function u : ! R de ned in a domain is monotone if oscB u osc@B u for every ball B . This de nition in fact goes back to Lebesgue in 1907 where he rst showed the relevance of the notion of monotonicity to elliptic PDEs in the plane. In order to handle very weak solutions of di erential inequalities, such as the distortion inequality, we need to extend this concept, dropping the assumption of continuity, and to the setting of Orlicz{Sobolev spaces. De nition. A real valued function u 2 W 1;P ( ) is said to be weakly monotone if for every ball B and all constants m M such that (9) jM , uj , ju , mj + 2u , m , M 2 W01;P (B) we have (10) m u(x) M for almost every x 2 B. For continuous functions (9) holds if and only if m u(x) M on @ B. Then (10) says we want the same condition in B, that is the maximum and minimum principles. Here, and in what follows we assume, unless otherwise stated, that the Orlicz function P satis es Z 1 P (t) dt (11) t3 = 1 1 and that the function t 7! t 58 is convex. For example Orlicz functions of the form
2 e log1 (1 + ) log2 (e + ) log3(ee + ) (logn ee + ) are of this form. 1;P ( ) consists of functions which, toThe Orlicz{Sobolev space Wloc gether with their rst derivatives, lie in the space LP ( ). Thus in the example given above we are looking in Zygmund type spaces just below 1;2 ( ). Wloc P ( ) =
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Lemma 5.1. Let be a bounded domain and suppose that u 2 W 1;P ( )\ C ( ) is weakly monotone. Then (12) min u u(x) max u @
@
for every x 2 . The paper [17] by Manfredi should be mentioned as the beginning of the systematic study of weakly monotone functions. We now recall a fundamental monotonicity result in the Orlicz{ Sobolev classes.
Theorem 5.1. The coordinate functions of mappings with nite distortion in W 1;P ( ) are weakly monotone. There is a particularly elegant geometric approach to the continuity estimates of monotone functions. The idea goes back to Gehring in his study of the Liouville theorem in space where he developed the Oscillation Lemma. We need the P {modulus of continuity P ( ) de ned for 0 < 1 as follows. For > 0 the value t of P at is uniquely determined by the equation Z 1=
P (st) ds s3 = P (1): 1 Certainly P is a non-decreasing function with (14) lim ( ) = 0: !0 P Given the transcendental nature of the equation one must solve, it is impossible in all but the most elementary situations, to calculate . Here are a few explicit formulas for ( ) which exhibit the correct asymptotics for near 0. P (t) = t2 ; ( ) = j log j, 12 More generally for all > 0 we have, P (t) = t2 log ,1 (e + t); > 0; ( ) j log j, 2 (13)
and nally
2 t P (t) = log(e + t) ; ( ) [log j log j], 12 ;
t2 P (t) = log(e + t) log ; ( ) [log log j log j], 12 ; log(3 + t) We now have the fundamental modulus of continuity estimate.
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Theorem 5.2. Let u 2 W 1;P (B) be weakly monotone in B = B(z0 ; 2R) . Then for all Lebesgue points a; b 2 B(z0 ; R) we have j a , b j (15) ju(a) , u(b)j 16RkrukB;P 2R : In particular, u has a continuous representative for which (15) holds for all a and b in the disk B(z0 ; R). In the statement above we have used Z 1 1 krukB;P = inf : (16) P (jruj) P (1)
jBj B to denote the P -average of ru over the ball B.
Theorem 5.3. 1Every mapping with nite distortion in the Orlicz{ ;P Sobolev class Wloc ( ), is continuous.
6. Liouville Type Theorem Here is a rst taste of the power of Theorem 5.1. Theorem 6.1. Let f : C ! C be a mapping of nite distortion whose di erential belongs to LP (C ). Then f is constant. The proof consists in showing that R krukB;P ! 0 as R ! 1 in (15) using the Dominated Convergence Theorem. 7. Solutions A principal solution is a homeomorphism h : C ! C with 1;1(C n E ), 1. a discrete set E (the singular set) such that h 2 Wloc 2. the Beltrami equation hz (z) = (z)hz (z) holds for a.e. z 2 C , and 3. we have the normalisation h(z) = z + o(1) at 1 It will become clear that the key to understanding the Beltrami equation and its local solutions is in the existence and uniqueness properties of the principal solutions. A function f , not necessarily a homeomorphism, is a very weak solution if it satis es: there is a discrete set Ef (the singular set) such that h 2 1 ;1 Wloc ( n Ef ). the Beltrami equation fz (z) = (z)fz (z) holds for almost every z 2
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The point is that one expects a solution f to be the composition of the principal solution h with a meromorphic function ' de ned on h( ), f = ' h; (the Stoilow factorisation theorem). Thus away from the poles, the solution f is locally as good as the principal solution. 7.1. Uniqueness of Principal Solutions. Here is the most general uniqueness result that we are aware of. Theorem 7.1. Every elliptic equation hz = H (z; hz ) admits at most one principal solution in the Sobolev-Orlicz class z + W 1;P (C ) We use the term z + W 1;P (C ) to denote the mappings h with jhz j + jhz , 1j 2 LP (C ). As far as the ellipticity is concerned, we assume that there is a measurable compactly supported function k : C ! B such that for almost every z 2 C and all ; 2 C jH (z; ) , H (z; )j k(z)j , j
Proof. Let h be a solution to the equation. Thus hz (z) = 0 for z suciently large. The point is that given two principal solutions h1 and h2 , the mapping f = h1 , h2 has nite distortion and its di erential Df = Dh1 , Dh2 belongs to LP (C ). To see this note j(h1 , h2)z j = jh1z , hz2 j = jH (z; h1z ) , H (z; h2z )j k(z)j(h1 , h2 )z j whence J (z; h1 ,h2 ) 0. It follows that f is constant from the Liouville theorem. The normalisation at 1 implies that this constant is 0. 8. Stoilow Factorisation We now state that if a Beltrami equation admits a homeomorphic solution, then all other solutions in the same class are obtained from this solution via composition with a holomorphic mapping. Theorem 8.1. Suppose we are given a homeomorphic solution h 2 1 ;P Wloc ( ) to the Beltrami equation (17) a:e:
hz = (z)hz
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1;P ( ) takes the form Then every solution f 2 Wloc f (z) = (h(z)); z 2
where : h( ) ! C is holomorphic.
9. Failure of Factorisation Now an example which shows that for fairly nice solutions one cannot expect a factorisation theorem even in the case of bounded distortion. Theorem 9.1. Let K > 1 and q0 < K2K+1 . Then there is a Beltrami coecient supported in the unit disk with the following properties. kk1 = KK ,+11 , The Beltrami equation hz = hz admits a Holder continuous so1;2 (C ), lution h 2 z + W 1;q0 (C ) which fails to be in Wloc The solution h is not quasiregular, and therefore not the principal solution, nor obtained from the principal solution by factorisation. 10. Distortion in the Exponential Class Theorem 10.1. There exists a number p0 > 1 such that every Beltrami equation hz (z) = (z) hz (z) a:e: C with Beltrami coecient such that (z) , 1 (z) j(z)j K K (z) + 1 B and (18) eK 2 Lp(B) with p p0 , admits a unique principal solution h 2 z + W 1;2 (C ). There are examples to show that in order for there to be a principal solution in the natural Sobolev space z + W 1;2(C ) it is necessary that the exponent p at (18) is large, at least p 1. As a matter of fact, somewhat more is true in Theorem 10.1. The higher the exponent of integrability of epK the better the regularity of the solution. That is even beyond L2 , such as L2 log L with any 0, see [16]. The situation is di erent if the integrability exponent of eK is smaller than the critical exponent p0. Here the principal solution need not be in z + W 1;2(C ), but we still obtain a satisfactory class of solutions.
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Theorem 10.2. Suppose the distortion function K = K (z) for the Beltrami equation is such that eK 2 Lp (B) for some positive p. Then the equation admits a unique principal solution h (19) h 2 z + W 1;Q(C ); Q(t) = t2 log,1(e + t) 1;Q( ) solution is factorisable. Moreover, every Wloc
11. Distortion in the Subexponential Class We assume here that the Beltrami coecient is supported in the unit disk B. Theorem 11.1. There is a number p 1 such that every Beltrami equation whose distortion function has K ( z ) exp 1 + log K (z) 2 Lp(B) for p > p, admits a unique principal solution h 2 z + W 1;Q(C ) with Orlicz function Q(t) = t2 log,1 (e + t). Moreover we have Modulus of Continuity; (20) jh(a) , h(b)j2 log log(1CK+ 1 ) ja,bj for all a; b 2 2B. Inverse; The inverse map g = h,1 (w) has nite distortion K = K(w) and log K 2 L1 (C ) Factorization; each solution g 2 Wloc1;Q( ) to the equation gz = (z)gz ; a:e:
admits a Stoilow factorisation (21) g(z) = h(z) where is holomorphic in h,1 ( ). In particular, all non-constant 1;Q( ) are open and discrete. solutions in Wloc 12. Existence Theory. Various reductions show the important case to be when the Beltrami coecient is compactly supported in the unit disk B. Then any solution is analytic outside the unit disk.
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12.1. Results from Harmonic Analysis. The existence proof presented here exploits a number of substantial results in harmonic analysis. The arguments clearly illustrate the important r^ole that the higher integrability properties of the Jacobians have to play. The critical exponent p0 in Theorem 10.1 depends only on the constants in three inequalities which we now state. The rst is a direct consequence of [8]. Theorem 12.1. (Coifman, Lions, Meyer, Semmes) The Jacobian determinant J (x; ) of a mapping 2 W 1;2 (C ) belongs to the Hardy space H 1(C ) and we have the estimate Z
kJ (x; )kH ( ) C1 jD j2
(22)
1 C
C
Next we have from [9] Theorem 12.2. (Coifman, Rochberg) Let be a Borel measure in C such that its Hardy{Littlewood maximal function M (x; ) is nite at a single point (and therefore at every point). Then log M (x; ) 2 BMO(C ) and its norm is bounded by an absolute constant, (23) k log M (x; )kBMO C2 : Finally we shall need the constant C3 which appears in the H 1-BMO duality theorem of Fe erman, [12]. Theorem 12.3. (Fe erman) For K 2 BMO(C ) and J 2 H 1(C ) we have (24)
Z
K (x) J (x)
dx
C3kK kBMO kJ kH
1
Having these prerequisites we can reveal that the exponent in Theorem 10.1 is (25) p0 = 8C1C2C3: 13. Sketch of Proof for Theorem 10.1 We again refer the reader to [14] for more details, but the basic ideas can be found here. Let K be the distortion function. Set
Ap =
Z
C
pK (z ) e
, ep dz < 1
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We approximate by smooth functions via molli cation. We have K K and the uniform bound Z
(26)
C
pK (z ) e
,
ep
K (z ) e
dz e
Z
C
pK (z ) e
, ep dz = Ap
, Put = 2 to see Next, the maximal function is nite everywhere, (27) M (z; e K ) = e + M (z; e K , e )) and this last term is a constant plus a function in L2 (C ). Now consider the BMO functions K (z) = 1 log M (z; e K ) (28) p
2 L2 (C ). of e K (z)
By Theorem 12.2, the BMO norm of this function does not depend on , (29) kK kBMO 2Cp 2 Moreover this function pointwise majorises the distortion function. We need a uniform L2 bound for K . Clearly 1 Z K (z)2 dz j2Bj 2B Z = 21 log2 [M (z; e K )]2 4 j2Bj 2B Z 41 2 log2 j21Bj [M (z; e K , e ) + e ]2 2B Z 1 2 C 2 2
K
2 4 2 log 2e + j2Bj (e , e ) C 2 Here we have used the L inequality for the maximal operator. This gives (30)
Z
2B
jK (z)j2 dz C4 log2 (1 + Ap)
where C4 is an absolute constant. Let us now return to the molli ed Beltrami equation, (31) fz = (z)fz We look for a C 1 {solution of (31) in the form (32) fz (z) = e(z) ; fz (z) = (z)e(z)
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TADEUSZ IWANIEC AND GAVEN MARTIN
where 2 W 1;p (C ; C ), for some p > 2, is compactly supported. The necessary and sucient condition for is that (e )z = ( e )z or equivalently (33) z = z + ( )z This equation is uniquely solved using the Beurling{Ahlfors transform, that is the singular integral operator de ned by Z 1 d ^ d (Sg)(z) = , 2i g((z)j, )2 C Note that z = Sz and so equation (33) reduces to (34) (I , S )z = ( )z As kS k2 = 1 and as k k1 < 1, there is p > 2 such that k k1kS kp < 1 In this case the operator I , S has a continuous inverse. Thus (35) z = (I , S ),1( )z 2 Lp (C ) and also (36) z = Sz 2 Lp (C ) Note that z vanishes outside the support of which is contained in B(0; 2). Also z = Sz = O(z,2) as z ! 1. Thus (z) Cz asymptotically, for a suitable constant C . In fact Z 1 (37) (z) = (Tz )(z) = 2i z,(z) d ^ d C where T is the complex Riesz Potential. Hence is Holder continuous with exponent 1 , p2 , by the Sobolev Imbedding Theorem. Now the solution f of equation (32) is unique up to a constant as fz = 0 outside 2B and as fz , 1 2 Lp (C ). That is, f is a principal solution to the Beltrami equation (31). It is important to realise here that the Jacobian of f is strictly positive, (38) J (z; f ) = jfz j2 , jfz j2 = (1 , j j2)e2 > 0 The Implicit Function Theorem tells us that f is locally one-to-one. Another observation to make is that limz!1 f (z) = 1. It is an elementary topological exercise to show that f : C ! C is a global homeomorphism of C . It's inverse is C 1{smooth of course.
WHAT'S NEW FOR THE BELTRAMI EQUATION ?
145
We now digress for a second to outline the existence proof in the classical setting where K (z) K < 1. As the sequence K is uniformly bounded we nd there is an exponent p = p(K ) > 2 such that (39) kfz kp + kfz , 1kp CK where CK is a constant independent of . Hence the Sobolev Imbedding Theorem yields the uniform bound (40) jf (a) , f (b)j CK ja , bj1, p2 + ja , bj The same inequality holds for the inverse map and hence p p,2 j a , b j (41) jf (a) , f (b)j 2 CK + ja , bj p,2 We may assume that f (0) = 0. As the p{norms of fz and fz , 1 are uniformly bounded, we may assume that each converges weakly in Lp(C ) after possibly passing to a subsequence. From the uniform continuity estimates and Ascoli's Theorem, we may further assume f ! f locally uniformly in C . Obviously f satis es the same modulus of continuity estimates and is therefore a homeomorphism. Moreover, it follows that the weak limits of fz and fz , 1 must in fact be equal to fz and fz , 1 respectively. Hence f is a homeomorphism in the Sobolev class z + W 1;p(C ), that is fz and fz , 1 in Lp(C ). Finally observe that ! pointwise almost everywhere, and hence in Lq (C ), where q is the Holder conjugate of p. The weak convergence of the derivatives shows that f is a solution to the Beltrami equation. Back to the more general setting. If we followed the above argument we nd the Lp bounds are useless as we cannot keep them uniform. We therefore seek an alternative route via a Sobolev-Orlicz class where uniform bounds might be available. We note the elementary inequality (42) (juj + jvj)2 2K (juj2 , jvj2) + 4K 2jv , wj2 ,1 juj and whenever u; v; w are complex numbers such that jwj KK +1 K 1. We apply this inequality pointwise with u = z ; v = z ; w = z and K = K (z) as de ned at (28), where (43) (z) = f (z) , z 2 W 1;2(C ); K = K (z) and use equations (31), (29) we can write (j z j + j z j)2 2K (j z j2 , j z j2) + 4(K )2j j2
146
TADEUSZ IWANIEC AND GAVEN MARTIN
and hence
jD (z)j2 2K J (z; ) + 4j K j2
(44)
Next we integrate this and use Theorems 12.1 and 12.3 to obtain Z C
j
j 2C3kK kBMO k
D 2
J (z; )
Z
kH + 4
Z
1
2B
jK j2
4C1Cp 2C3 jD j2 + 4C4 log2 (1 + Ap) C
where in the latter step we have used the uniform bounds at (29) and (30). It is clear point why we have chosen p0 = 8C1C2C3 at (25). R at this 2 The term C jD j in the right hand side can be absorbed in the left hand side. After doing this we obtain the uniform bounds in L2 Z
(45)
C
jD j2 8C4 log2(1 + Ap)
which read as
k
D
(46)
kL ( ) C5 log
Z
2 C
B
epK
and in turn leaves us with the local estimate for the mapping f (z) = (z) + z, namely (47)
k
Df
kL (BR ) C5 R + log 2
Z
B
epK
where BR = B(0; R). As f is monotone (being a C 1 homeomorphism) we can apply the modulus of continuity estimate of Theorem 5.2, (48)
,R
pK Be jf (a) , f (b)j C6 R +1 log log 2 e + jaR,bj
for all a; b 2 BR . Now consider the inverse map to f . Let us denote it by h = (f ),1 : C ! C . As both f and h are smooth di eomorphisms we
WHAT'S NEW FOR THE BELTRAMI EQUATION ?
nd
Z
BR
j
Z
j dw =
Dh (w) 2
B Z R
=
Z
K (w; h )J (w; h ) dw
h (BR ) C
147
K (z; f ) dz
(K (z) , 1) dz + jh (BR )j
Cp
R2 +
Z
B
(K (z) , 1) dz
In this last inequality we have put in the uniform bound jh (BR)j CpR2 . One interesting way to see this estimate (though perhaps not the easiest) is via the Koebe distortion theorem. Anyway, we have (49)
Z
B(0;R)
j
j C
Dh 2
(R2 +
Z
B
K)
and consequently we have the continuity estimate at (5.2) for h , 2R K CR (50) jh (x) , h (y)j2 B R log e + jx,yj For f this reads as R , CR2 B K (51) jf (a) , f (b)j R exp ja , bj2 whenever a; b 2 B(0; R) and R 1. The uniform W 1;2 bounds, and the continuity estimates from above and below now enable us to pass to the limit. We nd f ! f and h ! h = f ,1 locally uniformly in C and Df and Dh converging weakly in L2loc (C ). As in the classical setting this implies that f is a homeomorphic solution to the Beltrami equation. Moreover fz ; fz , 1 2 L2 (C ) and the same is true of the inverse function References [1] L.V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand, Princeton 1966; Reprinted by Wadsworth Inc. Belmont, 1987. [2] K. Astala, Area distortion of quasiconformal mappings, Acta Math., 173, (1994), 37{60. [3] K. Astala, T. Iwaniec, P. Koskela and G.J. Martin, Mappings with BMO bounded Distortion, Math. Annalen, 317, (2000), 703{726. [4] J. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 63, (1977), 337-403.
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TADEUSZ IWANIEC AND GAVEN MARTIN
[5] B. Bojarski, Homeomorphic solutions of Beltrami systems, Dokl. Akad. Nauk. SSSR, 102, (1955), 661{664. [6] M.A. Brakalova and J.A. Jenkins, On solutions of the Beltrami equation, J. Anal. Math., 76, (1998), 67{92. [7] R. Coifman and G. Weiss, Analyse Harmonique Non-commutative sur Certain Espaces Homogenes, Lecture Notes in Math., 242, Springer-Verlag, 1971. [8] R.R. Coifman, P.L Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72, (1993), 247{286. [9] R.R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc., 79, (1980), 249{254. [10] R.R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. Math., 103, (1978), 569{645. [11] G. David, Solutions de l'equation de Beltrami avec kk1 = 1, Ann. Acad. Sci. Fenn. Ser. AI Math., 13, (1988), 25{70. [12] C. Fe erman, Characterisations of bounded mean oscillation, Bull. Amer. Math. Soc., 77, (1971), 587{588. [13] F.W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc., 103, (1962), 353{393. [14] T. Iwaniec and G.J. Martin The Beltrami Equation, Memoirs of the Amer. Math. Soc., to appear. [15] T. Iwaniec and G.J. Martin Geometric Function Theory and Non-linear Analysis, to appear, Oxford University Press. [16] T. Iwaniec , P. Koskela, G.J. Martin and C. Sbordone Mappings of exponentially integrable distortion, to appear, [17] J. Manfredi, Weakly monotone functions, J. Geometric Analysis, 3, (1994), 393-402. [18] C.B. Morrey, On the solutions of quasi-linear elliptic partial di erential quations, (1938), 126{166. [19] S. Muller, A surprising higher integrability property of mappings with positive determinant, Bull.Amer. Math. Soc. 21 (1989), 245{248. [20] Rochberg and Weiss Analytic families of Banach spaces and some of their uses, Recent progress in Fourier analysis (El Escorial, 1983), 173{201, North-Holland Math. Stud., 111, North-Holland, Amsterdam-New York, 1985. [21] V. Ryazanov, U. Srebro and E. Yacubov, BMO{quasiregular mappings, To appear, J. D'Analyse Math. [22] E.M. Stein, Note on the class L log L, Studia Math., 32, (1969), 305{310. T. Iwaniec, Department of Mathematics, Syracuse University, Syracuse NY, USA
E-mail address :
[email protected]
G.J. Martin, Department of Mathematics, The University of Auckland, Auckland, NZ
E-mail address :
[email protected]
SOME SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS ASSOCIATED WITH LIE GROUPS PALLE E. T. JORGENSEN To Derek Robinson on the occasion of his 65th birthday Abstract. In this note we survey results in recent research papers
on the use of Lie groups in the study of partial di erential equations. The focus will be on parabolic equations, and we will show how the problems at hand have solutions that seem natural in the context of Lie groups. The research is joint with D.W. Robinson, as well as other researchers who are listed in the references.
1. Introduction When the Hamiltonian of a quantum-mechanical system is related to a Lie algebra, it is often possible to use the representation structure of the Lie algebra to decompose the Hilbert space of the quantummechanical system into simpler (irreducible) pieces. For example, if a Hamiltonian commutes with the generators of a Lie algebra, the Hilbert space of the system can be decomposed into irreducibles of the Lie algebra, and the Lie algebra elements themselves can be used as elements in a set of commuting observables. We have aimed at making the present paper accessible to a wide audience of non-specialists, stressing the general ideas and motivating examples, as opposed to technical details. 2000 Mathematics Subject Classi cation. Primary 35B10; Secondary 22E25, 22E45, 31C25, 35B27, 35B45, 35C99, 35H10, 35H20, 35K10, 41A35, 43A65, 47F05, 53C30. Key words and phrases. approximating variable coecient partial di erential equation with constant coecients, t ! 1 asymptotics, boundary value problem, Gaussian estimates, heat equation, Hilbert space, homogenization, nilmanifold, parabolic, partial di erential equations, scaling and approximation of solution, spectrum, strati ed group. This research was partially supported by two grants from the U.S. National Science Foundation, and by the Centre for Mathematics and its Applications (CMA) at The Australian National University (ANU). This paper is an expanded version of a lecture given by the author at the National Research Symposium on Geometric Analysis and Applications at the ANU in June of 2000. 149
150
PALLE E. T. JORGENSEN
The class of such Hamiltonians is quite large: see [JoKl85] and [Jor88]. In this introduction we will review those Hamiltonians H whose interaction terms are polynomial in the position variables. Such Hamiltonians are directly and naturally related to nilpotent Lie algebras. The nilpotent case is studied in Section 2. The spectrum of H is obtained by decomposing the physical space on which the Hamiltonian H acts into irreducible representations of the underlying nilpotent group. Sometimes this decomposition is decisive, as is the case with a particle in a constant magnetic eld, where the decomposition leads to a harmonic-oscillator Hamiltonian. Sometimes the decomposition leads to a new Hamiltonian that requires further analysis, as is the case with a particle in a curved magnetic eld. The time evolution of the system is obtained by solving the heat equation of the underlying nilpotent Lie group. By writing the Hamiltonian as a quadratic sum of Lie-algebra elements and then using the representation of these Lie-algebra elements arising from the regular representation, it is possible to write e,tH as the convolution of a kernel (which is a solution of the heat equation) with a representation acting on the physical Hilbert space; see [Jor88]. The simplest case of this spectral picture is as follows: Consider a nonrelativistic spinless particle of mass m in an external magnetic eld B (x). The Hamiltonian for such a system is given by e 2 1 (1.1) H = 2m p , c A ; where p = hi r and A is the vector potential satisfying B = r A. Consider the commutators h e i he e pi , c ai ; pj , c aj = , i c "ijk bk ; h e i h @bj h (1.2) pi , c ai ; bj = i @x i bij ; h e i h @bjki h pi , c ai; bjk = i @x i bijk ; i ... ... ; where A = (a1 ; a2; a3), B = (b1 ; b2; b3 ), x = (x1 ; x2; x3 ). If B is a polynomial in x, eventually the derivatives of B will give zero, so that the set of commutators closes. The resulting Lie algebra formed by real linear combinations of the elements (1.3) pi , ec ai; bi; bij ; : : :
PARTIAL DIFFERENTIAL EQUATIONS AND LIE GROUPS
151
is therefore a nilpotent Lie algebra, and the Hamiltonian (1.1) is qua, dratic in the rst three Lie algebra elements Xi := pi , ec ai , i = 1; 2; 3, from the list (1.3). By general theory, e.g., [Rob91], this Lie algebra is the Lie algebra g of some Lie group G, which we may take to be simply connected. We show further in [JoKl85] and [Jor88] that there is a unitary representation U of G on L2 (R 3 ) such that 3 2 ! X e 2mH = dU pi , c ai : i=1 If there is a constant of motion for the Lie-algebra elements pi , ec ai, then U is a direct integral R over a corresponding spectral parameter . We then get H = d H () where H has absolutely continuous spectrum, while each H () has purely discrete spectrum. If 0 ( ) 1 ( ) is the spectrum of H (), then each 7! i ( ) is real analytic, and we get the following typical spectral picture.
2( )
1( ) 0( )
In this paper we will focus attention on a more restricted case wherein the coecients are periodic. As shown in Section 3, this case shares the spectral band structure with the polynomial-magnetic- eld case. We show that in the periodic case the regularity of the coecients may be relaxed, and in fact, our spectral-theoretic results will be valid when the operator has L1 -coecients. 2. Periodic operators We begin by recalling some elementary de nitions and facts about strati ed Lie groups from [FoSt82]. A real Lie algebra g is called
152
PALLE E. T. JORGENSEN
strati ed if it has a vector-space decomposition
(2.1)
g=
r M k=1
g(k) ;
for some r, which we shall take nite here, all but a nite number of the subspaces g(k) are nonzero, g(k); g(l) g(k+l) (2.2) for all k; l 2 N , and g(1) generates g as a Lie algebra. Thus a strati ed Lie algebra is automatically nilpotent, and if r is the largest integer such that g(r) 6= 0, then g is said to be nilpotent of step r. A Lie group is de ned to be strati ed if it is connected and simply connected and its Lie algebra g is strati ed. Let G be a strati ed Lie group and exp: g ! G the exponential map. The Campbell{Baker{Hausdor formula establishes that exp (X ) exp (Y ) = exp (H (X; Y )) ; where H (X; Y ) = X + Y + [X; Y ] =2 + a nite linear combination of higher-order commutators in X and Y . Thus X; Y ! H (X; Y ) de nes a group multiplication law on the underlying vector space V of g which makes V a Lie group whose Lie algebra is g and the exponential map exp: g ! V is simply the identity. Then V with the group law is di eomorphic to G. Next let dk denote the dimension of g(k) and d the of g and for each k choose a vector-space basis X (k) = (dimension k) (k) X1 ; : : : ; Xdk of g(k) such that X1; : : : ; Xd = X1(1) ; : : : ; Xd(rr) is a basis of g. If 1; : : : ; d is the dual basis for g , i.e., if k (Xl) = k;l, de ne k = k exp,1. Then 1; : : : ; d are a system of global coordinates for G, and the product rule on G becomes k (xy) = k (x) + k (y) + Pk (x; y) ; x; y 2 G; where Pk (x; y) is a nite sum of monomials in i (x), i (y) for i < k with degree between 2 and m. It follows that both left and right Haar measure on G can be identi ed with Lebesgue measure d1 dd . If Xi denotes one of the (abstract) Lie generators, we denote by Ai the corresponding right-invariant vector eld on G, i.e., Ai on a test function on G is given by A(il) = dL (Xi), or more precisely, (l) d (exp (,tX ) g) j ; g 2 G; (2.3) Ai (g) = dt i t=0 and similarly A(ir) = dR (Xi) given by (r) d (g exp (tX )) j : (2.4) Ai (g) = dt i t=0
PARTIAL DIFFERENTIAL EQUATIONS AND LIE GROUPS
153
Since we can pass from left to right with the adjoint representation, the formulas may be written in one alone, and we will work with A(il), and denote it simply Ai . If 1 j d1 we will need the functions yj on G de ned by (2.5)
yj exp
d X k=1
k Xk
!!
= j :
These functions satisfy the following system of di erential equations: (2.6) ,A(il)yj = A(ir) yj = i;j : It follows by the standard ODE existence theorem that the functions yi on G are determined uniquely by (2.6) and the \initial" conditions yi (e) = 0. Also note that (2.6) is consistent only for the di erential equations de ned from a sub-basis A1 ; : : : ; Ad , and that they would be overdetermined had we instead used a basis: hence the distinction between subelliptic and elliptic. In addition, we have given a discrete subgroup , in G such that M = G=, is compact. It is well-known that it then has a unique (up to normalization) [Jor88, Rob91] invariant measure . The corresponding Hilbert space is L2 (M; ), and the invariant operators on G pass naturally to invariant operators on M ; see [BBJR95]. Let X1 ; : : : ; Xd be the generating Lie-algebra elements. Then the corresponding invariant vector elds on G will be denoted A1; : : : ; Ad , and those on M will be denoted B1 ; : : : ; Bd . Functions ci;j 2 L1 (G) are given, and we form the quadratic form 1
1
1
1
(2.7)
h (f ) =
d X 1
i;j =1
hAif j ci;j Aj f i :
If further (2.8) ci;j (g ) = ci;j (g) for g 2 G; 2 ,; then we have a corresponding form hM on M = G=,. Introducing , (2.9) c"i;j (x) = ci;j ",1x ; " > 0; we get for each " a periodic problem corresponding to the period lattice ",. To speak about ", for " 2 R + , we must have an action of R + on G which generalizes the familiar one " : (x1 ; : : : ; xd) 7,! ("x1 ; : : : ; "xd ) d of R . It turns out that this can only be done if G is strati ed, and so in particular nilpotent; see [FoSt82], [Jor88]. In that case it is possible to
154
PALLE E. T. JORGENSEN
construct a group of automorphisms f"g"2R of G which is determined by the di erentiated action d" on the Lie algebra g. If g is speci ed as in (2.1){(2.2), then , d" X (1) = "X (1) ; X (1) 2 g(1); " 2 R + : (2.10) Let H , respectively H", be the selfadjoint operators associated to the period lattices , and ", (see [BBJR95] or [Rob91]), and let St = e,tH , St" = e,tH" . We now turn to the homogenization analysis of the limit " ! 0 which leads to our comparison of the variable-coecient case to the constant-coecient one. It should bePstressed that in the Lie case, even the \constant-coecient" operator i;j Ai c^i;j Aj is not really constantcoecient, as the vector elds Ai are variable-coecient. Take even the simplest example where G is the three-dimensional Heisenberg group of upper triangular matrices of the form 01 x z1 (2.11) g = @0 1 yA ; x; y; z 2 R: 0 0 1 In this case, dim g(1) = 2, and dim g(2) = 1, with g(2) spanned by the central element in the Lie algebra. Di erentiating matrix multiplication (2.11) on the left as in (2.3), we get the following three identities: @ + y @ = ,dL (X ) ; A1 = @x 1 @z @ = ,dL (X ) ; A2 = @y 2 @ = ,dL (X ) ; A3 = @z 3 where the rst vector eld is of course variable coecients. We will use standard tools [ZKO94] (see also [Dau92], [Tho73], [Wil78]) on homogenization. Theorem 2.1. [BBJR95] Suppose the system ci;j 2 L1 is given and assumed strongly elliptic. Then there is a C0-semigroup S^t on L2 (G; dx) with constant coecients, where dx is left Haar measure, such that
"
lim
St , S^t f 2 = 0 "!0 for all f 2 L2 (G; dx) and t > 0. The constant coecients of the limit operator c^i;j may be determined as follows: We show in [BBJR95] that if (2.12) ci;j (g) := h (gi , yi; gj , yj ) +
PARTIAL DIFFERENTIAL EQUATIONS AND LIE GROUPS
155
and if C (g) is the corresponding quadratic form, then the problem (2.13) inf C (g) =: C^ g
has a unique solution, i.e., the in mum is attained at f1 ; : : : ; fd such that ^ (2.14) C (f ) = C: The order relation which is used in the in mum consideration (2.13) is the usual order on hermitian matrices: For every g, the matrix C (g) := (ci;j (g))di;j=1 is hermitian, and the matrix inequality C (g) C^ may thus be spelled out as follows: X X zi ci;j (g) zj zic^i;j zj for all z1; : : : ; zd 2 C : 1
1
1
i;j
i;j
Solvability of this variational problem is part of the conclusion of our analysis in [BBJR95], i.e., the existence of the minimizing functions f1; : : : ; fd . Then the coecients of the homogenized operator can also be computed with the aid of the coordinates yi, i = 1; : : : ; d1, introduced in (2.5) and (2.6). One has the representation 1
(2.15)
c^i;j =
Z
Y
dy
d X 1
k;l=1
(Ak (fi (y) , yi)) ck;l (y) (Al (fj (y) , yj ))
= hY (fi , yi; fj , yj ) ; where h denotes the sesquilinear form associated with H , and the subscript Y refers to the region of integration. Speci cally, Y is a fundamental domain for the given lattice , in G. For example, we may take Y to be de ned by \ (2.16) Y= x 2 G ; jxj x ,1 ;
2,
and j j de ned relative to a geodesic distance d, jxj := d (x; e), x 2 G. Then S (i) 2, Y = G, and (ii) meas (Y 1 \ Y 2) = 0 whenever 1 6= 2 in ,. (These are the axioms for fundamental domains of given lattices, but we stress that (2.16) is just one choice in a vast variety of possible choices.) The simplest case of the construction is G = R, and it was rst considered in [Dav93, Dav97] by Brian Davies. This is the simplest
156
PALLE E. T. JORGENSEN
possible heat equation, and we then have the conductivity represented by a periodic function c, say c (x + p0) = c (x) ; x 2 R ; where p0 is the period. Then H = , dxd c (x) dxd , and it can be checked that 1 Z p dx ,1 c^ = p : 0 0 c (x) Theorem 2.2. [BBJR95] Adopt the assumptions of Theorem 2.1. Then D=2 ^ lim t ess sup Kt (x ; y) , Kt (x ; y) = 0 t!1 0
jxj2 +jyj2 at
for each a > 0 where jxj = dc (x ; e), and where
dc (x ; y) = sup
(
(x) , (y) ; 2 Cc1 (G) ; d X 1
i;j =1
)
ci;j (Ai ) (Aj ) 1 pointwise
and Ai refers to the Lie action of the vector eld Ai on from (2.3). It is our aim here only to sketch the ideas, and the reader is referred to our papers for details, but we stress that the proof is based on homogenization, see, e.g., [BLP78], [ZKON79], [Koz80], and [AvLi91] The number D is the homogeneous degree de ned from the given grading, or strati cation, g(i) of the nilpotent Lie algebra g. As spelled out in [Jor88] and [FoSt82], there are numbers i depending on the Lie-structure coecients such that X D = i dim g(i) : i
To be speci c, the numbers i are determined in such a way that we get a group of scaling automorphisms f"g"2R of g, and therefore on G, and it is this group which is fundamental in the homogenization analysis. Speci cally, extending (2.10), " : g ! g is de ned by , (2.17) " X (i) = "i X (i); X (i) 2 g(i) ; and then extended to g by linearity via (2.1), in such a way that (2.18) " ([X; Y ]) = [" (X ) ; " (Y )] ; X; Y 2 g; " 2 R + : Hence if (2.2) holds, then it follows from (2.17) and (2.18) that i = i for i = 1; 2; : : : . In the case of the Heisenberg Lie algebra g, we have +
PARTIAL DIFFERENTIAL EQUATIONS AND LIE GROUPS
157
[X; Y ] = Z as the relation on the basis elements; Z is central. Then g(1) = span (X; Y ), g(2) = R Z , 1 = 1, 2 = 2, so D = 4. Let Kt and K^ t be the respective integral kernels for the semigroups St and S^t , and set and Then
jjK jjp = xess 2G
Z
dy jK (x; y)j
p
G
1=p
jjK jj1 = x;yess2G jK (x; y)j :
Theorem 2.3. [BBJR95] Adopt the assumptions of Theorem 2.1. Then D=2 ^ ^ lim t K lim K t , Kt = 0; t , Kt = 0: t!1 t!1 1 1 3. G = R d The case G = R d was considered in [BJR99], where we further showed that the limit St" ! S^t then holds also in the spectral sense. In that case, we scale by " = 1=n, n ! 1, and then identify the limit operator as having absolutely continuous spectral type, and we prove spectral asymptotics. (A general and classical reference for periodic operators is [Eas73].) Starting with an equation which is invariant under the Zd-translations, we then use the Zak transform [Dau92] to write St = e,tH as a direct integral over Td (= R d =Zd), viz., (3.1)
St =
Z Td
St(z);
and we establish continuity of z 7! St(z) in the strong topology [BJR99, Lemma 2.2]. Pick a positive C 1-function on R d of integral one, and set Z (n) d ci;j (x) = n dy (ny) ci;j (x , y) ; Rd
and form the corresponding C0-semigroup St(n) = e,tH n ; where H (n) is de ned from c(i;jn). We then show in [BJR99] that St(n) approximates St , not only in the strong topology, but also in a spectraltheoretic sense. Using this, we establish the following connection be, z (z ) , tH , tH i i tween St = e and St = e in (3.1). Setting z = e ; : : : ; e d , we get ( )
( )
1
158
PALLE E. T. JORGENSEN
Theorem 3.1. If n (z) denotes the eigenvalues of Hz then
(3.2) Nlim N 2 n (w) ; wN = z; n = 0; 1; : : : !1
nD
E
o
= (n , ) j C^ (n , ) ; n 2 Zd ; where the limit is in the sense of pointwise convergence of the ordered sets, and where C^ = (^ci;j ) is the constant-coecient homogenized case. The rate of convergence of the eigenvalues in (3.2) can be estimated further by a trace norm estimate. We refer the reader to [BJR99] for details of proof, but the arguments in [BJR99] are based in part on the references [Aus96], [DaTr82], [Eas73], and [ZKON79]. In addition, we mention the papers [Aus96], [AMT98], and [TERo99], which contain results which are related, but with a di erent focus. Finally, we mention that our result from [BJR99], Theorem 3.1, has since been extended in several other directions: see, e.g., [Sob99] and [She00]. Acknowledgements. We are grateful to Brian Treadway for excellent typesetting and graphics production, and to the participants in the National Research Symposium at The Australian National University for fruitful discussions, especially A.F.M. ter Elst. References
[Aus96] [AMT98] [AvLi91] [BBJR95] [BLP78] [BJR99] [DaTr82] [Dau92] [Dav93]
P. Auscher, Regularity theorems and heat kernel for elliptic operators, J. London Math. Soc. (2) 54 (1996), 284{296. P. Auscher, A. McIntosh, and P. Tchamitchian, Heat kernels of second order complex elliptic operators and applications, J. Funct. Anal. 152 (1998), 22{73. M. Avellaneda and F.-H. Lin, Lp bounds on singular integrals in homogenization, Comm. Pure Appl. Math. 44 (1991), 897{910. C.J.K. Batty, O. Bratteli, P.E.T. Jorgensen, and D.W. Robinson, Asymptotics of periodic subelliptic operators, J. Geom. Anal. 5 (1995), 427{443. A. Bensoussan, J.-L. Lions, and G.C. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978. O. Bratteli, P.E.T. Jorgensen, and D.W. Robinson, Spectral asymptotics of periodic elliptic operators, Math. Z. 232 (1999), 621{650. B.E.J. Dahlberg and E. Trubowitz, A remark on two-dimensional periodic potentials, Comment. Math. Helv. 57 (1982), 130{134. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 61, Society for Industrial and Applied Mathematics, Philadelphia, 1992. E.B. Davies, Heat kernels in one dimension, Quart. J. Math. Oxford Ser. (2) 44 (1993), 283{299.
PARTIAL DIFFERENTIAL EQUATIONS AND LIE GROUPS [Dav97] [Eas73] [FoSt82] [Jor88] [JoKl85] [Koz80] [Rob91] [She00] [Sob99] [TERo99] [Tho73] [Wil78] [ZKO94] [ZKON79]
159
E.B. Davies, Limits on Lp regularity of self-adjoint elliptic operators, J. Di erential Equations 135 (1997), 83{102. M.S.P. Eastham, The Spectral Theory of Periodic Di erential Equations, Scottish Academic Press, Edinburgh, Chatto & Windus, London, 1973. G.B. Folland and E.M. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, Princeton, 1982. P.E.T. Jorgensen, Operators and Representation Theory: Canonical Models for Algebras of Operators Arising in Quantum Mechanics, North-Holland Mathematics Studies, vol. 147, Notas de Matematica, vol. 120, North-Holland, Amsterdam{New York, 1988. P.E.T. Jorgensen and W.H. Klink, Quantum mechanics and nilpotent groups, I: The curved magnetic eld, Publ. Res. Inst. Math. Sci. 21 (1985), 969{999. S.M. Kozlov, Asymptotics of fundamental solutions of second-order divergence di erential equations, Mat. Sb. (N.S.) 113(155) (1980), no. 2(10), 302{323, 351 (Russian), English translation: Math. USSRSb. 41 (1982), no. 2, 249{267. D.W. Robinson, Elliptic Operators and Lie Groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1991. Z. Shen, The periodic Schrodinger operators with potentials in the Morrey{Campanato class, preprint, University of Kentucky, 2000. A.V. Sobolev, Absolute continuity of the periodic magnetic Schrodinger operator, Invent. Math. 137 (1999), 85{112. A.F.M. ter Elst and D.W. Robinson, Second-order subelliptic operators on Lie groups, I: Complex uniformly continuous principal coecients, Acta Appl. Math. 59 (1999), 299{331. L.E. Thomas, Time dependent approach to scattering from impurities in a crystal, Comm. Math. Phys. 33 (1973), 335{343. C.H. Wilcox, Theory of Bloch waves, J. Analyse Math. 33 (1978), 146{ 167. V.V. Z ikov, S.M. Kozlov, and O.A. Olenik, Homogenization of Di erential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994, translated by G.A. Yosi an from the Russian Usrednenie differencial~nyh operatorov, \Nauka", Moscow, 1993. V.V. Z ikov, S.M. Kozlov, O.A. Olenik, and H.T. Ngoan, Averaging and G-convergence of di erential operators, Uspekhi Mat. Nauk 34 (1979), no. 5(209), 65{133, 256, Russian Math. Surveys 34 (1979), no. 5, 69{ 148.
Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419, U.S.A.
E-mail address :
[email protected] URL: http://www.math.uiowa.edu/~jorgen/
PRINCIPAL SERIES AND WAVELETS CHRISTOPHER MEANEY Abstract. Recently Antoine and Vandergheynst [1, 2] have pro-
duced continuous wavelet transforms on the -sphere based on a principal series representation of ( 1). We present some of their calculations in a more general setting, from the point of view of Fourier analysis on compact groups and spherical function expansions. n
S O n;
1. Coherent States We begin with Antoine and Vandergheynst's de nition of a coherent state, as presented in [1, 2]. Here G is a locally compact group. Suppose that X is a homogeneous space of G, X = G=H , equipped with a G-invariant measure. Let (U; L (Y )) be a unitary representation of G on some Lebesgue space L (Y ). Assume there is a Borel cross section : X ,! G; (x) H = x; 8x 2 X: Say that 2 L (Y ) is admissible mod(H; ) when 2
2
2
Z
jhU ( (x)) j'ij dx < 1; 8' 2 L (Y ) : 2
X
2
The orbit of an admissible vector under (X ), fU ( (x)) : x 2 X g
is called a coherent state. Note that there are other variations on the theme of \restricted square integrability", such as the case described in [3]. 1991 Mathematics Subject Classi cation. 43A90,22E46,43A75,42C40. Key words and phrases. Semisimple Lie group, coherent state, continuous wavelet transforms, principal series, Plancherel formula, admissible vectors. This is the content of my lecture at the National Research Symposium on Geometric Analysis and Applications in Canberra, June 2000. In the past year my research was partially supported by the ARC Small Research Grants Scheme. 160
PRINCIPAL SERIES AND WAVELETS
161
2. Frames Suppose now that is an admissible vector in L (Y ). De ne a linear operator 2
A; : L (Y ) ,! L (Y ) 2
by
hA; ' j' i = 1
2
Z
X
2
h' jU ( (x)) i hU ( (x)) j' i dx; 8' ; ' 2 L (Y ): 1
2
1
2
2
When this has a bounded inverse, say that the coherent state is a frame. When the orbit of under (X ) is a frame of L (Y ) there is the continuous wavelet transform, 2
W : L (Y ) ,! L (X ) 2
2
de ned by
W ' (x) = h'jU ( (x)) i ; 8' 2 L (Y ): This operator is one-to-one and its range H is complete with respect to the inner-product:
hW 'jW iH = W 'jW A,; L2 X ; ; ' 2 L (Y ): 2
1
2
(
)
Hence there is a unitary isomorphism W : L (Y ) ,! H . 2
3. The setting For the calculations which we will describe here, the ingredients are: G is a noncompact connected semisimple Lie group with nite centre and Cartan involution . K is the corresponding maximal compact subgroup. G = KAN is an Iwasawa decomposition. M is the centralizer of A in K . X = G=N . Y = K=M . U is a certain principal series action of G on L (K=M ), to be de ned below. Assume that (K; M ) is a Gel'fand pair. See Knapp's book for details [5, page 119]. 2
162
CHRISTOPHER MEANEY
4. Decompositions There are Iwasawa projections K : G ! K , A : G ! A, N : G ! N , for which
8g 2 G:
g = K(g)A(g)N(g);
The Haar measure on G is given in terms of that of K and right Haar measure of AN , [5, page 139] with
dg = dk dr (an): The measure on K is normalized so that Z
K
dk = 1:
There is a mapping log : A ! a with
8a 2 A:
exp(log(a)) = a; For each 2 a let
a = e
a
(log( ))
; 8a 2 A:
5. Invariant Integration There is the special functional 2 a determined by the structure of the group G. For f 2 Cc(G) the integral formula for Haar measure on G is Z
G
f (x) dx =
Z Z Z
f (kan) a dndadk: 2
K A N
See [6, Prop. 7.6.4] for details. We can use KA to parametrize G=N and the G-invariant integral on G=N is given by Z
G=N
F (y)dy =
Z Z
F (kaN )a dadk 2
K A
for F 2 Cc(G=N ). Hence, we take the Borel section : G=N ! G to be (kaN ) = ka; 8a 2 A; k 2 K:
PRINCIPAL SERIES AND WAVELETS
163
6. Induced Representations Consider the space of continuous covariant functions: 8 9 f : G ! C continuous < = f (gman) = a,f (g); ; : I(G) = :f : 8g 2 G; m 2 M; a 2 A; n 2 N Left translation by elements of G preserves the property of covariance: , (U (g)f ) (x) = f g, x ; 8g; x 2 G; f 2 I(G): 1
U (g) : I(G) ,! I(G); 8g 2 G: For a covariant function f 2 I(G), f (x) = f (K(x)A(x)N(x)) = A(x), f (K(x)); 8x 2 G: Equip I(G) with the inner product Z
hf jf i = 1
and norm
2
kf k =
The completion of I(G) is
K
Z
f (k)f (k) dk 1
2
jf (k)j dk 2
K
1=2
:
HU = L (K=M ): 2
The action of G on HU is an example of a principal series representation, see section 8.3 of Wallach's book [6]. For our purposes, the essential fact is that U jK is the regular representation of K on a subspace of L (K ). If f 2 L (K=M ), extend it to be an element of HU by assigning f (kan) = a,f (k): Notice that if f 2 L (K=M ), U (g)f (k) = A(g, k),f (K(g, )k); k 2 K; g 2 G: For each g 2 G the action of U (g) extends to a continuous linear operator on HU . It is a unitary representation: 2
2
2
1
1
Z
hU (g)f jU (g)f i = (U (g)f )(k)(U (g)f )(k) dk 1
=
Z
K
2
K
1
2
A(g ,1k ),2 f1 (K(g ,1 k))f2 (K(g ,1k)) dk: = hf1 jf2 i
Lemma 1. The representation (U; HU ) is unitary. When restricted to K , it is the action of K by left translation on L (K=M ). 2
164
CHRISTOPHER MEANEY
7. Fourier analysis on the compact group K We review some basic facts about analysis on compact groups. Let b K be the dual object of K , consisting of a maximal set of inequivalent irreducible unitary representations ( ; V ) of K . For each integrable function f on K there is the Fourier series: X f (x) = d f (x):
2Kb
Convolution with a character is Z f (x) = f (y) tr( (y, ) (x)) dy = tr(fb( ) (x)) 1
K
where the Fourier coecient is Z Z , b f ( ) = f (x) (x ) dx = f (x) (x) dx: 1
K
K
The Fourier coecients are linear transformations fb( ) 2 HomC (V ; V ): Fourier coecients of convolutions are products of Fourier coecients: (f
g)^( ) =
Z Z
f (x)g(x, y) (y, ) dxdy 1
ZK ZK
1
= f (x)g(x, y) (y, xx, ) dxdy K K = bg( )fb( ): De ne left translation on K by , 8x; y 2 K; x f (y ) = f (x y ); and the composition with inversion f _(x) = f (x, ); 8x 2 K: Fourier coecients of left translates satisfy 1
1
1
1
1
(x
f )^ ( ) =
Z
f (x, y) (y, xx, ) dy = fb( ) (x, ) 1
K
1
1
1
Fourier coecients of adjoints satisfy (g_)^( ) = gb( ): The L (K ) inner product can be viewed as a convolution: 2
Z
K
f (x)g(x) dx =
Z
f (x)g_(x, ) dx = f g_(1): 1
K
PRINCIPAL SERIES AND WAVELETS
165
For f; g 2 L (K ), the Fourier series of their convolution is absolutely convergent, see [4], X f g(x) = d f g (x) 2
f and g in L (K ):
2Kb
2
Z
K
f g(x) =
f (x)g(x) dx =
kf k = 2 2
X
2Kb
X
2Kb
X
2Kb
d tr gb( )fb( ) (x) ; d tr(fb( )gb( ));
2
d
fb( )
: 2
In particular, for each 2 Kb , kfb( )k 2 = d kf k : See Appendix D of Hewitt and Ross [4] for details about the norms k k p ; 1 p 1: If h 2 L (K ) then f 7! f h; L (K ) ,! L (K ); is a bounded linear operator which commutes with left translation. Similarly, f 7! h f; L (K ) ,! L (K ); is a bounded linear operator which commutes with right translation. The norm of both of these operators is
sup
bh( )
: 2
2 2
1
2Kb
2
2
2
2
1
8. Homogeneous Spaces Now we return to dealing with functions on K=M , which we identify with right-M -invariant functions on K . For each 2 Kb , let V M = fv 2 V : (m)v = v; 8m 2 M g and P : V ,! V M , the orthogonal projection on to this subspace. Let be the normalized Haar measure on M . Its Fourier coecients are b b( ) = P ; 8 2 K:
166
CHRISTOPHER MEANEY
If f 2 L (K=M ) then b f = f ; =) fb( ) = P fb( ); 8 2 K: We are restricting our attention to the case where (K; M ) is a Gel'fand pair, which means that , b dim V M 1; 8 2 K: Lemma 2. If (K; M ) is a Gel'fand pair and f 2 L (K=M ), then for all 2 Kb , rank(fb( )) 1 and (V M )? ker(fb( ) ): Lemma 3. If (K; M ) is a Gel'fand pair and f 2 L (K=M ), then for all 2 Kb , fb( )fb( ) = kfb( )k 2 P : Lemma 4. If (K; M ) is a Gel'fand pair and f 2 L (K=M ), then for all 2 Kb , kfb( )k p = kfb( )k 2 ; 1 p 1: Lemma 5. If (K; M ) is a Gel'fand pair and h 2 L (K=M ), then the norm of the operator f 7! f h; L (K ) ,! L (K=M ); is o n o np b b b sup kh( )k 2 : 2 K = sup d kh k : 2 K : 1
1
1
2
1
1
2
2
2
, dim V M
In this lemma, if so we need only
= 0 then bh( ) =,0 and M take the supremum over those for which dim V = 1. 9. Admissible Vectors In [2] the unitary representation (U; HU ) of G is said to be squareintegrable modulo N if there is a non-zero vector for which Z Z
jhU (ka)j ij a dadk < 1 2
K A
2
for all 2 HU . Such an is called admissible. Notice that this can be rearranged to say Z Z
U (a)
2
jU (k, ) a dadk < 1 K A for all 2 HU . Recall that U jK is left translation. 1
2
PRINCIPAL SERIES AND WAVELETS
We then nd that Z
jhU (ka) j i j dk 2
K
167
2 Z Z = (U (a)) (x) (kx) dx dk ZK K = (U (a)) _(k) 2 dk
K
= (U (a) ) _ Using the Plancherel formula for this, X
(U (a) ) _ = d tr (U (a))^( ) b( )b( ) (U (a))^( ) 2 2
2 2
=
X
d k(U (a))^ ( )k 2 kb( )k 2 2
2
We arrive at the general version of Antoine and Vandergheynst's criterion for admissibility. Theorem 1. If 2 HU = L (K=M ) has the property that 2
sup
Z
k(U (a))^( )k a da < 1 2
2Kb A
2
2
then is admissible. Since the functions here are right-M -invariant, the only non-zero parts of the Fourier series correspond to those for which P 6= 0. Theorem 2. If 2 HU = L2 (K=M ) is admissible and there are constants 0 < c1 c2 for which
c
Z
1
k(U (a))^ ( )k a da c 2
A
2
2
2
for all 2 Kb with P 6= 0, then the corresponding coherent state is a frame. We can reword this to see that the criterion for to give rise to a frame for L2 (K=M ) is that there are constants 0 < c1 c2 for which
c d 1
Z
A
for all 2 Kb with P 6= 0.
k(U (a)) k a da c ; 2 2
2
2
168
CHRISTOPHER MEANEY
10. Spherical Functions Let Kb M denote the set of those 2 Kb with P 6= 0. For each 2 Kb M de ne the spherical function ' = = : If f 2 L (K=M ) its Fourier series is X d f ' : 1
2Kb M
When K=M = S n, this is the usual spherical harmonic expansion. To use the criterion for a frame, we need estimates on
d
Z
uniformly in 2 Kb M .
A
k(U (a)) ' k a da; 2 2
2
11. Zonal Functions A special case occurs when is bi-M -invariant, since it is then expanded in a series X = d c ' with c = hj' i :
2Kb M
But U (a) is also bi-M -invariant and its expansion is X U (a) = d c (a)'
2Kb M
with
c (a) = hU (a)j' i = jU (a, )' : Since the spherical functions ' are matrix entries of irreducible representations, ( 0 ' ' = ' =d if = 0 0 if 6= ; and k' k = 1=d . Hence, Theorem 2 says that a bi-M -invariant function produces a frame for L (K=M ) when there are positive constants c c for which 1
0
2 2
1
2
2
0
for all 2 Kb M .
Z
jc (a)j a da c 2
A
2
2
PRINCIPAL SERIES AND WAVELETS
169
12. Antoine and Vandergheynst The results in [2] are concerned with the case where: G = SOe(1; 3), K = SO(3), M = SO(2), and K=M =S . A = (0; 1) with multiplication, X = SO(3) A, = 1. Kb M = f0; 1; 2; 3; : : : g, dn = 2n +1, and the spherical functions 'n are normalized ultraspherical polynomials. Suppose we use spherical coordinates (; ) to parametrize S . Proposition 3.4 of [2] states that if 2 L (S ) is admissible and 2
2
2
2
Z 2
(; ) d 6= 0 then gives rise to a frame. This is achieved using the spherical harmonic expansion of U (a) and the asymptotics of the zonal spherical functions, to get the inequality in Theorem 2 above. In [2] there is presented a sucient condition on a function 2 L (S ) so that it satis es the hypotheses of Theorem 1. These are similar to the moment conditions in the Euclidean space setting, see Proposition 7 in [3]. Proposition 3.6 [2] states that if 2 L (S ) satis es Z Z (; ) sin() dd = 0 1 + cos() then it is admissible. 0
2
2
2
2
2
0
0
References 1. J.-P. Antoine and P. Vandergheynst, Wavelets on the -sphere and related manifolds, J. Math. Phys. 39 (1998), no. 8, 3987{4008. 2. , Wavelets on the 2-sphere: a group-theoretical approach, Appl. Comput. Harmon. Anal. 7 (1999), no. 3, 262{291. 3. J. E. Gilbert, R. A. Kunze, and C. Meaney, On derived intertwining norms for the Lorentz group, Representation theory and harmonic analysis (Cincinnati, OH, 1994), Amer. Math. Soc., Providence, RI, 1995, pp. 57{73. 4. Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Springer-Verlag, New York, 1970, Die Grundlehren der mathematischen Wissenschaften, Band 152. 5. Anthony W. Knapp, Representation theory of semisimple groups, Princeton University Press, Princeton, N.J., 1986, An overview based on examples. 6. Nolan R. Wallach, Harmonic analysis on homogeneous spaces, Marcel Dekker Inc., New York, 1973, Pure and Applied Mathematics, No. 19. n
Dept. Mathematics, Macquarie University, North Ryde, NSW 2109, Australia
E-mail address :
[email protected]
SINGULARITIES AND THE WAVE EQUATION ON CONIC SPACES RICHARD B. MELROSE AND JARED WUNSCH
Introducing polar coordinates around a point in Euclidian space reduces the Euclidian metric to the degenerate form (1) dr2 + r2 d!2 where r is the distance from the point and d!2 is the round metric on the sphere. If X is an arbitrary manifold with boundary, the class of conic metrics on X is modeled on this special case. Namely, a conic metric is a Riemannian metric on the interior of X such that for some choice of the de ning function x of the boundary (x 2 C 1(X ) with @X = fx = 0g, x 0, dx 6= 0 on @X ), the metric takes the form g = dx2 + x2 h on X = X n@X; near @X: Here h is a smooth symmetric 2-cotensor on X such that h0 = hj@X is a metric on @X: In fact a general conic metric can be reduced to a form even closer to (1) in terms of an appropriately chosen product decomposition of X near @X; that is, by choice of a smooth di eomorphism F (2) [0; )x @X ,! O X; O an open neighborhood of @X : The normal variable in x 2 [0; ) is then a boundary de ning function, at least locally near @X; and the slices F x=x0 have given di eomorphisms to @X: Now such a product decomposition can be chosen so that (3) F g = dx2 + x2 hx; in x < ; where hx is a family of metrics on @X . This reduced form is closely related to the behavior of geodesics near the boundary. Up to orientation and parameterization there is a unique geodesic reaching the boundary at a given point p: In particular the normal bration of X near @X given by the segments F ([0; ) fpg), p 2 @X , consists of geodesics which hit the boundary, each at the corresponding point p: We shall discuss here the behavior of solutions to the wave equation (4) (Dt2 , )u = 0 on R X 170
SINGULARITIES AND THE WAVE EQUATION ON CONIC SPACES
171
when X is endowed with a conic metric, is the associated (positive) Laplacian on functions, and Dt = ,i@=@t. For simplicity we take X to be compact. It is only really important that @X be compact. Our primary concern is to describe the phenomenon of the propagation of singularities for solutions to (4). To do so it is necessary to understand the behavior of solutions in a way related to the functional analytic domain of : For the moment we simply say that we are dealing with `admissible' solutions. This condition is explained further below. In the interior of X the propagation of singularities, described precisely in terms of the notion of wavefront set, was treated in detail by Hormander ([4]). We paraphrase Hormander's result here as \Singularities travel along null bicharacteristics, which in the case of the wave equation project to time-parameterized geodesics." Thus, in the microlocal sense of singularities described by the wave front set, a bicharacteristic segment, which covers a light ray, either consists completely of singularities for a given solution or the solution has no singularity along it. This quite adequately describes the propagation of singularities except where a light ray hits the boundary at some point and at some time. Here a `splitting' of singularities will usually take place. This is generally called a di ractive e ect. The contrapositive of this e ect can be succinctly stated as follows: \If no singularity reaches the boundary at time t then no singularity leaves at time t:" The point here is that the regularity along any one of the `radial' rays leaving the boundary at a given time is related, in general, to the singularities on all the incoming rays (although there are two separate components, as described below) arriving at the boundary at that time. Thus, even if singularities arrive at the boundary at time t along just one ray, they will in general depart along all rays leaving the boundary at time t: There are, however, some important exceptions to this general spreading of singularities. For instance, if X is a conic manifold with `trivial' conic metric de ned by the blowup of a point in a smooth Riemannian manifold. In this case, admissible solutions are just the lifts of solutions in the usual sense and, because of Hormander's theorem on interior singularities, the singularities are carried outward only on the one ray continuing the incoming ray in the original manifold.
172
RICHARD B. MELROSE AND JARED WUNSCH
For a general conic metric there is a similar notion of the `geometric continuation' of an incoming geodesic which hits the boundary. For a trivial conic metric obtained from a blowup, the boundary metric h0 is the standard metric on the sphere. The geometrically related incoming and outgoing rays hit this sphere at antipodal points; these can also be thought of as the points separated by geodesics of length on the unit sphere. In the case of a general conic metric we may mimic this by de ning the relation (5) G(p) = fq 2 @X ; 9 a geodesic in @X for h0 of length with end points p; qg: In general of course, G(p) is not smooth. Generically it is a hypersurface with Lagrangian singularities; it is always the projection of a smooth Lagrangian relation. A geometric re nement of the di raction result is obtained by considering the order of singularities with respect to Sobolev spaces and an additional `second microlocal' regularity condition. For simplicity suppose that the (admissible) solution u is singular only near @X and only near a single incoming ray hitting the boundary at time t and at the point p: In the past (for t < t) we may suppose that the solution is locally in some Sobolev space H s: Suppose further that the singularities of the solution are not too strongly focused on @X insofar as tangential smoothing raises the overall regularity, that is, for some k; ` > 0, s+` (6) (@X + 1),k u 2 Hloc in t < t near @X; where @X is the Laplacian on @X with respect to the metric h0 , extended to act on a neighborhood of X using the product decomposition (2). Under these two assumptions and the additional requirement that (7) 0 < ` < n2 ; we obtain the following `geometric theorem'. \If an admissible solution is singular only near an incoming ray arriving at @X at time t and (6) and (7) hold, then on outgoing rays with initial point in the complement of G(p), s+`, 21 , (8) 8 > 0 in t > t near @X:" u 2 Hloc When slightly generalized and re ned, as described below, this result applies to the fundamental solution p sinpt
SINGULARITIES AND THE WAVE EQUATION ON CONIC SPACES
p G(p){
173
incoming singularity outgoing regularity
Figure 1. Given the tangential smoothing property,
there is greater regularity on outgoing rays from points not -related to incoming rays carrying singularities. with pole close to @X and with ` < n,1 2 . The di ractive theorem merely tells us that if the pole is speci ed at (x; p) at t = 0, then singularities cannot emanate from @X except at time t = x. On the other hand, while `strong' singularities can emanate from all points in G(p); this geometric theorem tells us that the solution is microlocally more regular on rays starting from @X at t = x but with initial point outside G(p): We can in fact use the conormality of the fundamental solution to obtain a sharper result than (8): the analogue of (8) for the fundamental solution yields H s+l, regularity, i.e. we obtain one-half derivative of improvement over the general case. In the special case in which the metric g takes precisely the `product' form (9) g = dx2 + x2 h(y; dy); near the boundary, Cheeger and Taylor [1, 2] have given an explicit analysis of the fundamental solution constructed by separation of variables. (See also the discussion by Kalka-Meniko [6].) They Sobolev regularity they obtain is the same, and is therefore optimal in general. They also show that the outgoing solution is conormal and compute the precise order. A version of the results of Cheeger-Taylor has been established in the analytic category by Rouleux [14]. Lebeau [7, 8] has also obtained a di ractive theorem in the setting of manifolds with corners in the analytic category. Detailed proofs of the results in this paper will appear in [11]. The rst author acknowledges partial support from NSF grant DMS9625714. The second author was partially supported by an NSF VIGRE instructorship at Columbia University and by an NSF postdoctoral fellowship, and is grateful to
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Alberto Parmeggiani and Cesare Parenti for invaluable conversations. The authors wish to thank the referee for several helpful suggestions.
1. Friedrichs extension To describe the admissibility condition, near the boundary, for solutions to (4) we rst describe the domain of the Laplacian for a conic metric. We take the Friedrichs extension of . By de nition, is associated to the DirichletZ form (10) F (u; v) = hdu; dvig dg ; u; v 2 Cc1(X ) X
Hence, dg is the metric volume form; in this case dg = 'xn,1 dx dh0 near @X; n = dim X ; ' 2 C 1; ' > 0 : The inner product in (10) is that induced, by duality, by the metric on T X : Following Friedrichs we de ne, n o D(1=2 ) = clos Cc1(X ) w.r.t. F (u; u) + kuk2L2g ; whenever X is a compact conic manifold with boundary of dimension n 2: This is a Hilbert space with dense injection D(1=2 ) ,! L2g (X ) so there is a dual injection L2g (x) ,! (D(1=2 ))0: The natural operator : D(1=2 ) ! (D(1=2 ))0 is determined by (u; ')L2g = F (u; ') 8 u; ' 2 D(1=2 ): Then the Friedrichs extension of is the unbounded operator with domain D() = u 2 D(1=2 ); u 2 L2g (x) : It is a self-adjoint, non-negative operator and in this case has discrete spectrum of nite multiplicity. This allows its complex powers to be de ned by reference to an eigenbasis. The real powers are isomorphisms o the null space, which consists precisely of the constants. Each of the powers is therefore a Fredholm map s : D(s) ! L2g (X ) 8 s 2 R : with null space the constants and range the orthocomplement of the constants. The domains form a scale of Hilbert spaces, and D(s) ,! D(t ) is dense 8 s t with D(0) = L2g (X ).
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For our purposes it is also important to note when the domains consist of extendible distributions, i.e. those dual to C_1(X ). This is the case only for s > , n4 and more precisely (11) C_1(X ) ,! D(s) ,! C ,1(X ) are dense inclusions for , n4 < s < n4 : The limits of this range correspond to the occurrence of formal solutions of u = 0: 2. Wave group The Cauchy problem for the wave equation (Dt2 , )u = 0; on R X (12) ujt=0 = u0; Dt ujt=0 = u1 has a unique solution u 2 C 0(R ; D(1=2 )) \ C 1(R ; L2g (x)) 8 (u0; u1) 2 E = E1 = D(1=2 ) L2g (X ) : These ` nite energy solutions' are the main object of study here. More s ,1 ,1 2 generally, with the equation interpreted in C (R ; D( )) the Cauchy problem has a unique solution u 2 C 0(R ; D( s2 )) \ C 1(R ; D( s2 , 12 )) (13) 8 (u0; u1) 2 Es = D( 2s ) D( 2s , 21 ) : The regularity hypothesis on the solution can be weakened to 1 (R ; D( s2 , 21 )) u 2 L2loc (R; D( s2 )) \ Hloc without changing the unique solvability. Notice that these calculations are consistent under decrease of s: Furthermore, partial hypoellipticity in t shows that the solution to (12) satis es ,k (R ; D( s2 + k2 )) 8 k 2 R : (14) u 2 Hloc An admissible solution to the wave equation is one that satis es q p (15) u 2 Hloc (R ; D( 2 )) for some q; p 2 R ,p,2 (R ; D( q2 ,1 )). Such a solution automatically with (4) holding in Hloc satis es (14) for some s: These statements can be reinterpreted in terms of the wave group u u ( t ) 0 (16) U (t) : u 7! D u(t) ; U (t) : Es ! Es 8 s: 1 t
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3. Hormander's theorem Let M be a manifold without boundary. The wave front set of a distribution u 2 C ,1(M ) is a closed subset of the cosphere bundle WF(u) S M: It may be de ned by decay properties of the localized Fourier transform, or the FBI (Fourier{Bros{Iagonitzer) transform, or by testing with pseudodi erential operators. The projection (WF(u)) M is exactly the C 1 singular support, the complement of the largest open subset of M to which u restricts to be C 1. A re ned notion of wavefront set is the Sobolev-based wavefront set, denoted WFs; this is a closed subset of S M , where now the projection is the complement of the largest open subset of M to which u restricts to be H s. If u satis es a linear di erential equation, Pu = 0; then WF(u) (P ) S M when (P ) is the characteristic variety of P , the set on which its (homogeneous) principal symbol, p; vanishes. If p is real then the symplectic structure on T M , or the contact structure on S M; de nes a `bicharacteristic' direction eld VP on S M , tangent to (P ): The integral curves of VP are called bicharacteristics; those lying in (P ) are called null bicharacteristics. Theorem 1 (Hormander). Let P be a (pseudo)-di erential operator with real principal symbol. If Pu = 0 then WF (u) (P ) is a union of maximally extended null bicharacteristics. The same result also holds with WF replaced by WFs for any s: In our case, M = R X so T M = T R T X . The principal symbol of the d'Alembertian is 2 , j j2g , where is the dual variable to t and j jg is the (dual) metric on T X . Then (P ) = + (P ) [ ,(P ) S M when I (P ) = R S X are the disjoint parts of (P ) in > 0 and < 0. In this representation of (P ) the null bicharacteristics are geodesics on X , lifted canonically to S X , with t as ane parameter. Thus, for the wave equation over X , Hormander's theorem does indeed reduce to the informal propagation statement described above. Combined with standard results relating the singularities of the solution to singularities of the initial data, Hormander's theorem applied to the wave equation on a conic manifold yields complete information on
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the behavior of singularities except along bicharacteristics lying above geodesics which hit the boundary. 4. Diffractive theorem On parametrized geodesic segments with an end point on the boundary, the de ning function x is either strictly increasing or strictly decreasing near the boundary. For each sign of and for each t 2 R the bicharacteristics covering such geodesics which hit the boundary at t = t and along which t is increasing (resp. decreasing) as x decreases, form a smooth submanifold of T (ft < tg X ) (resp. T (ft > tg X )). We denote these `radial' surfaces (near @X ) by R;I (t) and R;O (t) (P ) where is the sign of and I; O refers to whether these are `incoming' or `outgoing' and hence, equivalently, whether they lie in t < t or t > t: Theorem 2 (Di ractive theorem). If u is an admissible solution to (4) then for any t 2 R , s 2 R , = , R;I (t) \ WFs(u) = ; ) R;O (t) \ WFs(u) = ;: Here, WFs(u) is the wave front set computed relative to the Sobolev space H s; locally in the interior. This is a precise form of the di ractive result described informally above. Notice that the singularities for di erent signs of are completely decoupled. This does not, however, represent any re nement in terms of propagation along the underlying geometric rays, since all geodesics are covered by bicharacteristics with xed and of either sign. The proof of this result is discussed brie y below in x7. 5. Geometric theorem Consider a geodesic on X which hits the boundary at a point p 2 @X: An open set of perturbations of the geodesic, meaning geodesics starting near some interior point on the geodesic and with initial tangent close to the tangent to the geodesic, will miss the boundary. A limit of such curves as the perturbation vanishes consists of three segments. The rst is the incoming geodesic segment. The second is a geodesic segment in the boundary, of length . The third is the outgoing geodesic from the end point of the boundary segment, which is therefore a point in G(p) as de ned in (5) (see Figure 2). Thus it is reasonable to suppose that, amongst the outgoing bicharacteristics leaving the boundary at time t; those with initial points in G(p) will be more
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Figure 2. A sequence of geodesics nearly missing the
boundary, and the three segments which they approach. closely related to an incoming bicharacteristic with end point p arriving at time t: We call these the geometrically-related bicharacteristics (or geodesics). For instance, if there are incoming singularities on a single ray the singularities on the `non-geometrically-related' outgoing bicharacteristics might be expected to be weaker than the incoming singularity. However, this is not in general the case. To obtain such a geometric re nement of the di raction result we need to impose an extra `nonfocusing' assumption. Theorem 3 (Geometric theorem). Let u be an admissible solution to (4) and let = . Suppose that R;I (t) \ WFs u = ; near @X . Suppose additionally that for some k and 0 < ` < n2 (17) WFs+`(1 + @X ),k u \ R;I (t) = ;: For any 0 < r < ` , 1=2; if no incoming bicharacteristic hitting the boundary at time t at a point in G(p) with sgn = is in WFs+r u, then the outgoing bicharacteristic with initial point p 2 @X and sgn = is not in WFs+r, u for any > 0. If in addition to (17) we have (18) WFs+`(xDx + (t , t)Dt)(1 + @X ),k u \ R;I (t) = ;: then the same conclusion follows for all 0 < r < l. Thus the additional assumption (17) allows regularity on outgoing rays to be deduced from regularity in the incoming geometricallyrelated rays up to the corresponding level above `background' regularity.
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As already noted, this result may be applied to the fundamental solution with initial point near the boundary. If the initial pole of the fundamental solution is suciently close to the boundary then there is a unique short geodesic segment from it to the boundary, arriving at a point p: If t is the length of the segment then, provided t is small enough, (17) and (18) hold with s < , n2 + 1 for any ` < n,1 2 . It follows that on R;O (t); the outgoing set, the fundamental solution is in H 1=2,, for all > 0, microlocally near the non-geometrically related rays, those with end point not in G(p), whereas the general regularity is H , n2 +1, for all > 0. This is a gain of `nearly' n,1 2 derivatives over the background regularity. In this way we extend part of the result of Cheeger and Taylor [1, 2] in the product case (9) to the general conic case. Inspection of the fundamental solution constructed in [1] reveals the `nearly' n,1 2 di erence in smoothness between geometric and non-geometric rays to be sharp. 6. Spherical conormal waves Around a given point q in a compact Riemann manifold there are `spherical' conormal waves which are singular only on the spherical surfaces r = t, for small t of both signs. These just correspond to conormal data at t = 0 at the ( ctive) cone point q. An important example is the fundamental solution, in which case the result follows from Hadamard's construction. In the more general case of a conic manifold with boundary there are similar contracting, and then expanding, conormal waves. Theorem 4. If u is an admissible solution near @X and t = 0 which is conormal to t = ,x for t < 0 then it is conormal to t = x, near the boundary, for small t > 0. These conormal solutions to the wave equation in the general conic case are at the opposite extreme to those considered in the Geometric Theorem above. Namely, they are already smooth in the tangential variables, so no tangential smoothing in the sense of (6) is possible. Further analysis of the structure of these waves shows that the principal symbols undergo a transition at x = 0, the boundary, given by the scattering matrix for the model cone with the same boundary metric. Since this scattering matrix should have full support in general, this provides counterexamples to any extension of the geometric theorem in which the tangential smoothing condition is dropped.
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7. Methods The basic method we use is microlocal, but non-constructive. It is a direct extension of one of the proofs by Hormander of the interior propagation theorem. This `positive' commutator method is itself a microlocalization of the energy method for hyperbolic equations. In it a `test' pseudodi erential operator, A, is applied to the equation and the essential positivity of the symbol of the commutator 1i [P; A] gives a local regularity estimate on the solution. To extend this method to cover behavior of solutions near the boundary we replace the ordinary notions of wavefront set, pseudodi erential operators and microlocalization with versions appropriately adapted to the geometry. When considering the Laplacian itself on the manifold with boundary with conic metric, the appropriate notion is that of a weighted b-pseudodi erential operator (see [13]). This for instance allows the precise description of the domains of the powers of which is used at various points in the argument. However, for the wave operators for the conic Laplacian the appropriate notion corresponds to the `edge' calculus of pseudodi erential operators discussed originally by Mazzeo [9], arising from a ltration of the boundary (see also Schulze [15]). In this case, the manifold with boundary is X R and the bers of the boundary are the surfaces t = const. Thus t is the base variable of the bration. To the edge calculus of pseudodi erential operators, given by microlocalization from the di erential operators generated by xDx, Dy (where the y's are tangential variables) and xDt , we associate a notion of wavefront set. We can prove the propagation theorem analogous to that of Hormander for this `edge' wavefront set. However, in this new sense, Dt2 , is not globally of principal type but rather has two radial surfaces. These correspond to the end points of bicharacteristics arriving at, and leaving from, the boundary. At these surfaces there are restrictions on the propagation results, very closely related to those for scattering Laplacians in [10]. The construction of the test operator A; which away from the radial surfaces is essentially given by owout along the geodesic spray on @X; becomes more delicate at the radial surfaces. Positivity relies on the precise form of the singularity of the Hamilton vector eld there. These propagation estimates form the basis of both the di ractive and geometric theorems. In the former we combine the estimates with a variant of the one-dimensional FBI transform, scaled with respect to the normal variable x. This reduces the di ractive result to an iterative
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application of a uniqueness theorem for the Laplacian on the model, non-compact cone. To obtain the geometric theorem, showing that the outgoing singularities on non-geometrically related rays are weaker than the incoming ones, we use a division theorem. The additional hypothesis of microlocal tangential smoothing is shown to imply that the solution actually lies in a weighted Sobolev space with a higher x weight (hence more `divisible' by x) than is given, a priori, by energy conservation. This allows the microlocal propagation results indicated above to be pushed further at the outgoing radial surface and so yields the extra regularity. 8. Applications and extension The propagation of singularities results of the type discussed above should allow estimates of the spectral counting function as shown originally by Ivrii ([5], see also [12] and [3]). We expect these methods to extend to more complicated geometries, including manifolds with corners and iterated conic spaces. References [1] Je Cheeger and Michael Taylor, On the di raction of waves by conical singularities. I, Comm. Pure Appl. Math. 35 (1982), no. 3, 275{331, MR84h:35091a. , On the di raction of waves by conical singularities. II, Comm. Pure [2] Appl. Math. 35 (1982), no. 4, 487{529, MR84h:35091b. [3] L. Hormander, The analysis of linear partial di erential operators, vol. 3, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985. [4] Lars Hormander, On the existence and the regularity of solutions of linear pseudo-di erential equations, Enseignement Math. (2) 17 (1971), 99{163. MR 48 #9458 [5] V. Ivrii, On the second term in the spectral asymptotics for the LaplaceBeltrami operator on a manifold with boundary, Funct. Anal. Appl. 14 (1980), 98{106. [6] M. Kalka and A. Meniko , The wave equation on a cone, Comm. Partial Differential Equations 7 (1982), no. 3, 223{278, MR83j:58110. [7] G. Lebeau, Propagation des ondes dans les varietes a coins, Seminaire sur les E quations aux Derivees Partielles, 1995{1996, E cole Polytech., Palaiseau, 1996, MR98m:58137, pp. Exp. No. XVI, 20. [8] Gilles Lebeau, Propagation des ondes dans les varietes a coins, Ann. Sci. E cole Norm. Sup. (4) 30 (1997), no. 4, 429{497, MR98d:58183. [9] R. Mazzeo, Elliptic theory of di erential edge operators I, Comm. in P.D.E. 16 (1991), 1615{1664. [10] R.B. Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Spectral and scattering theory (Sanda, 1992) (M. Ikawa, ed.), Marcel Dekker, 1994, pp. 85{130. [11] R.B. Melrose and J. Wunsch, Propagation of singularities for the wave equation on conic manifolds, In preparation.
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[12] Richard Melrose, The trace of the wave group, Microlocal analysis (Boulder, Colo., 1983), Amer. Math. Soc., Providence, R.I., 1984, pp. 127{167. MR 86f:35144 [13] Richard B. Melrose, The Atiyah-Patodi-Singer index theorem, A K Peters Ltd., Wellesley, MA, 1993. MR 96g:58180 [14] Michel Rouleux, Di raction analytique sur une variete a singularite conique, Comm. Partial Di erential Equations 11 (1986), no. 9, 947{988. MR 88g:58182 [15] B.-W. Schulze, Boundary value problems and edge pseudo-di erential operators, Microlocal analysis and spectral theory (Lucca, 1996), Kluwer Acad. Publ., Dordrecht, 1997, pp. 165{226. MR 98d:47112 Department of Mathematics, MIT, Cambridge MA 02139 Department of Mathematics, SUNY at Stony Brook, Stony Brook NY 11794
SOME REMARKS ON OSCILLATORY INTEGRALS GERD MOCKENHAUPT
1. Introduction The purpose of this note is to describe some results about oscillatory integral operators. Speci cally we are interested in bounds in Lebesgue spaces of operators given by
T f (x) =
Z
Rk
ei'(x;) f ( ) d;
with '(x; ) a real-valued smooth function on Rn Rk ; k n. Obviously T is bounded as maps from Lqcomp to Lploc. What is of interest here is the dependence of the norm for increasing . This will of course depend on the conditions we put on the phase function '. To guarantee that ' lives on an open subset of Rn Rk it is natural to start with the condition (1) rank d dx' = k; x 2 Rn and 2 Rk : We will assume this condition throughout this note. For work related to weaker assumptions see, e.g. [21] and [18]. One of the questions we will ask is: What is the optimal (q; p)-range for which the operator T has norm of order ,n=p? In particular we would like to understand how this range will depend on k. To put things in perspective let us begin by describing what is known for the case k = n: A model phase function here is '(x; ) = x , for x; 2 Rn. Then T is a localized version of the Fourier transform and p q the (Lcomp; Lloc)-boundedness properties are covered by the Hausdor Young inequality. For general phase functions satisfying (1) the L2 theory of Fourier integral operators gives kjTkjLqcomp!Lploc C ,n=p; with p = q0 2 the dual exponent of q, i.e. 1=q0 + 1=q = 1. Next we consider the case k = n , 1: A basic result was obtained by E. M. Stein in the sixties. He discovered, for n 2, that the Fourier 1991 Mathematics Subject Classi cation. 42B15. Key words and phrases. oscillatory integrals, restriction theorems. The support of the Australian Research Council is greatfully acknowledged. 183
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GERD MOCKENHAUPT
transform has the following restriction property: For the unit sphere S n,1 in Rn and d a rotationally invariant measure
Z
jfb( )j2 d( ) C kf k2Lp0 (Rn ); for some p0 > 1. By localizing to a ball of radius in Rn, the dual of this (2)
S n,1
inequality states that the operator T with phase function '(x; ) = x ( ), where : U ! S n,1 parameterizes a coordinate neighborhood of the unit sphere in Rn, has norm of order ,n=p as a map from L2 (U ) to Lploc for some p < 1. Improvements on the range of exponents p were made by P. Tomas [26] and E.M. Stein. Moreover, it was shown by E.M. Stein (see [24]) that for nonlinear phase functions ' the norm of T has order ,n=p as an operator from L2comp to Lploc for p 2(n + 1)=(n , 1), provided ' satis es the following curvature condition: for each x 2 Rn the hypersurface parameterized by (3) 7! rx'(x; ) has nonvanishing Gaussian curvature. This (L2 ; Lp)-result is sharp in the sense that p = 2(n + 1)=(n , 1) is critical. Moreover, due to an example of J. Bourgain [2, 4] under the conditions (3) and (1) Stein's result can not be improved in case n is odd if we require q = 1 (see also [15]). 1 However under some further p conditions, remarkable improvements have been made by J. Bourgain [1]. His method, which led to further improvements in [27, 5] 2 n p = k and [25], showed in particular for the situation of the QQ p = 4n,k 2k unit sphere described above, QQ QQ that for certain exponents Q p less then the critical L2 Q1 q=2 q exponent 2(n +1)=(n , 1)n that Figure (p; q)-range for k = n and k = n , 1. kTkL1(U )!Lploc C , p : It is expected that the (q; p)-range for which this inequality holds is determined by: p > 2n=k and p (2n , k)=kq0 (see Figure). For n = 2 the norm of T is essentially well understood due to work of L. Carleson and P. Sjolin [6] provided the curvature condition (1) and (3) are satis ed. We note that for the expected bounds the crucial point is to understand for the operators T the (L1; Lp)-bounds. Our main concern here are the cases k < n , 1. There have been some results in the past addressing the problem of (Lqcomp; Lploc)-bounds :
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for oscillatory integral operators in these cases. However, these results mainly discussed the cases k = 1; n , 2 or n=2 for n even (see e.g. [7], [8], [12], [11], [19], [20]). For di erent k some results are obtained in [10] and [17]. A natural question which we ask here is the following: Suppose (1) holds. Under which conditions on the phasenfunction ' does T map Lqcomp(Rk ) to Lploc(Rn) with norm of order , p in the full range (4) p 2n k, k q0 and p > 2kn for k < n? One of our results will be that we can expect these optimal bounds only when k n=2. We will also see that in some situations where the phase function is linear in the x-variables an analogue of the SteinTomas result holds, i.e. optimal (L2 ; Lp)-bounds hold, but the (L1; Lp)bounds fail to hold in the range given in (4). This apparently appears only if k < n , 2. We should mention that one of the main diculties which distinguishes the case k < n , 1 from k = n , 1 lies in the fact that, although a stationary phase argument shows that for 2 C 1(Rk ) and most x 2 Rn the decay of T (x) is of order ,k=2, in general isotropic bounds for Tf (x) decay slower (see e.g. [9]). 2. A necessary curvature condition Here we derive a necessary condition on the phase function ' such that T is bounded in the full range described in the above gure (for k n). First we observe that if T has norm of order ,n=p, then for each x0 the operator with phase function 'R(x; ) = R ('(x0 +x=R; ), '(x0; )) satis es the same bounds uniformly in R > 0. Hence, the operator with the linearized phase function (x; ) 7! x rx'(x0 ; ) has the same bounds. By reparameterizing the k-dimensional submanifold 7! rx'(x0 ; ) over the tangent plane at a given point using translation invariance we may assume that x rx'(x0 ; ) has the form (x1 ; x2 ( )), with (0) and d (0) both vanishing. A further scaling argument {replacing x1 ! Rx1 ; x2 ! R2x2 and ! =R and letting R ! 1{ shows that the phase function x1 + x2 ~( ), here ~ is the second order part of the Taylor expansion of , gives rise to an operator Z ~ (x) = Tf ei( x +x ~() ) f ( ) e,jj =2 d; Rk
1
2
2
which is bounded from Lq (Rk ) to Lp(Rn) for (q; p) on the line p = (2n , k)=kq0 provided that T has norm of order ,n=p on this line.
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Write
x2 ~( ) = 21 Q(x2 ); P ,k x B and B 2 Sym(k), where Sym(k) denotes with Q(x2 ) = nj=1 2;j j j the space of symmetric matrices on Rk . To emphasize the dependence on Q the operator T~ will be denoted by TQ in the following and we refer to the submanifolds parameterized by H : Rk 3 7! (; B1 ; : : : ; Bn,k ) 2 Rn as the associated quadratic submanifold MQ. If TQ maps L1(Rk ) to Lp(Rn), then in particular for the constant function 1 we have G = T~Q1 2 Lp. A computation gives (5)
kGkpp = C
Z
Rn,k
j det(E + iQ(x2 ))j,p=2+1 dx2 ;
here E denotes the unit matrix in Sym(k). To ensure that the above integral is nite for some p < 1 we need that the symmetric matrices Bj ; 1 j n , k, are linearly independent which requires that n k(k + 3)=2. To nd a further restriction we show Proposition 2.1. If the function G above is in Lp(Rn), for all p > 2n=k, then Z (6) j det Q(x)j, d(x) < 1 for all < n ,k k ; S n,k, with the uniform measure on the unit sphere S n,k,1. Proof. To see this we use polar coordinates in (5) and write x2 = ry , and r = jxj. Then j det(E + iQ(x))j2 = det(E + Q(x)2 = 1 + r2 c21 + : : : c2k,1r2k,2 + det Q(y)2r2k : Suppose that supj;y jcj (y)j c and let L(y) = maxf1; c=j det Q(y)jg. Then we get the following lower bound on kGkpp for p = 2n=k + 2, > 0: Z Z1 rn,k,1 jrk det Q(y)j(n,k)=k+ drd(y); 1
S n,k,1 L(y)
which evaluates to (6) by integrating the inner integral. As a consequence we show: Corollary 2.2. Suppose the function G de ned above is in Lp (Rn) for all p > 2n=k. Then the following hold: If k is odd, then k n2 .
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If k is even and the subspace fQ(x)jx 2 Rn,k g intersects the cone of positive de nite matrices in Sym(k), then k n=2.
Proof. The idea here is to nd a hypersurface on the unit sphere in Rn,k where the function det Q vanishes at least of order 1. Assuming k < n= R 2, i.e. (n , k)=k > 1, Proposition (2.1) implies that the integral Sn,k, j det Qj,1 d is nite. Since the polynomial det Q(x) = det(x1 B1 + + xn,k Bn,k ), with Bi 2 Sym(k), is homogenous of degree k for some power > 0 the function jxj j det Q(x)j,1 must be integrable over the unit ball in Rn,k . We can assume that det Q(x) does not vanish identically and that B1 is a diagonal matrix with entries 1. If k is odd we can write locally det Q(x) = (x1 , '(x2 ; : : : ; xn,k )) (x) where '; are real continuous functions and '(0) = 0. Hence , for all > 0, j det Qj,1 is not locally integrable on the unit ball in Rn,k and therefore k n=2. To show the second part we may assume that B1 = E . Then x1 ! Q(x1 ; x2 ; : : : ; xn,k ) is the characteristic polynomial of the symmetric matrix Q(0; x2; : : : ; xn,k ) and therefore has only real zeros. So again det Q(x) = (x1 , '(x2 ; : : : ; xn,k )) (x). As before we nd that k has to be n=2. The condition in the proposition above may be phrased in an invariant way. Consider the submanifold M parameterized by 7! rx'(x0; ) and x a point P = rx'(x0 ; 0). We assume that M carries the induced Euclidean metric. Let NP (M ) be the normal plane at P 2 M , TP (M ) be the tangent plane at P , v 2 NP (M ) and let GP (v) be the Gaussian curvature at P of the orthogonal projection of M (along v) into Rv TP (M ). Then (6) states that 1
(7)
Z
S n,k,1 Np (M )
jGP (v)j, d(v) < 1 for all < n ,k k ;
where denotes a nontrivial rotationally invariant measure on the unit sphere in NP (M ).
\
3. Restriction to quadratic submanifolds In the following we show some positive results for the operators TQ. We write TQf = f dQ, where dQ is the measure on Rn with support on MQ de ned by
Q(f ) =
R Theorem 3.3. If TQ is bounded
Z
f (; H ( )) e,jj =2 d: 2
Rk
, S n,k,1 j det Q(x)j d (x) < 1 for 2 k p n from L (R ) to L (R ) for p 2 2nk,k .
= n,k k . Then
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GERD MOCKENHAUPT
Proof. It is enough to show (and in fact equivalent) that the composition TQTQ f = dc f maps the dual space Lp0 (Rn) into Lp(Rn) for p 2 2nk,k . Our strategy is now to de ne an analytic family Tz which evaluates at z = 0 to TQ and is bounded from L1 to L1 on the line
0; we nd, using polar cz k1: coordinates x2 = ry; r = jx2 j the following bound for kK Z Z 1 (1 + rk j det Q(y)j)z iry n , k , 1 C supn,k e r dr dy ,( n , k + kz ) n , k , 2R S 0 1
2
1
R
2
2
1
1
2
1
1
Since we are assuming Sn,k, j det Q(y)j, n,k k d(y) < 1; we see that the above integral is bounded by a constant times Z1 1 sup j ,(n , k + kz) rn,k,1(1 + rk )z eixr dr j: x2R 0 n , k On the line
SOME REMARKS ON OSCILLATORY INTEGRALS
189
cancels with the Gamma function in front of the last integral. Hence jKcz j is bounded. We note that if we would have been working with the analytic family K~ z (x1 ; x2) = ,(n , 1k + kz) det(E + Q(x2 )2)z=2 dc(x1; x2 ) then the above method gives the following R Corollary 3.4. If Sn,k, j det Q(x)j, d(x) < 1 for all < n,k k . Then TQ is bounded from L2 (Rk ) to Lp (Rn) for p > 2 2nk,k . 1
Arguing similarly one can show that if for a suitable polynomial p(z) the -distributions
z (f ) = p(z)
Z
Rn,k
j det Q(y)jz f (y) dy;
has a bounded Fourier transform on the line <(z) = , n,k k then TQ is bounded from L2 (Rk ) to Lp(Rn) (p has only to annihilate nitely many poles of bz ). Using this observation one can show that in certain cases one has optimal (L2 ; Lp)-bounds for TQ, although the (L1; Lp)bounds fail to hold for some p > 2n=k. We provide a few examples in the following. First we de ne for (x; X ) 2 Rk Sym(k) = Rn with n = k(k2+3) (9)
Tf (x; X ) =
Z
Rk
ei(x+X) f ( ) e,jj =2 d: 2
Then we have the following theorem, whose rst part was independently shown in [10] and for the special case k = 2 in [7]. Theorem 3.5. The operator T has the following properties: (1) T is bounded from L2 (Rk ) to Lp(Rn) i p 2 2nk,k . (2) T is unbounded as an operator from L1(Rk ) to Lp(Rn) for p 2(k + 1) (= 2kn + k , 1). For the proof we will need the Fourier transform of j det X jz ; X 2 Sym(k); z 2 C. This has been computed rst by T. Shitani and more recently by Faraut and Satake [13] using the theory of Jordan algebras. To state the result, we note rst that Sym(k) \ GL(k; R) decomposes under the operation (g; X ) ! gXgt into k +1 GL(k; R)-orbits, j ; j = 0; : : : ; k, where j is the cone of symmetric matrices of signature (k , j; j ). Let 0 be the orbit of the unit matrix E 2 Sym(k). Associated
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GERD MOCKENHAUPT
to 0 is the Gamma function
e,trace(X ) (det X )s, k dX
k k, Y ,(s , j ,2 1 ): = (2) 0j k
, (s) = 0
Z
+1 2
0
(
4
1)
For 0 i k we de ne Zeta distributions
i(f; s) =
Z
i
f (X ) j det X js dX:
The poles here lie on the arithmetic progression 12 Z \ (,1; ,1]. We have X i(f;b s , k +2 1 ) = (2),k(k+1)=2 eiks=2 , (s) ui;j (s) j (f; ,s); 0ik 0
where ui;j is a polynomial of degree k in e,is. Putting s = k+1 2 (1 , z ), then it easily follows that the Fourier transform of 1 1 k +1 , ( 2 (1 , z)) j det X jz k is a bounded function in X 2 Sym(k) on the imaginary line 2(k +1). In fact, since kT 1kpp = C Sym(k) j det(E + iX )j,p=2+1 dX , we nd using generalized polar coordinates +1 2
0
Z
Y
Rk 1j k
j1 + ij j,p=2+1
Y
1i<j k
ji , j j d1 : : : dk < 1
Now, the worst decay of the integrant is along the coordinate axes. Checking exponents it follows that the last integral is nite if and only if p > 2(k + 1). We remark that one can show that the operator (9) is a bounded operator from L1(Rk ) to L2k+2(BR ) with norm of order (log R) k , where BR is a ball of radius R in Rn (note that 2k + 2 is an even integer). As a second example we consider for m > 1 the set M (m; C) of complex m m-matrices which we might consider as a real subspace 1 2 +2
SOME REMARKS ON OSCILLATORY INTEGRALS
191
of Sym(4m) via the following real linear map 0 0 0 X Y1 B 0 0 ,Y X CC 2 Sym(4m); Q : M (m; C) 3 Z = X + iY ! B @tX ,tY 0 0 A tY tX 0 0 where X; Y denote the real resp. imaginary part of Z . Note that for 2 C we have det(E + iQ(Z )) = det(2E + Z Z )2. It has been shown by E.M. Stein [23] that the Fourier transform of the Zeta distribution
(f; s) = 1
(s)
Z
M (m;C)
j det Z js f (Z )dZ
j det Z j,s,2m, where (s) = (s) (s , 2) : : : (s ,
is given by the 2m + 2), (s) = ,(, 2s )=,( 2+2 s ). Let k = 4m and de ne for (x; Z ) 2 Rk M (m; C) = Rn, with n = 2m(m + 2), the oscillatory integral operator Z Tf (x; Z ) = ei(x+Q(Z ))) f ( ) e,jj =2 d: 2
Then we have
Rk
Theorem 3.6. The operator T has the following properties: (1) T is bounded from L2 (Rk ) to Lp(Rn) i p 2 2nk,k . (2) T is unbounded as an operator from L1(Rk ) to Lp(Rn) for p 2m + 1 (= 2kn + k4 , 1).
Using polar coordinates associated to the Cartan decomposition corresponding to the symmetric space SU (n; n)=S (U (n) U (n)) it is not hard to check that we haveQT 1 2 Lp i Z h1 : : : hm 1i<jm(h2i , h2j )2 Q (1 + h2)p,2 dh1 : : : dhm Rm i 1im is nite, i.e. p > 2m +1. This con rms the second part of the theorem. These examples suggest that sharp L2-restriction estimates should hold for most quadratic submanifolds. It would be interesting to nd out for which sets inside Sym(k)n,k for which sharp L2 -restriction fails (so far we have only some insight in the case n , k = 2; 3). In the above examples k was always < n=2. However, there even in case k = n , 3 examples for which we have optimal (L2; Lp)-bounds but the (L1; Lp)-bounds fail for some p > 2n=k: An example is provided by Q(x) = (x1 + x3 )n2,3 + x1 (12 + 32 + + n2,5)x3 (22 + 42 + : : : n2,4) + 2x2(12 + 34 + + n,5n,4)
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GERD MOCKENHAUPT
It can be shown that the Fourier transform of the corresponding distribution is essentially a cone multiplier of order 3=(2n , 6) hence bounded function. But we do not have (L1; Lp)-boundedness for all p > 2n=(n , 3). To see this one has to check when j det(E + iQ(x))j,p=2+1 is integrable, where Q is the Hessian of the quadratic form ! q(x; ).pNow, E + iQ(x1 , x3 ; x2; x1 + x3 ) has eigenvalues 1+2ix1; 1+ i(x1 x22 + x23) and we nd using polar coordinates in the x2; x3 -variables that the L1 -norm of j det(E + iQ(x))j,p=2+1 is bounded from below by a multiple of Z Z r dr 1 p ,1 p dx1: jx ,rj1 (1 + jx + rj) n, (1 + jx , rj) n, ,1 x >0 (1 + jx1 j) 1 1 The last integral is nite i p2 , 1 + n,2 4 ( p2 , 1) > 2, i.e. p > 2 nn+2 ,2 . For more details and a description of how this examples arises in the context of nonregular orbits under certain Lie group actions we refer to [16]. Finally we mention the following theorem for the case k = n , 2 (see [16] and [7]). Theorem 3.7. If B1 ; B2 2 Sym(n , 2) are linear independent then for the operator TQ corresponding to Q(x1 ; x2 ) = x1 B1 + x2 B2 the following statements imply each other (1) TQ is bounded from L2 (Rk ) to Lp(Rn) for p 2 nn+2 ,2 . 2 n p n (2) TR Q1 2 L (R ) for p > n,2 . (3) S j det Q(x)j, < 1 for < n,2 2 . 2
1
1
2
4
2
4
2
1
References [1] J. Bourgain. Besicovitch type maximal operators and applications to Fourier analysis. Geom. and Func. Anal, 1:147{187, 1991. [2] J. Bourgain. p estimates for oscillatory integrals in several variables. Geom. and Func. Anal, 1:321{374, 1991. [3] J. Bourgain. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Geom. and Func. Anal., 3(2):107{262, 1993. [4] J. Bourgain. Some new estimates on oscillatory integrals. Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), 83{112, Princeton Math. Ser., 42, Princeton Univ. Press. [5] J. Bourgain. On the dimension of Kakeya sets and related maximal inequalities Geom. and Func. Anal., 9(2):256{282, 1999. [6] L. Carleson and P. Sjolin. Oscillatory integrals and a multiplier problem for the disc Studia Math., 44:287{299, 1972. [7] M. Christ. Restriction of the Fourier transform to submanifolds of low codimension. Thesis, University of Chicago, 1982. [8] M. Christ. On the restriction of the Fourier transform to curves: endpoint results and degenerate curves. Trans. Amer. Math. Soc., 287:223{238, 1985. L
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[9] J.J. Duistermaat. Oscillatory integrals, Lagrange immersions and unfoldings of singularities. Comm. Pure and Appl. Math., 27:207{281, 1974. [10] L. de Carli and A. Iosevich. Some sharp restriction theorems for homogeneous manifolds preprint [11] S.W. Drury. Restriction of Fourier transforms to curves. Ann. Inst. Fourier, Grenoble, 35(1):117{123, 1985. [12] S.W. Drury and K. Guo. Some remarks on the restriction of the Fourier transform to surfaces. Math. Proc. Cambridge Philos. Soc., 113(1):153{159, 1993. [13] J. Faraut and I. Satake. The functional equation of Zeta distributions associated with formally real Jordan algebras. Tohoku Math. Jour., 36:469{482, 1984. [14] A. Greenleaf and A. Seeger. Fourier integral operators with fold singularities. J. Reine Angew. Math., 455:35{56, 1994. [15] W. Minicozzi and C. Sogge. Negative results for Nikodym maximal functions and related oscillatory integrals in curved spaces. Math. Res. Lett., 4(2-3):221237, 1998. [16] G. Mockenhaupt, Bounds in Lebesgue spaces of oscillatory integrals, Habilitationschrift, Universitat Siegen, 1996. [17] G. Mockenhaupt. A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc., 25(1):31{36, 1991. [18] D.H. Phong and E.M. Stein. The Newton polyhedron and oscillatory integral operators. Acta Math., 179(1):105{152, 1997. [19] E. Prestini. Restriction theorems for the Fourier transform to some manifolds in Rn . Proc. of Symp. in Pure Math., 35:101{109, 1979. [20] A. Ruiz. On the restriction of Fourier transforms to curves. Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II, 186{212, Wadsworth Math. Ser., Wadsworth, Belmont, Calif., 1983. [21] A. Seeger. Degenerate Fourier integral operators in the plane. Duke Math. J., 71(3):685{745, 1993. [22] A. Seeger and C.D. Sogge. Bounds for eigenfunctions of di erential operators. Indiana Univ. Math. J., 38(3):669{682, 1989. [23] E.M. Stein. Analysis on matrix spaces and some new representations of ( C ). Ann. Math., 86:461{490, 1967 [24] E.M. Stein. Harmonic Analysis: Real-variable methode, orthogonality and oscillatory integrals. Princeton Math. Ser., 43, Princeton Univ. Press, 1993. [25] T. Tao, A. Vargas and L. Vega. A bilinear approach to the restriction and Kakeya conjectures. J. Amer. Math. Soc., 11:967{1000, 1998. [26] P.A. Tomas. A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc., 81:477{478, 1975. [27] T.H. Wol . Recent work connected with the Kakeya problem. Prospects in Mathematics (Princeton, NJ, 1996), Amer. Math. Soc., Providence, RI, 1999. S L n;
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
E-mail address :
[email protected]
YET ANOTHER CONSTRUCTION OF THE CENTRAL EXTENSION OF THE LOOP GROUP MICHAEL K. MURRAY AND DANIEL STEVENSON Abstract. We give a characterisation of central extensions of a Lie group G by C in terms of a di erential two-form on G and a
di erential one-form on G G. This is applied to the case of the central extension of the loop group.
1. Introduction Let G and A be groups. A central extension of G by A is another group G^ and a homomorphism : G^ ! G whose kernel is isomorphic to A and in the center of G^ . Note that because A is in the center of G^ it is necessarily abelian. We will be interested ultimately in the case that G = (K ) the loop group of all smooth maps from the circle S 1 to a Lie group K with pointwise multiplication but the theory developed applies to any Lie group G. 2. Central extension of groups Consider rst the case when G is just a group and ignore questions of continuity or di erentiability. In this case we can choose a section of the map . That is a map s : G ! G^ such that (s(g)) = g for every g 2 G. From this section we can construct a bijection : A G ! G^ by (g; a) = (a)s(g) where : A ! G^ is the identi cation of A with the kernel of . So we know that as a set G^ is just the product A G. However as a group G^ is not generally a product. To see what it is note that (s(g)s(h)) = (s(g))(s(h)) = gh = (s(gh)) so that s(g)s(h) = c(g; h)s(gh) where c : G G ! A. The bijection : A G ! G^ induces a product on A G for which is a homomorphism. To calculate this The rst author acknowledges the support of the Australian Research Council. The second author acknowledges the support of the Australian Research Council. 194
CENTRAL EXTENSION
195
product we note that (a; g) (b; h) = (a)s(g)(b)s(h) = (ab)s(g)s(h) = (ab)c(gh)s(gh): Hence the product on A G is given by (a; g) ? (b; h) = (abc(g; h)gh) and the map is a group isomorphism between G^ and A G with this product. Notice that if we choose a di erent section s~ then s~ = sh were h : G ! A. It is straightforward to check that if we pick any c : G G ! A and de ne a product on A G by (a; g) ? (b; h) = (abc(g; h)gh) then this is an associative product if and only if c satis es the cocycle condition c(g; h)c(gh; k) = c(g; hk)c(h; k) for all g, h and k in G. If we choose a di erent section s~ then we must have s~ = sd for some d : G ! A. If c~ is the cocycle determined by s~ then a calculation shows that (1) c(g; h) = c~(g; h)d(gh)d(g),1d(h),1: We have now essentially shown that all central extensions of G by A are determined by cocycles c modulo identifying two that satisfy the condition (1). Let us recast this result in a form that we will see again in this talk. De ne maps di : Gp+1 ! Gp by (2) 8 > i = 0; <(g2; : : : ; gp+1); di(g1; : : : ; gp+1) = >(g1; : : : ; gi,1gi; gi+1; : : : ; gp+1); 1 i p , 1; :(g1; : : : ; gp); i = p: If M p(G; A) = Map(Gp; A) then we de ne : M p(G; A) ! M p,1 (G; A) by (c) = (c d1 )(c d2),1(c d3) : : : . This satis es 2 = 1 and de nes a complex M 0 (G; A) ! M 1 (G; A) ! M 2 (G; A) ! : : : The cocycle condition can be rewritten as (c) = 1 and the condition that two cocycles give rise to the same central extension is that c = c~(d). If we de ne : M p(G; A) ! M p,1(G; A) H p(G; A) = kernel image : M p+1 (G; A) ! M p (G; A)
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MICHAEL K. MURRAY AND DANIEL STEVENSON
then we have shown that central extensions of G by A are classi ed by H 2(G; A). 3. Central extensions of Lie groups In the case that G is a topological or Lie group it is well-known that there are interesting central extensions for which no continuous or di erentiable section exists. For example consider the central extension Z2 ! SU (2) = Spin(3) ! SO (3) of the three dimensional orthogonal group SO(3) by its double cover Spin(3). Here SU (2) is known to be the three sphere but if a section existed then we would have SU (2) homeomorphic to Z2 SO(3) and hence disconnected. From now on we will concentrate on the case when A = C . Then G^ ! G can be thought of as a C principal bundle and a section will only exist if this bundle is trivial. The structure of the central extension as a C bundle is important in what follows so we digress to discuss them in more detail. 3.1. C bundles. Let P ! X be a C bundle over a manifold X . We denote the bre of P over x 2 X by Px. Recall [1] that if P is a C bundle over a manifold X we can de ne the dual bundle P as the same space P but with the action p g = (pg,1) and, that if Q is another such bundle, we can de ne the product bundle P Q by (P Q)x = (Px Qx )=C where C acts by (p; q)w = (pw; qw,1). We denote an element of P Q by p q with the understanding that (pw) q = p (qw) = (p q)w for w 2 C . It is straightforward to check that P P is canonically trivialised by the section x 7! p p where p is any point in Px: If P and Q are C bundles on X with connections P and Q then P Q has an induced connection we denote by P Q. The curvature of this connection is RP + RQ where RP and RQ are the curvatures of P and Q respectively. The bundle P has an induced connection whose curvature is ,RP . Recall the maps di : Gp ! Gp,1 de ned by (2). If P ! Gp is a C bundle then we can de ne a C bundle over Gp+1 denoted (P ) by (P ) = 1,1 (P ) 2,1(P ) 3,1 (P ) : : : : If s is a section of P then it de nes (s) a section of (P ) and if is a connection on P with curvature R it de nes a connection on (P ) which we denote by (). To de ne the curvature of () let us denote
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by q (Gp) the space of all di erentiable q forms on Gp. Then we de ne a map (3) : q (Gp) ! q (Gp+1) P by = pi=0 di , the alternating sum of pull-backs by the various maps di : Gp+1 ! Gp. Then the curvature of () is (R). If we consider ((P )) it is a product of factors and every factor occurs with its dual so ((P )) is canonically trivial. If s is a section of P then under this identi cation (s) = 1 and moreover if is a connection on P then () is the at connection on (P ) with respect to ((s)). 4. Central extensions Let G be a Lie group and consider a central extension ^! C ! G G: Following Brylinski and McLaughlin [2] we think of this as a C bundle G^ ! G with a product M : G^ G^ ! G^ covering the product m = d1 : G G ! G. Because this is a central extension we must have that M (pz; qw) = M (p; q)zw for any p; q 2 P and z; w 2 C . This means we have a section s of (P ) given by s(g; h) = p M (p; q) q for any p 2 Pg and q 2 Ph. This is well-de ned as pw M (pw; qz) qz = pw M (p; q)(wz),1 qz = p M (p; q) q. Conversely any such section gives rise to an M . Of course we need an associative product and it can be shown that M being associative is equivalent to (s) = 1. To actually make G^ into a group we need more than multiplication we need an identity e^ 2 G^ and an inverse map. It is straightforward to check that if e 2 G is the identity then, because M : G^ e G^ e ! G^ e, there is a unique e^ 2 G^ e such that M (^e; e^) = e^. It is also straightforward to deduce the existence of a unique inverse. Hence we have the result from [2] that a central extension of G is a C bundle P ! G together with a section s of (P ) ! G G such that (s) = 1. In [2] this is phrased in terms of simplicial line bundles. Note that this is a kind of cohomology result analogous to that in the rst section. We have an object (in this case a C bundle) and of the object is `zero' i.e. trivial as a C bundle. For our purposes we need to phrase this result in terms of di erential forms. We call a connection for G^ ! G, thought of as a C bundle, a connection for the central extension. An isomorphism of
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MICHAEL K. MURRAY AND DANIEL STEVENSON
central extensions with connection is an isomorphism of bundles with connection which is a group isomorphism on the total space G^ . Denote by C (G) the set of all isomorphism classes of central extensions of G with connection. Let 2 1 (G^ ) be a connection on the bundle G^ ! G and consider the tensor product connection (). Let = s(()). We then have that ( ) = ((s))(()) = (1)(2()) =0 as 2 () is the at connection on 2(P ). Also d = s (d()) = (R). Let ,(G) denote the set of all pairs ( ; R) where R is a closed, 2i integral, two form on G and is a one-form on G G with (R) = d and ( ) = 0. We have constructed a map C (G) ! ,(G). In the next section we construct an inverse to this map by showing how to de ne a central extension from a pair ( ; R). For now notice that isomorphic central extensions with connection clearly give rise to the same ( ; R) and that if we vary the connection, which is only possible by adding on the pullback of a one-form from G, then we change ( ; R) to ( +(); R+d). 4.1. Constructing the central extension. Recall that given R we can nd a principal C bundle P ! G with connection and curvature R which is unique up to isomorphism. It is a standard result in the theory of bundles that if P ! X is a bundle with connection which is at and 1 (X ) = 0 then P has a section s : X ! P such that s() = 0. Such a section is not unique of course it can be multiplied by a (constant) element of C . As our interest is in the loop group G which satis es 1 (G) = 0 we shall assume, from now on, that 1 (G) = 0. Consider now our pair (R; ) and the bundle P . As (R) = d we have that the connection (w) , ( ) on (P ) ! G G is at and hence (as 1(GG) = 0) we can nd a section s such that s((w)) = . The section s de nes a multiplication by s(p; q) = p M (p; q) q: Consider now (s) this satis es (s)(((w))) = (s ((w)) = ( ) = 0. On the other hand the canonical section 1 of ((P )) also satis es this so they di er by a constant element of the group. This means that there is a w 2 C such that for any p, q and r we must have M (M (p; q); r) = wM (p; M (q; r)):
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Choose p 2 G^ e where e is the identity in G. Then M (p; p) 2 G^ e and hence M (p; p) = pz for some z 2 C . Now let p = q = r and it is clear that we must have w = 1. So from ( ; R) we have constructed P and a section s of (P ) with (s) = 1. However s is not unique but this is not a problem. If we change s to s0 = sz for some constant z 2 C then we have changed M to M 0 = Mz. As C is central multiplying by z is an isomorphism of central extensions with connection. So the ambiguity in s does not change the isomorphism class of the central extension with connection. Hence we have constructed a map ,(G) ! C (G) as required. That it is the inverse of the earlier map follows from the de nition of as s (()) and the fact that the connection on P is chosen so its curvature is R. 5. Conclusion: Loop groups In the case where G = L(K ) there is a well known expression for the curvature R of a left invariant connection on L(^K ) | see [5]. We can also write down a 1-form on L(K ) L(K ) such that (R) = d and ( ) = 0. We have: Z 1 R(g)(gX; gY ) = 42 hX; @ Y id S1 Z (g1; g2)(g1X1; g2X2) = 41 2 hX1; (@ g2)@ g2,1id: S1 Here h ; i is the Killing form on k normalised so the longest root has length squared equal to 2 and @ denotes di erentiation with respect to 2 S 1. Note that R is left invariant and that is left invariant in the rst factor of G G. It can be shown that these are the R and arising in [3]. In [4] we apply the methods of this talk to give an explicit construction of the `string class' of a loop group bundle and relate it to earlier work of Murray on calorons. References
[1] J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization, Progr. Math., 107, Birkhauser Boston, Boston, MA, 1993. [2] J.-L. Brylinski and D. A. McLaughlin, The geometry of degree-four characteristic classes and of line bundles on loop spaces. I, Duke Math. J. 75 (1994), no. 3, 603{638;
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MICHAEL K. MURRAY AND DANIEL STEVENSON
[3] M. K. Murray, Another construction of the central extension of the loop group, Comm. Math. Phys. 116 (1988), 73{80 [4] M.K. Murray and D. Stevenson, Higgs elds, bundle gerbes and string structures. In preparation. [5] A. Pressley and G. Segal, Loop groups. Oxford, Clarendon Press, 1986. Department of Pure Mathematics, University of Adelaide, Adelaide, SA 5005, Australia
E-mail address, Michael K. Murray: [email protected] E-mail address, Daniel Stevenson: [email protected]
SPECTRUM OF THE RUELLE OPERATOR AND ZETA FUNCTIONS FOR BROKEN GEODESIC FLOWS LUCHEZAR STOYANOV
1. Introduction Let K be an obstacle in n, where n 3 is odd, i.e. K is a compact subset of n with C 1 boundary @K such that
= nnK is connected. One of the main objects of study in scattering theory (by an obstacle) is the so called scattering matrix S (z) related to the wave equation in with Dirichlet boundary condition on . This is (cf. [LP], [M] or [Z]) a meromorphic operator-valued function S (z) : L2 (Sn,1) ,! L2 (Sn,1) with poles fj g1 j =1 in the half-plane Im(z ) > 0. A variety of problems in scattering theory deal with nding geometric information about K from the distribution of the poles fj g. In what follows we describe one particular problem of this type. The obstacle K is called trapping if there exists an in nitely long bounded broken geodesic (in the sense of Melrose and Sjostrand [MS]) in the exterior domain . For example, if contains a periodic broken geodesic (this is always the case when K has more than one connected component), then K is trapping. It follows from results of Lax-Phillips (1971) and Melrose (1982) that if K is non-trapping, then fz : 0 < Im(z) < g contains nitely many poles j for any > 0 (cf. the Epilogue in [LP] for more precise information). In the rst edition of their monograph Scattering Theory published in 1967, Lax and Phillips conjectured that for trapping obstacles there should exist a sequence fj g of scattering poles such that Imj ! 0 as j ! 1. However M. Ikawa [I1] showed that this is not the case when K is a disjoint union of two strictly convex compact domains with smooth boundaries. It turned out that in this particular case the scattering matrix has poles approximately at the points k d + i, k = 0; 1; 2; : : : , where d is the distance between the two connected IR
IR
IR
IR
IR
201
202
LUCHEZAR STOYANOV
components K1 and K2 of K and > 0 is a constant depending only on the curvatures of @K at the ends of the shortest segment connecting K1 and K2 . Substantial new information concerning the distribution of poles in this case was later given by C. Gerard [G]. Ikawa modi ed the initial conjecture of Lax and Phillips in the following way.
Lax-Phillips Conjecture (LPC) (in the form given by M. Ikawa): If K is trapping, then there exists > 0 such that the strip fz : 0 < Im(z) < g contains in nitely many p^oles j . So far results concerning this conjecture are only known in the case when K has the form (1) K = K1 [ K2 [ : : : [ Ks ; where Ki are strictly convex disjoint compact domains in n (with C 1 boundaries) satisfying the following condition introduced by M.Ikawa: IR
(H )
Km\ convex hull(Ki [ Kj ) = ; for all m 6= i 6= j 6= m.
As we mentioned above, the LPC holds in the case s = 2. There are partial results in the case s 3 also due to Ikawa (cf. [I3], [I4]). Below we brie y describe Ikawa's approach in dealing with this case. The starting point is the distribution X u(t) = eijtjj ; t 6= 0 ; j
where as above fj g is the set of all poles of the scattering matrix S (z). Guillemin and Melrose [GM] showed that the sum of the principal singularities of u(t) on + is X m (2) (,1) T j det(I , P )j,1=2 (t , d ) ; IR
where runs over the set of periodic broken geodesics in K , d is the period (return time) of , T the primitive period (length) of , and P the linear Poincare map associated to . Applying the Laplace transform to (2), Ikawa [I3] introduced the following zeta function (which could be called the scattering zeta function) X Z (s) = (,1)m T jI , P j,1=2 e,sd ; s 2 C :
2
SPECTRUM OF THE RUELLE OPERATOR AND ZETA FUNCTIONS... 203
He then showed that the existence of analytic singularities of Z (s) implies the existence of a band 0 < Im(z) < containing an in nite number of scattering p^oles j (i.e. LPC holds). Clearly Z (s) is a Dirichlet series. Let s0 be its abscissa of absolute convergence. Assuming n = 3, Ikawa showed that there exists > 0 such that in the region s0 , < Re(s) s0 the analytic singularities of Z (s) coincide with these of the zeta function
Z0(s) =
1 X X
m=0
(,1)mr T em(,sT + ) ;
where runs over the set of primitive periodic broken geodesics in , r = 0 if has an even number of re ection points and r = 1 otherwise, and 2 is determined by the spectrum of the linear Poincare map related to . The function Z0(s) is rather similar to a dynamically de ned zeta function (cf. the survey of Baladi [Ba] for general information on this topic). One of the main tools to study this sort of zeta function is the so called Ruelle operator (well known e.g. from the study of Gibbs measures in statistical mechanichs and ergodic theory, cf. [R1] and [Si]). Ikawa [I4] succeeded to implement results of Pollicott (1986) and Haydn (1990) concerning the spectrum of the Ruelle operator and derived sucient conditions for Z0(s) (and therefore Z (s)) to have a pole in a small neighbourhood of s0 in C. From this he derived the following. IR
Theorem 1. ([I4]) If K is a nite union of disjoint balls in
3
with the same radius > 0 and > 0 is suciently small, then LPC holds. IR
The study of the scattering zeta function itself seems to be rather dif cult and very few results concerning it are known. Petkov [P] showed that Z (s) admits an analytic continuation in the domain Re(s) s0. Moreover, under some additional assumptions about the geometry of the obstacle K and assuming rational independence of the primitive periods of periodic orbits, Petkov proved that supt2 jZ (s0 + it)j = 1 (cf. [P] for some further results on the properties of Z (s)). Assuming that the broken geodesic ow has two periodic orbits 1 and 2 such that T 1 =T 2 is a diophantine number, Naud [N] proved that Z (s) has an analytic continuation to a domain of the form s0 , jCtj < Re(s) s0 ; jtj = jIm(s)j 1 ; for some constants C > 0 and > 0. IR
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LUCHEZAR STOYANOV
2. Spectrum of the Ruelle operator and dynamical zeta function
In the case when the obstacle K has the form (1), it is sometimes more convenient to work with the billiard ball map B (i.e. the shift along the broken geodesic ow from boundary to boundary) on the non ( n ) that wandering set M0 . The latter is the set of all points 2 S@K generate trapped broken geodesics in both directions. As a dynamical system B : M0 ,! M0 is naturally isomorphic to the Bernoulli shift on the symbolic space IR
=
1 Y
f1; 2; : : : ; sg ;
,1
the isomorphism being given by the natural coding of the geodesics by sequences fij g, where ij = 1; : : : ; s is the number of the connected component Kij containing the j th re ection point. Ikawa [I4] used the classical interpretation of the Ruelle operator as an operator acting on the space of Lipschitz functions on the symbol space . However, having in mind some signi cant recent developments, it seems more convenient to use a di erent model which is more closely related to the dynamics of the ow. In this section we describe some recent results of the author concerning the spectrum of the Ruelle operator and the dynamical zeta function related to the broken geodesic ow in the exterior of an obstacle of the form (1) in the 2. The results are similar to these of Dolgopyat [D] in the case of Anosov ows on compact manifolds. One would expect that similar results could be obtained (by using similar techniques) for obstacles in n, n 3. Such results would very likely lead to partial solutions of the LPC for cases much more general then the one considered in Theorem 1 above. From now on we assume that K is an obstacle of the form (1) in 2 . Let be the non-wandering set of the broken geodesic ow in . Clearly is the union of all orbits generated by elements of the set M0 ( ) and a suciently small > 0 let de ned above. For x 2 n S@K Ws(x) = fy 2 S ( ) : ( t (x); t(y)) for all t 0 ; ( t(x); t (y)) !t!1 0 g ; IR
IR
IR
Wu(x) = fy 2 S ( ) : ( t (x); t(y)) for all t 0 ; ( t (x); t(y)) !t!,1 0 g be the (strong) stable and unstable sets of size . It is easy to show that ( ) and every suciently small > 0, W s(x) and for every x 2 n S@K
SPECTRUM OF THE RUELLE OPERATOR AND ZETA FUNCTIONS... 205
Wu(x) are 1-dimensional submanifolds of S ( ). It is worth mentioning that both Ws(x) \ and Wu(x) \ are Cantor sets. Given > 0, set S( ) = fx = (q; v) 2 S ( ) : dist(q; @K ) > g ; = \ S( ) : In what follows in order to avoid ambiguity and unnecessary complications we will consider stable and unstable manifolds only for points x in S ( ) or ; this will be enough for our purposes. It follows from the genaral theory of horocycle foliations (cf. [S] or [KH]) that if > 0 is suciently small, there exists > 0 such that if x; y 2 and (x; y) < , then Ws(x)\ and [,;](Wu(y)\) intersect at exactly one point [x; y]. That is, there exists a unique t 2 [,; ] such that t ([x; y]) 2 Wu(x). Setting (x; y) = t, de nes the so called temporary distance function ([Ch1]). If z 2 and U Wu (z) and S Ws(z) are closed (i.e. containing their end points) curves containing z and such that U \ and S \ have no isolated points, then = [U \ ; S \ ] = f[x; y] : x 2 U \ ; y 2 S \ g is called a rectangle in . Notice that is "foliated" by leaves [x; S \ ] of stable manifolds. Let R = fRigki=1 be a family of rectangles such that each Ri is contained in a C 1 cross-section Di S( ) to the ow t . Thus, for each i, Ri = [Ui \ ; Si \ ], where Ui and Si are closed curves in Wu(zi) and Ws(zi), respectively, for some zi 2 . Set R=
[k
i=1
Ri :
The family R is called complete if there exists T > 0 such that for every x 2 there exist t1 2 [,T; 0) and t2 2 (0; T ] with t1 (x) 2 R and t2 (x) 2 R. Thus, (x) = t2 (x) > 0 is the smallest positive time with P (x) = (x)(x) 2 R, and P : R ,! R is the Poincare map related to the family R. Following [B] and [Ra] we will say that a complete family R = fRigki=1 of rectangles in S( ) is a Markov family of size 2 (0; =2) for the billiard ow t if: (a) Ri \ Rj = ; for i 6= j and for each i the sets Ui \ and Si \ are contained in the interior of the curves Ui and Si, respectively; (b) diam(Ri ) ; (c) For any i 6= j and any x 2 Ri \ P ,1 (Rj ) we have P ([x; Si \ ]) [P (x); Sj ] and P ([Ui \ ; x]) [Uj ; P (x)];
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LUCHEZAR STOYANOV
(d) For any i = 1; : : : ; k and x 2 Ri the function is constant on the set [x; Si \ ]. The existence of a Markov family R of an arbitrarily small size > 0 for t follows from the construction of Bowen [B] (cf. also Ratner [Ra]). It follows from a result of C. Robinson [Ro] that there exists an open neighbourhood V of in S ( ) and C 1 transverse foliations F u and F s on V such that Wu(x) \ V = F u(x) and Ws(x) \ V = F s(x) for any x 2 \ V . Fix a neighbourhood V and C 1 foliations F u and F s with these properties. Choosing > 0 suciently small, we may assume that our Markov family R satis es the following additional condition: (e) For each i the cross-section Di (and therefore the rectangle Ri) is contained in V . In what follows we assume that R = fRigki=1 is a xed Markov family for t satisfying the extra conditions (e) and (f). Then
U=
[k
i=1
Ui ;
is a nite disjoint union of compact curves in V . Using the foliations F u and F s in V , assuming again that V is suciently small, we can extend the product [x; y] over the whole Ui Si for any i as follows. Given x 2 Ui Wu(zi) = F u(zi) \ V and y 2 Si Ws(zi) = F s(xi ) \ V , the sets F s(x) and [,;](F u(y)) intersect at exactly one point [x; y]. The image of the C 1 map Ui Si 3 (x; y) 7! [x; y] is a 2-dimensional sumanifold of S ( ) which will be denoted by R~i = [Ui; Si]. The projection p : R~ = [ki=1 R~i ,! U along the leaves of F s is C 1. Then the Poincare map P : R~ ,! [ki=1Di and the corresponding root function : R~ ,! [0; 1) are well-de ned and C 1 . Thus, can be extended to a C 1 map : U ,! U by the same formula: = p P . In the same way one observes that (x; y) is well-de ned and C 1 for (x; y) 2 [ki=1 R~i R~i . Let C (U ) be the space of bounded continuous functions on U . For g 2 C (U ) denote kgk = kgk0 = supx2U jg(x)j. Given g, the Ruelle operator Lg : C (U ) ,! C (U ) is de ned in the usual way: (Lg h)(u) =
X
(v)=u
eg(v) h(v):
SPECTRUM OF THE RUELLE OPERATOR AND ZETA FUNCTIONS... 207
If g 2 C 1 (U ), then clearly Lg preserves the space C 1(U ). Here C 1 (U ) denotes the space of di erentiable function with bounded derivatives on U . A conjugacy between \ R and the Bernoulli shift on a symbolic space A is now de ned in the usual way. Let A = (Aij )ki;j=1 be the matrix given by Aij = 1 if P (Ri) \ Rj 6= ; and Aij = 0 otherwise. Then A = f(ij )1 j =,1 : 1 ij k; Aij ij+1 = 1 for all j g; and the Bernoulli shift ~ : A ,! A is given by ~ ((ij )) = ((i0j )), where i0j = ij+1 for all j . De ne : R ,! A by (x) = (ij )1 j =,1, where P j (x) 2 Rij for all j 2 Z. Notice that P : R ,! R is invertible, so P j is well-de ned for all integers j . The map is a homeomorphism when A is considered with the product topology, and ~ = P . The projection r of on A is given by r = . The subset \ U of R can be naturally identi ed with +A = f (ij )1 j =0 : 1 ij k ; Aij ij+1 = 1 for all j 0 g: Namely, if : A ,! +A is the natural projection, then + = j\U : \ U ,! +A is a bijection with ~ + = + , where ~ denotes the corresponding Bernoulli shift on +A . The dynamical zeta function of the broken geodesic ow t is de ned by Y (s) = (1 , e,s`( )),1 ;
where runs over the set of closed orbits of t and `( ) is the least period of . One can easily see that
1 0 1 X X (s) = exp B @ n1 n e,s n(x)CA ; n=1
(x)=x x2U \
where
n(x) = (x) + ((x)) + : : : + (n (x)) is the period (length) of the closed orbit generated by x. That is, we have ! 1 X 1 (s) = exp n Zn(,s ) ; where
n=1
Zn(,s ) =
X
n (x)=x x2U \
e,s n (x) :
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LUCHEZAR STOYANOV
The following lemma1 of Ruelle ([R2], cf. also [PoS]) partially explains the relationship between the Ruelle operator and the behaviour of the dynamical zeta function (s).
Lemma 1. (Ruelle) Let xj be an arbitrary point in Uj \ for every j = 1; : : : ; k. There exist constants C > 0, > 0, 2 (0; 1) such that
k , Zn(,s ) , X n (x ) C jIm(s)jnn L j ,s Uj j =1
for all n 1 and all s 2 C with Re(s) hT , , hT being the topological entropy of the ow t, and Uj is the characteristic function of Uj in U .
The suspended ow ~ r over rA = f(; t) : 2 A; 0 t r( ) g is de ned by ~sr (; t) = (; t + s), where we use the identi cation (; r( )) = (( ); 0) in A . Then r;A+ = f(; t) 2 rA : 2 +A g is a closed subset of rA which is invariant under the semi ow ~tr , t 0 (cf. [PP2] for details). A conjugacy between ~tr and t on is de ned by r ( s(x)) = ((x); s) for x 2 R and 0 s (x). Given 2 (0; 1), consider the metric d on A given by d (; ) = 0 if = and d (; ) = n if i = i for jij n and n is maximal with this property. In a similar way one de nes a metric d on +A . Using it, we get a metric on r;A+ by setting dr ((; t); (; s)) = d (; )+ jt , sj. The spaces of Lipschitz functions on +A and r;A+ with respect to the metrics just de ned will be denoted by F (+A ) and F (r;A+), respectively. In the present setting the well-known Perron-Ruelle-Frobenius theorem reads as follows. IR
Perron-Ruelle-Frobenius Theorem. Let f 2 F (+A ) be a realvalued function. (a) The Ruelle operator Lf : F (+A ) ,! F (+A ) has a simple eigenvalue = ePr(f ) , where Pr(f ) is the topological pressure of f , and a strictly positive eigenfunction h 2 F (+A ). (b) spec(Lf ) n fg is contained in a disk of radius strictly less than . (c) There exists a unique probability measure = f on +A such that
Z
Z
Lf (g) d = g d
for all g 2 F (+A ). 1
Its statement is slightly modi ed to suit the present context.
SPECTRUM OF THE RUELLE OPERATOR AND ZETA FUNCTIONS... 209
The measure f is the so called Gibbs measure related to the potential f ([R1], [Si]). The complex case is more complicated. It was estabslished by Pollicott [Po] that for any f = u + iv 2 F (+A ) the spectral radius of Lf as an operator on F (+A ) is not greater than ePr(u). Moreover, if Lf has an eigenvalue with jj = ePr(u) , then is simple and unique and the rest of the spectrum is contained in a disk of strictly smaller radius. If Lf has no eigenvalues with jj = ePr(u) , then the whole spectrum of Lf is contained in a disk of radius less than ePr(u) . The following result was obtained by using a modi cation of the technique developed by Dolgopyat [D] in order to prove a similar result in the case of Anosov ows on compact manifolds2, in particluar for geodesic ows on surfaces of negative curvature.
Theorem 2. ([St2]) There exist constant c0 < hT and 2 (0; 1) such that for Re(s) c0 and jIm(s)j >> 0 the spectral radius of the Ruelle operator L,s does not exceed . Using the above theorem and applying the argument of Pollicott and Sharp [PoS] in the case of geodesic ows on compact surfaces of negative curvature, one derives that the zeta function Y (s) = (1 , e,s`( ) ),1
of the billiard ow t has an analytic continuation in a half-plane Re(s) > c0 for some c0 < hT except for a simple pole at s = hT . Moreover, folowing [PoS] again, one derives that there exists c 2 (0; hT ) such that () = #f :R`( ) g = li(ehT ) + O(ec) as ! 1, where li(x) = 2x du= log u. The latter is a much stronger result than the standard Prime Orbit Theorem for open planar billiards in [Mor] (cf. [St1] for the higher dimensional case) derived by means of a result of Parry and Pollicott [PP1]. 3. Exponential decay of correlations In this section we continue to consider the case when K is an obstacle of the form (1) in 2, and we also use most of the notation from Sect.2. IR
In fact the primary aim of Dolgopyat was to establish exponential decay of correlations for such ows. See Sect.3 for more information. 2
210
LUCHEZAR STOYANOV
Let F 2 F () for some > 0. Denote by F~ the function on rA such that F~ r = r F . There exists = ( ) 2 (0; 1) such that F~ 2 F (rA). Let ~ be the Gibbs measure related to F~ with respect to the suspended ow ~r , and let P = Pr~r (F~ ) be the topological pressure of F~ with respect to ~ r . A function f~ 2 F (+A ) related to F~ is de ned by Z r() ~ f ( ) = F~ (; s) ds : 0
Let ~ be the Gibbs measure on A determined by the function f~ , Pr. Then (cf. [PP2]) Z 1 d~(; s) = (r) d~( )ds ; where (~g) = g~( ) d~( ) : A Moreover, we have Pr~ (f~ , Pr) = 0: The Gibbs measures ~ and ~ give rise to measures and on R and \ U , respectively, via the conjugacies r and . IfZf is the function on \ U such that f~ (x) = f , then f (x) = F ( s(x)) ds, so f 2 F ( \ U ). The 0 measures and are called the Gibbs measures related to F and f ,P , respectively. It follows from above that Pr t (F ) = P and Pr (f ,P ) = 0. Given a a Holder continuous potential F on and arbitrary A; B 2 F (), the correlation function of A and B is de ned by
Z
Z
A;B (t) = A(x)B ( t(x)) dF (x),
Z
A(x) dF (x)
B (x) dF (x) :
It is an important problem in smooth ergodic theory (and also in various areas in physics) to know whether such a function decyas exponentially fast as t ! 1. Using Theorem 2 above and the technique developed by Dolgopyat [D], one immediately gets the following.
Theorem 3. Let F be a Holder continuous function on in S ( )
and let F be the Gibbs measure determined by F on . For every > 0 there exist constants C = C ( ) > 0 and c = c( ) > 0 such that jA;B (t)j Ce,ctkAk kB k for any two functions A; B 2 F ().
Here khk denotes the Holder constant of h 2 F ().
SPECTRUM OF THE RUELLE OPERATOR AND ZETA FUNCTIONS... 211
We refer the reader to the recent survey article of Baladi [Ba] for general information and historical remarks on decay of correlations. Amongst the most recent achievements one should mention the important articles of Liverani [L], Young [Y], Chernov [Ch1] and Dolgopyat [D] answering long standing questions. It apppears that for billiards the only results of this type that have been known so far concern the corresponding discrete dynamical system (generated by the billiard ball map from boundary to boundary). To my knowledge, these results are: the subexponential decay of correlations established for a very large class of dispersing billiards by Bunimovich, Sinai and Chernov [BSC] and the exponential decay of correlations for some classes of dispersing billiards in the plane and on the two-dimensional torus established by Young [Y] and Chernov [Ch2] as consequences of their more general arguments. Theorem 3 above describes a non-trivial class of billiard (broken geodesic) ows with exponential decay of correlations for any Holder continuous potential. References [Ba] V. Baladi, Periodic orbits and dynamical spectra, Ergod. Th. & Dynam. Sys. 18 (1998), 255-292. [B] R. Bowen, Symbolic dynamics for hyperbolic ows, Amer. J. Math. 95 (1973), 429-460. [BSC] L. Bunimovich, Ya. Sinai,and N. Chernov, Statistical properties of twodimensional hyperbolic billiards, Russ. Math. Surveys 46 (1991), 47-106. [Ch1] N. Chernov, Markov approximations and decay of correlations for Anosov
ows, Ann. of Math. 147 (1998), 269-324. [Ch2] N. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phys. 94 (1999), [D] D. Dolgopyat, On decay of correlations in Anosov ows, Ann. of Math. 147 (1998), 357-390. [G] Gerard C.: Asymptotique des p^oles de la matrice de scattering pour deux obstacles strictement convexes. Bull. de S.M.F., Memoire n 31, 116 (1988). [GM] V.Guillemin and R.Melrose, The Poisson sumation formula for manifolds with boundary, Adv. in Math. 32 (1979), 204-232. [I1] M.Ikawa, On the poles of the scattering matrix for two strictly convex obstacles, J. math. Koyto Univ. 23 (1983), 127-194. [I2] M. Ikawa, Decay of solutions of the wave equation in the exterior of several strictly convex bodies, Ann. Inst. Fourier, 38 (1988), 113-146. [I3] M. Ikawa, On the distribution of poles of the scattering matrix for several convex bodies, pp. 210-225 in Lecture Notes in Math., vol. 1450, Springer, Berlin, 1990. [I4] M. Ikawa, Singular perturbations of symbolic ows and poles of the zeta functions, Osaka J. Math. 27 (1990), 281-300; Addendum: Osaka J. Math. 29 (1992), 161-174. [KH] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge 1995.
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[LP] P. Lax, R. Phillips, Scattering Theory, Revised Ed., Academic Press, London, 1989. [L] C. Liverani, Decay of correlations, Ann. of Math. 142 (1995), 239-301. [M] R. Melrose: Geometric Scattering Theory, Cambridge Univ. Press, 1994 [MS] R. Melrose, J. Sjostrand: Singularities in boundary value problems. Comm. Pure Appl. Math. 31 (1978), 593-617; 35 (1982),129-168. [Mor] T. Morita, The symbolic representation of billiards without boundary condition, Trans. Amer. Math. Soc. 325 (1991), 819-828. [N] F. Naud, Analytic continuation of the dynamical zeta function under a diophantine condition, Preprint 2000. [PP1] W. Parry and M. Pollicott, An analogue of the prime number theorem and closed orbits of Axiom A ows, Ann. Math. 118 (1983), 573-591. [PP2] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Asterisque 187-188, 1990. [P] V. Petkov, Analytic singularities of the dynamical zeta function, Nonlinearlity 12 (1999), 1663-1681. [Po] M. Pollicott, On the rate of mixing of Axiom A ows, Invent. Math. 81 (1985), 413-426. [PoS] M. Pollicott and R. Sharp, Exponential error terms for growth functions of negatively curved surfaces, Amer. J. Math. 120 (1998), 1019-1042. [Ra] M. Ratner, Markov partitions for Anosov ows on n-dimensional manifolds, Israel J. Math. 15 (1973), 92-114. [Ro] C. Robinson, Structural stability of C 1 ows, Lect. Notes in Math. 468 (1975), 262-277 [R1] D. Ruelle, Thermodynamic formalism, Addison-Wesley, Reading, Mass., 1978. [R2] D. Ruelle, Resonances for Axiom A ows, Commun. Math. Phys. 125 (1989), 239-262. [Si] Ya. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surveys 27 (1972), 21-69. [S] S. Smale, Di erentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. [St1] L. Stoyanov, Exponential instability and entropy for a class of dispersing billiards, Ergod. Th. & Dynam. Sys. 19 (1999), 201-226. [St2] L. Stoyanov, Spectrum of the Ruelle operator and exponential decay of correlations for open billiard ows, Amer. J. Math., to appear. [Y] L.-S. Yang, Statistical properties of systems with some hyperbolicity including certain billiards, Ann. Math. 147 (1998), 585-650. [Z] M. Zworski, Counting scattering poles, In: Spectral and Scattering Theory, M. Ikawa Ed., Marcel Dekker, New York 199 ; pp. 301-331. Luchezar Stoyanov, Department of Mathematics and Statistics, University of Western Australia, Perth WA 6709, Australia
E-mail address :
[email protected]
ON THE BANACH-ISOMORPHIC CLASSIFICATION OF L SPACES OF HYPERFINITE SEMIFINITE VON NEUMANN ALGEBRAS p
F. A. SUKOCHEV Abstract. We present a survey of recent results in the Banach
space classi cation of non-commutative Lp -spaces.
An important role in Banach space theory has always been played by the problem of classifying Banach spaces. This problem has many facets. In our present setting we address this problem by looking at a Banach space as a linear topological space. The natural maps then are continuous linear operators and we look for invariants under isomorphism (=bicontinuous one-to-one linear operator). In general, the development of Banach space theory has clearly shown that there is no hope left for a complete structural theory of Banach spaces, although one can still hope to have such a theory in some special cases. Our objective in the present talk is to describe recent results in this direction in the special case of non-commutative L spaces associated with semi nite von Neumann algebras. To place this work in its proper context we brie y review its origins, beginning with the work of both the mathematicians mentioned so far: Banach and von Neumann. Many fruitful directions in Banach space theory emerged from the famous book [B] by Banach, and the date of appearance of the French edition of this book (1932) is usually regarded as the date of birth of the theory itself. The nal chapter (XII) of this book discusses in depth the problems of comparison between the elements of the two families of Banach spaces (perhaps, the most important families of classical Banach spaces): the spaces l and L = L (0; 1), 1 p < 1. Recall that l is the space of all in nite complex-valued sequences a = (a )1=1 , such that p
p
p
p
p
X := ( ja j ) 1
kak
p
lp
n
n
1=p
< 1;
n=1
Research supported by the Australian Research Council. 213
n
214
F. A. SUKOCHEV
and L = L (0; 1) is the space of all (equivalence classes of) Lebesgue measurable functions f on (0; 1) such that p
p
kf k
Lp
Z := (
1
jf (t)j dt)1 < 1: p
0
=p
The result which is of direct relevance to our present theme can be formulated as follows.
Theorem 1 [B], (1932) There exists an isomorphic embedding of the space L into l if and only if p = q = 2. p
q
In other words, these two families are pairwise non-isomorphic. In contrast to Banach's book, the paper of J. von Neumann [N] is almost completely unknown, even to experts. It appeared in 1937 ( ve years after Banach's book), in an obscure Russian journal, which ceased to exist almost immediately after its rst volume was printed. From the present point of view, the theory of non-commutative L -spaces began from this paper. Let me brie y describe (a version of) von Neumann's construction of L -spaces associated with the von Neumann algebra M of all n n complex matrices. As a linear space it is identi ed with M . Given the matrix A = (a ) =1 2 M , let jAj = (AA)1 2 . Fixing the standard trace Tr on M , we set kAk := Tr(jAj )1 ; 1 p < 1: It is established in [N] that k k is a norm on M and it is customary to denote the space (M ; kk ) by C . In the modern terminology, the space C is a non-commutative L -space associated with von Neumann algebra (M ; Tr). It seems clear from [N] that von Neumann was well aquainted with Banach's book and after having constructed the n2-dimensional space matrix space C , he remarks that another natural way to metrize the n2-dimensional linear space M is to identify standard matrix units e ; 1 i; j n with the rst n2 coordinate vectors of the space2 l , in other words to convert M into the n2 -dimensional space l . This leads to an immediate problem: 2whether these two n2-dimensional spaces, C (non-commutative) and l (commutative), coincide. In [N], von Neumann easily established that the spaces C and l 2 are nonisometric. From the viewpoint adopted in our present setting, the natural question would be whether the Banach-Mazur distance between C and l 2 p
p
n n
n
ij
=
n
i;j
n
p
=p
p
p
n
n
n
p
p
n
p
p
n
n
p
n
ij
p
n p
n
n p
n p
n
n
p
p
n
n
p
p
ON THE BANACH-ISOMORPHIC CLASSIFICATION OF L SPACES ... 215 p
is uniformly bounded. Recall that the Banach-Mazur distance d(X; Y ) between Banach spaces X and Y is d(X; Y ) := inf fkT k kT ,1k j T : X ,! Y isomorphismg; where we adopt the convention inf ; = +1. This question was answered only 30 years later, in the negative, by McCarthy (see [M] and also comments and additional references in Pisier's paper [P]). Before formulating McCarthy's result, recall rst that the in nite-dimensional analogues of the spaces C are Schatten-von Neumann ideals C ; we may also refer to them as to non-commutative L -spaces associated with von Neumann algebra B(H) of all bounded linear operators on the Hilbert space H equipped with the standard trace Tr. Recall that a compact operator A 2 B(H) belongs to C if and only if kAk := Tr(jAj )1 < 1: n
p
p
p
p
p
=p
Cp
Theorem 2 [M], (1967) There exists a constant C > 0 such that for any n 2 N and n2 -dimensional subspace X of L we have
j , j:
1 1
d(C ; X ) > Cn 3 n
p
p
p
1 2
The following consequence is straightforward.
Corollary 3 [M], (1967) There exists an isomorphic embedding of
the space C into L if and only if p = 2. p
p
The converse to the result of Corollary 3 was obtained by Arazy and Lindenstrauss [AL].
Theorem 4 [AL], (1975) There exists an isomorphic embedding
of the space L into C if and only if p = 2. p
p
216
F. A. SUKOCHEV
One by-product of the Arazy-Lindenstrauss arguments was an identi cation of yet another member of the L -family, the spaces S ; 1 p < 1. The de nition is very simple S := (1=1 C ) ; i.e. each element x 2 S is represented by an in nite sequence (x )1=1 with x 2 C and p
p
n
p
p
n
n
p
kxk
Sp
n
p
p
X := ( kx k
n
1
n
n Cp
n
)1 : =p
n=1
This space can be easily viewed as a subspace of C , in our terminology we may say that S is the L -space associated with the von Neumann algebra (1=1M ) (von Neumann subalgebra of B(H)) equipped with the standard trace Tr. p
p
p
n
n
Theorem 5 [AL], (1975) There exists an isomorphic embedding of the space C into S if and only if p = 2. p
p
Thus development from 1932 till 1975 has clearly shown that the following four families of in nite-dimensional separable L -spaces are non-isomorphic: p
(a) The L -spaces associated with the von Neumann algebra l1 = L1(N ) with the trace given by counting measure on N , i.e. the spaces l; p
p
(b) The L -spaces associated with the von Neumann algebra L1 = L1(0; 1) with trace given by Lebesgue measure on (0; 1), i.e. the spaces L; p
p
(c) The L -spaces associated with the von Neumann algebra (1=1M ) with trace Tr, i.e. the spaces S ; p
n
n
p
(d) The L -spaces associated with the von Neumann algebra B(H) with trace Tr, i.e. the spaces C . p
p
ON THE BANACH-ISOMORPHIC CLASSIFICATION OF L SPACES ... 217 p
Let us now overview the situation. Let M be an in nite dimensional semi nite von Neumann algebra acting on a separable Hilbert space, let be a normal faithful semi nite trace on M, and let L (M; ); 1 p < 1 be the Banach space of all -measuarble operators A aliated with M such that (jAj ) < 1 with the norm kAk := ( (jAj ))1 , where jAj = (A A)1 2 (see e.g. [FK]). It is quite natural to ask the following question: What is the linear-topological classi cation of the spaces L (M; ); p 6= 2? It is natural to subdivide the above question to further subcategories accordingly to various classi cation schemes for von Neumann algebras. Tne following two results (obtained jointly with V. Chilin) are a simple application of Pelczynski's decomposition method. p
p
p
p
=p
=
p
Proposition 6 [SC1], (1988) Let M be a commutative von Neumann algebra with a normal faithful semi nite trace . Then L (M; ), p= 6 2 is Banach isomorphic to one of the spaces l or L . p
p
p
Further, recall that a von Neumann algebra M is called atomic if every nonzero projection in M majorizes a nonzero minimal projection.
Proposition 7 [SC1], (1988) Let M be an atomic von Neumann algebra with a normal faithful semi nite trace . Then L (M; ); p 6= 2 is Banach isomorphic to one of the spaces l , S or C . p
p
p
p
The next logical step is the description of L -spaces associated with von Neumann algebras of type I with separable predual. It is wellknown that such an algebra can be represented as a countable l1-direct (H), where sum of von Neumann algebras of the type A M and A B A , A are commutative von Neumann algebras with separable preduals. One can easily see that the following Banach spaces are actually L -spaces associated with von Neumann algebras of type I : the direct sums L S and L C , the Lebesgue-Bochner spaces L (S ) and L (C ), as well as the space C L (S ). The following result (announced in [SC2]) actually shows that these examples actually exhaust the list of such spaces. p
n
n
n
p
p
p
p
p
p
p
p
p
p
p
p
218
F. A. SUKOCHEV
Theorem 8 [SC2], (1990) Let M be a type I von Neumann algebra with a normal faithful semi nite trace . Then L (M; ); p 6= 2 is Banach isomorphic to one of the following spaces l ; L ; S ; C ; S L ; L (S ); C L ; L (C ); C L (S ): Moreover, the spaces p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
l ; L ; S ; C ; S L p
p
p
p
p
p
are pairwise Banach non-isomorphic and non-isomorphic to any of the four remaining spaces L (S ); C L ; L (C ); C L (S ): p
p
p
p
p
p
p
p
p
Thus, the number of distinct L -families has been raised from 4 to 5. However the question whether the remaining 4 spaces are pairwise distinct proved to be very hard. The following result proved to be one of the necessary ingredients. p
Theorem 9 [S1], (1996) Let N be a nite von Neumann algebra with nite, normal, faithful trace 1 , let M be an in nite von Neumann algebra with semi nite, normal, faithful trace 2 . Then for p > 2 there is no Banach isomorphic embedding of C into L (N ; 1), whence L (N ; 1) and L (M; 2) are non-isomorphic for all p 2 (1; 1); p 6= 2. p
p
p
p
This result was subsequently used (together with other methods) in the rst part of the following theorem, which raises the number of distinct families of re exive L -families to 8. The second part of Theorem 10 below yields a complete linear-topological classi cation of the preduals to von Neumann algebras of type I . The proof of the second part is based on the study of the Dunford-Pettis property in von Neumann algebras and its preduals. p
Theorem 10 [S2], (2000) Let M be an in nite-dimensional von Neumann algebra of type I acting in a separable Hilbert space H , let be a normal faithful semi nite trace on M, let L (M; ); p 2 [1; 1); p = 6 2 be the L -space associated with M. Then p
p
ON THE BANACH-ISOMORPHIC CLASSIFICATION OF L SPACES ... 219 p
(a) the space L (M; ) is isomorphic to one of the following nine spaces: l ; L ; S ; C ; S L ; L (S ); C L ; L (C ); C L (S ); (L) and if (E; F ) is a pair of distinct spaces from (L), which does not coincide with the pair (L (C ); C L (S )), then E is not isomorphic to F ; (b) all nine spaces from (L) are pairwise non-isomorphic, provided p = 1. p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
However the question whether the L (C ) and C L (S ) are Banach distinct for 1 < p < 1 remained unresolved. The technique of [S1] for dealing with embeddings of C for p > 2 has not been sucient. The breakthrough has come with the following joint result of the author with U. Haagerup and H. Rosenthal. p
p
p
p
p
p
Theorem 11 [HRS1], [HRS2] (2000) Let N be a nite von Neumann algebra with nite, normal, faithful trace 1 , let M be an in nite von Neumann algebra with semi nite, normal, faithful trace 2 . Then for 1 p < 2 there is no Banach isomorphic embedding of C into L (N ; 1), whence L (N ; 1) and L (M; 2) are non-isomorphic for all p 2 [1; 1); p = 6 2. p
p
p
p
Theorem 11 is crucial in the proof of the following theorem which completes the isomorphic classi cation of separable L -spaces associated with von Neumann algebras of type I for p > 1; it yields more than just the non-isomorphism of L (C ) and C L (S ) and strengthens the second part of Theorem 10. p
p
p
p
p
p
Theorem 12 [HRS1], [HRS2] (2000) Let N be a nite von Neumann algebra with a xed faithful normal tracial state on N and 1 p < 2. Then L (C ) is not isomorphic to a subspace of C L (N ; ). p
p
p
p
Thus, all nine spaces listed in L (see Theorem 10) are pairwise nonisomorphic also for 1 p 6= 2 < 1. Much more follows via application of (a strengthened version of) Theorem 11. Let M be a hyper nite (i.e., M is a weak closure of a union of an increasing sequence of nite dimensional von Neumann
220
F. A. SUKOCHEV
algebras) semi nite von Neumann algebra. In this setting, M can be decomposed as M M 1 M 1 , where M , M 1 , M 1 are hyper nite von Neumann algebras of types I , II1 , and II1 respectively. Further, using disintegration and deep results of A. Connes [C], the algebras M 1 (respectively, M 1 ) can be realized as a countable l1 , where A is as direct sum of von Neumann algebras of the form A B above and B the unique hyper nite factor R of type II1 (respectively, the unique hyper nite factor R0 1 = R B(H) of type II1). Again via Pelczynski's decomposition method (and results of A. Connes) we arrive at the following classi cation result. I
II
II
II
I
II
II
II
;
Proposition 13 [HRS1], [HRS2] (2000) If M is a hyper nite semi nite von Neumann algebra with a normal faithful semi nite trace and 1 p < 1, then L (M; ) is isomorphic to one of the following p
thirteen spaces:
l ; L ; S ; C ; S L ; L (S ); C L ; L (C ); C L (S ) ; p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
L (R); C L (R); L (R) L (C ); L (R0 1 ): p
p
p
p
p
p
p
;
However, the question whether all spaces listed above are pairwise non-isomorphic is much harder. It required the full strength of Theorem 11 combined with very recent results of M. Junge [J].
Theorem 14 [HRS1], [HRS2] (2000) Let N be a nite von Neumann algebra with a xed faithful normal tracial state and 1 p < 2. Then L (R0 1 ) is not isomorphic to a subspace of L (N ) L (C ). p
;
p
p
p
Theorem 15 [HRS1], [HRS2] (2000) If M is a hyper nite semi nite von Neumann algebra with a normal faithful semi nite trace and 1 p < 1, then L (M; ) is Banach isomorphic to precisely one of p
the spaces listed in Proposition 13.
ON THE BANACH-ISOMORPHIC CLASSIFICATION OF L SPACES ... 221 p
[AL] [B] [C] [FK] [HRS1] [HRS2] [J] [M] [N] [P] [SC1] [SC2] [S1] [S2]
References
J. Arazy and J. Lindenstrauss, Some linear topological properties of the spaces Cp of operators on Hilbert spaces, Compositio Math. 30 (1975), 81-111. S. Banach, Theorie des operations lineaires, Warszawa (1932). A Connes, Classi cation of injective factors, Ann. of Math. (2) 104 (1976), 73-115. T. Fack and H. Kosaki, Generalized s-numbers of -measurable operators, Paci c J. Math. 123 (1986), 269-300. U. Haagerup, H.P. Rosenthal and F.A. Sukochev, On the Banachisomorphic classi cation of Lp spaces of hyper nite von Neumann algebras, C. R. Acad. Sci. Paris Ser. I Math. 331 (2000), 691-695. U. Haagerup, H.P. Rosenthal and F.A. Sukochev, Banach embedding properties of non-commutative Lp spaces, to appear. M. Junge, Non-commutative Poisson process, to appear. C. McCarthy, Cp , Israel J. Math. 5 (1967), 249-271. J. von Neumann, Some matrix-inequalities and metrization of matricspace, Rev. Tomsk Univ. 1 (1937), 286-300. G. Pisier, Some results on Banach spaces without local unconditional structure, Compositio Math. 37 (1978), 3-19. F. Sukochev and V. Chilin, Isomorphic classi cation of separable noncommutative Lp -spaces on atomic von Neumann algebras, Dokl. Akad. Nauk. UzSSR (1) (1988), 9-11 (Russian). F. Sukochev and V. Chilin, Symmetric spaces on semi nite von Neumann algebras, Soviet Math. Dokl. 42 (1991), 97-101. F. Sukochev, Non-isomorphism of Lp -spaces associated with nite and in nite von Neumann algebras, Proc. Amer. Math. Soc. 124 (1996), 1517-1527. F. Sukochev, Linear topological classi cation of separable Lp -spaces associated with von Neumann algebras of type I , Israel J. Math. 115 (2000), 137-156.
Department of Mathematics and Statistics, School of Informatics and Engineering, The Flinders University of South Australia, GPO Box 2100, Adelaide, SA 5001, Australia
E-mail address :
[email protected]
INTRODUCING QUATERNIONIC GERBES. FINLAY THOMPSON Abstract. The notion of a quaternionic gerbe is presented as
a new way of bundling algebraic structures over a four manifold. The structure groupoid of this bration is described in some detail. The Euclidean conformal group R+ SO(4) appears naturally as a (non-commutative) monoidal structure on this groupoid. Using this monoidal structure we indicate the existence of a canonical quaternionic gerbe associated to a conformal structure on a four manifold.
It is natural to think that quaternionic algebra and four dimensional geometry should be closely linked. Certainly complex algebra and analysis provide indispensable tools for exploring two dimensional Riemaniann geometry. However, despite many attempts, quaternionic algebra has not been usefully applied to the di erential geometry of four manifolds. The most commonly held view is that quaternionic algebra is too rigid to be useful in studying four manifolds. It is generally assumed that the natural setting for quaternionic di erential geometry is hyperKahler or hypercomplex. [10] The purpose of this talk/article is to present the notion of a quaternionic gerbe, and to demonstrate that they appear naturally as a quaternionic algebraic structure on four manifolds. This work appears as part of an e ort to realize the goal of \doing four dimensional geometry and topology with quaternionic algebra." Although quaternionic structures are de ned [8] for all 4n dimensional manifolds, the basic structures and diculties are already present in only four dimensions. The notion of \quaternionic curve" has been equated with that of a \self dual conformal" structure.[2] Note that even this class of manifolds is strictly larger than the hyperKahler manifolds. Here we restrict our attention to smooth oriented four manifolds, including hyperKahler and self dual conformal manifolds. It is proposed that a \quaternionic structure" on a four manifold is essentially a Euclidean conformal structure. This compares favourably 1
1 Except perhaps Atiyah's notes on solutions to the Yang-Mills equations on the
four sphere, [1]
222
QUATERNIONIC GERBES
223
with the two dimensional case where xing a complex structure is equivalent to xing a conformal structure. 1. The Problem. The most obvious de nition of a quaternionic structure on a four manifold M requires the existence of three integrable complex structures, I; J; K 2 End(TM ), such that, I = J = K = IJK = ,1: In terms of holonomy, this implies a reduction of the frame bundle's structure group to H , the group of unit quaternions. Note that H = GL(1; H ), which generalises the complex case in an obvious way. The problem comes when we consider Berger's list [2] of holonomy groups for Riemannian manifolds: real O(n); SO(n); complex U (n); SU (n); quaternionic Sp(n) Sp(1); Sp(n) exceptional G ; Spin(7) The quaternionic-Kahler series Sp(n) Sp(1) is clearly related to quaternionic algebraic structures, however it is not contained in GL(n; H ). Does this mean that quaternionic-Kahler manifolds are not quaternionic? In reaction to this apparent contradiction, S. Salamon de ned quaternionic manifolds (see [8]) as having a holonomy reduction to GL(n; H ) Sp(1). Then quaternionic-Kahler implies quaternionic, as you might expect. There are two interesting low dimensional coincidences in Berger's list. The rst U (1) = SO(2) tells us the complex Kahler curves are simply Riemannian surfaces. Moreover, because xing a conformal structure on a Riemannian surface corresponds to a holonomy reduction to R SO (2), and R SO (2) = C = GL(1; C ), geometrically speaking, xing a conformal structure is equivalent to xing a complex structure on two dimensional manifolds. The second coincidence Sp(1) Sp(1) = SO(4) seems similar, with \complex" replaced by \quaternionic". We also have, GL(1; H ) Sp(1) = H Sp(1) = R Sp(1) Sp(1) = R SO(4): The implication is that xing a quaternionic structure is equivalent to xing a conformal structure on four manifolds. But what exactly do we mean by a \quaternionic structure"? 2
2
2
2
+
+
+
+
224
FINLAY THOMPSON
1.1. The Impasse. The algebra of quaternions appears naturally as the generator of the Brauer group of the reals, Br(R ) = fR ; H g. The group structure is given by the tensor product, moduli \matrix" algebra. It is not necessary to go into the details of the Brauer group here, instead we simply note that H generates Br(R ) because of the following algebra isomorphism, : H R H ' EndR(H ); where (p q) : v 7! p v q for any p; q; v 2 H . Note that we have used both the left and the right module structures in de ning . The Euclidean conformal group R SO(4) has a natural quaternionic presentation using the isomorphism . Let i : H H ! H H be the canonical map associated to the tensor product. The image of the multiplicative group H H under these maps is precisely the conformal group. We have the following exact sequence of groups, i 1 ,,,! R ,,,! H H ,,,! R SO (4) ,,,! 1 , where R ! H H acts as r 7! (r; r ). Proposition . The Euclidean conformal group in four dimensions appears in a natural and quaternionic way as, R SO (4) = fp q = i(p; q ) j p; q 2 H g: Proof: The Euclidean norm of x 2 H is jxj = x x. Let p q = i(p; q). Then, jp q(x)j = jp x qj = pp x q q x p = jpjjqjjxj = jxj +
+
1
+
2
The above presentation of the conformal group, using the isomorphism : H H ! End(H ), places equal emphasis on the left and right module structures of H on itself. Indeed, the isomorphism is the H -bimodule structure on H ! It is then natural to consider the full bimodule structure as the important structure that we want to integrate over four manifolds. However this way is blocked. Proposition . The automorphisms of H considered as a H -bimodule are all scale multiples of the identity, AutH -bimodule (H ) = R Id : Proof: This is simply a consequence of Shur's lemma applied to the representations of M (R ). +
4
QUATERNIONIC GERBES
225
Thus a four manifold with an integrable H -bimodule structure de ned on the tangent bundle has holonomy contained in R Id, which forces the manifold to be ane. So we reach an impasse: The Euclidean conformal group has a natural quaternionic presentation using the H -bimodule structure on H . The automorphisms of H as a H -bimodule are simply scale multiples of the identity. We show that quaternionic gerbes provide a way of going past this impasse. 1.2. The Suggested Solution. The central idea is to use a more sophisticated way of \gluing" local data together. Although H has very few automorphisms when considered as an H bimodule, it does have an interesting group of automorphisms as an R -algebra, Aut(H ) = Inn(H ) = SO(3): Note that all the automorphisms are inner. The suggestion is to consider the set of linear maps in End(H ) that commute with the H bimodule structure, up to inner automorphisms. Such a map: f : H ! H is required to satisfy the equation, f (p v q) = (p)f (v) (q); where ; are inner automorphisms associated to f . It turns out that the set of all such generalised automorphisms is precisely the Euclidean conformal group, i(H H ) = R SO(4). The idea of allowing the two actions to commute up to an automorphism is natural in category theory. A gerbe is a special kind of sheaf of categories and provides a rich enough language to handle the inner automorphisms coherently. An excellent presentation of the theory of Abelian gerbes has been presented by Jean-Luc Brylinski in \Loop Spaces, Characteristic Classes and Geometric Quantisation." [5]. Nigel Hitchen, studying special Lagrangian sub-manifolds in dimension three, has also made use of Abelian gerbes. Hitchen's approach [6] stresses the idea that Abelian gerbes certain cohomology classes. Michael Murray has presented [7] Abelian gerbes in a di erent light as bundle gerbes. However the theory we are looking for is non-Abelian. L. Breen has de ned non-Abelian gerbes [3, 4] for arbitrary Lie groups and has developed the theory of 2-gerbes. Breen's work applies quite well to our present situation. +
+
226
FINLAY THOMPSON
2. The Structure Groupoid Just as there is a structure group associated to a principal bundle, a gerbe has an associated groupoid. In this section we will describe the structure groupoid associated to a quaternionic gerbe. Following Breen [3], we can associate to any crossed module a groupoid. We start from the crossed module de ned by, : H ! SO(3): where is the natural map onto the inner automorphisms, (p) = p p, , and SO(3) acts on H as automorphisms. Recently R. Brown and collaborators have been relating groupoids and crossed modules to algebraic topology. (see [12]) De nition . The quaternionic structure groupoid H is de ned: objects of H are elements of SO(3), any element p 2 H is considered a morphism p 2 H( ; ) p: ! whenever (p) = . For any two morphisms p : ! and q : ! , the composition is given by the map, q p = qp : ! = (qp) = (q)(p) : It is easy to check that all of the axioms of a small category are satis ed. In addition, because H is a division algebra, all the morphisms are invertible so that H is a groupoid. Note that the set of all morphisms in H consists of pairs (p; ) 2 H SO(3). We will abuse notation a little and say that H = H SO(3) as sets. Although we have used the left SO(3)-action on itself, we have not used the group structure on SO(3). 1
2.1. Tensor product on H. The small category H has a monoidal structure coming from the group structure on SO(3),
: H H ! H: We use the tensor product symbol because the central example of a monoidal structure on a category is that of the tensor product in vector spaces. This tensor product is not commutative, however it is associative. For any ; 2 SO(3), =
QUATERNIONIC GERBES
227
For any two maps p : ! (p) and q : ! (q) we de ne p q as, p q = p [q] : ! (p [q]) : To see that is well de ned we should check that tensor product of the ranges is the range of the tensor product of the maps, (p [q]) = (p)( [q]) = (p) (q) , = ((p) ) ((q) ) Note that is simply the semi-direct group structure coming from the action of SO(3) on H via inner automorphisms, (H; ) = H o SO(3): where (p; ) (q; ) = (p [q]; ). Moreover, this semi-direct product is isomorphic to the Euclidean conformal group, H o SO (3) ' R SO (4): 1
+
2.2. H -bimodules. We can also represent the groupoid H as a category of quaternionic bimodules in such a way that the tensor product is really a tensor product. In order to do this we need to de ne carefully what we mean by an H -bimodule. De nition . An H -bimodule is a vector space V with two commuting actions of the quaternions. Or equivalently, a bilinear map, : H H ! End(V ): Given an H -bimodule (V; ) we can present the action as, : H H ! End(V ); by using the universal property of the tensor product. In this form we see that the H -bimodules are simply the modules over End(H ) = M (R ), the algebra of four by four matrices. Therefore the only simple module is R . For our purposes we restrict ourselves to real four dimensional H -bimodules. The objects of H will be identi ed with the four dimensional H -bimodules. Although all such objects are structurally identical, we will distinguish between di erent quaternionic structures on the same underlying vector space. Proposition . Let V be a H -bimodule in H. Then there is an H bimodule isomorphism : H ! V . 4
4
228
FINLAY THOMPSON
Proof: V is a simple module over H , in two di erent ways. Comparing these actions we de ne for each v 2 V , v 6= 0, a map H ! H : p 7! pv by the rule, p v = v pv : The map p 7! pv is an R -algebra automorphism for all v,
v (pq)v = (pq) v = p (q v) = p (v qv ) = (p v) qv = v (pv qv ) So we have de ned a map V ! Aut(H ) = SO(3). To show surjectivity we start by xing some v and de ne [p] = pv . For any in Aut(H ) the tansitivity of SO(3) implies that = for some 2 SO(3). All automorphisms are inner, so there is some r 2 H such that = (r) = r r, . Then we observe, p (v r, ) = (v pv ) r, = (v r, ) (rpv r, ) = (v r, ) [pv ] = (v r, ) [p] = (v r, ) [p] 1
1
1
1
1
1
1
1
and we see that v r, maps onto . It is easy to see that the map V ! SO(3) bres through the projection V ! P (V ), where P (V ) is the real projective space of one dimensional subspaces in V . Let e 2 V be chosen in the preimage of the identity of SO(3). Then it is clear that p e = e p for all p 2 H , and, 1
:H !V p!pe is the isomorphism of H -bimodules.
We see from the above proof that each H -bimodule structure creates an identi cation of P (V ) with SO(3). Recall that as a smooth manifold SO(3) = RP = P (V ). We can go in the other direction as well. Let V be a right H -module and : V ! SO(3) be a H equivariant map, 3
(xp) = (p), (x): Then we de ne a left H action on V as, px = x (x)[p] 1
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The left action commutes with the right action and so V is an H bimodule, p(xq) = xq (xq)[p] = xq(q), (x)[p] = x (x)[p]q = (px)q We distinguish the di erent objects in H by using the di erent identi cations P (V ) ! SO(3) associated H -bimodule structures. Two objects di er by an element of SO(3). Now the tensor product can actually be represented as a tensor product. If V and W are the H -bimodules associated to objects and in H, then the H -bimodule associated to = is V H W . A quaternionic gerbe consists of the structure groupoid bred over a four manifold. To see how we do that, we need a closer look at the theory of sheaves of categories. 1
3. Sheaves of Categories or Stacks. A gerbe is a special kind of sheaf of categories. Our objective in this section is to present enough of the general theory so that we can understand what is the nature of gerbes, and how they can be useful. We will not present a self contained account here, instead we refer the reader to [5]. A presheaf of categories involves the interplay of locally de ned \objects" and \morphisms". A stack requires that the objects satisfy a descent property, up to an isomorphism. The concept is is quite exible, but still very precise. The isomorphisms that glue together the object data must satisfy additional coherence identities. 2
3.1. Local Homeomorphisms. Instead of working with the category of open sets on a manifold X , we work with local homeomorphisms: continuous map f : Y ! X such that, any y 2 Y has an open neighbourhood U whose image f (U ) is open in X , and, the restriction of f to U gives a homeomorphism between U and f (U ). De nition . The category of spaces over X , CX , has, objects are local homeomorphisms to X , f : Y ! X; a morphism g : (f : Y ! X ) ! (h : Z ! X ) is a local homeomorphisms, g : Y ! Z such that f = h g . 2 For us the terms \stack" and \sheaf of categories" refer to the same concept.
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An important example to keep in mind is associated ` to an open cover fUig of X . The canonical projection f : Y = i Ui ! X from the disjoint union onto X is a local homeomorphism. 3.2. Presheaves of Categories. In the same way that a presheaf of sets is simply a functor from CX to the category of sets, a presheaf of categories over X is a functor C from the category of spaces over X , CX , to the (bi)-category of small categories, functors and natural transformations. Or, more explicitly, to every local homeomorphism f : Y ! X we associate a small category, C (f : Y ! X ) to every arrow of local homeomorphisms k : (Z; g) ! (Y; f ) we associate a functor, C (k) = k, : C (f : Y ! X ) ! C (g : Z ! X ); to every composition k l : (W; h) ! (Z; g) ! (Y; f ) we associate an invertible natural transformation, k;l : l, k, ) (kl), This data must satisfy the following coherence condition, k;l m, l, k, ,,,! m, (lk), ? ? ? ? l;m y ylk;m 1
1
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1
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1
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k;lm (lm), k, ,,,! (lkm), It would be possible to de ne a presheaf of categories with the requirement that l, k, is strictly identical to (kl), . However that does not take advantage of the extra exibility provided. We will see later how quaternionic gerbes make use of this exibility. 3.3. Descent for Morphisms. Let C be a presheaf of categories. We say that the morphisms satisfy descent if for any two objects A; B in C (f : Y ! X ), the presheaf of sets on Y de ned by, Hom(A; B )(k : Z ! Y ) = Hom(k, (A); k, (B )) is actually a sheaf on Y . We can explain this in terms of objects and maps more directly. Let V X be a open neighbourhood, and let A; B be objects in C (V ) = C (V ,! X ). Now let fUig be a open cover of V . Take a collection of morphisms i : AjUi ! B jUi , where i 2 C (Ui). The morphisms satisfy descent if, ijUij = j jUij 8i; j; 1
1
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1
1
1
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implies the existence a unique morphism : A ! B in C (V ) such that i = jUi for all i. In the above we have denoted AjUi for the \restriction" of A to Ui. Of course the restriction is really a functor C (V ) ! C (V \ Ui ), and that functor is not necessarily trivial or obvious. However the presentation becomes much easier to if we make use of these small abuses of the notation. 3.4. Descent for Objects. The objects satisfy a much more complicated descent property, making use of the natural transformations appearing in the de nition. Let C be a presheaf of categories. Let V be any open set in X and f : Y ! V be any surjective local homeomorphism. The descent data for any A 2 C (Y ) consists of an isomorphism : p, (A) ! p, (A) in C (Y X Y ) such that, p, ( ) p, ( ) p, ( ) = Idp,1 1 A in H(Y X Y X Y ). We say that the objects satisfy descent if every pair (A; ) as above implies the existence of an object A0 2 C (V ) and an isomorphism : f , (A0) ! A in C (Y ) such that the following diagram in C (Y X Y ) commutes, 2
1 12
1 23
1 31
1
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1
( )
3
1
,1 f;p f;p 1 2 , 1 ,1 0 p f (A ) ,,,,,! p,1f ,1(A0 )
?? y
1
? ? y
2
p, (A) ,,,! p, (A) This rather complicated prescription can also be understood in terms of open sets in the normal sense. Let fUig be an open cover of V X . The descent data is equivalent to a set of local objects Ai 2 C (Ui ), and isomorphisms ij : AijUij ! Aj jUij in C (Ui \ Uj ). The isomorphisms are required to satisfy, ik jUijk = ij jUijk jk jUijk ; in the category over the triple intersection, C (Ui \ Uj \ Uk ). Again note that we have implicitly used the natural transformations by glossing over the restrictions. De nition . A stack (or sheaf of categories) on X is a presheaf of 1
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categories where objects and morphisms satisfy the descent conditions above. 3 The natural projections Y X Y ! Y are denoted p1 and p2 , and the three natural projections Y X Y X Y ! Y X Y are denoted by p12 ,p23 and p13 .
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4. Quaternionic Gerbes Now we re ne the notion of a stack to that of a gerbe by imposing three more conditions: 1. gerbes take values in groupoids, the full sub-category of small categories whose morphisms are invertible. 2. gerbes are locally non-empty. This means that there exists a surjective local homeomorphism f : Y ! X such that C (Y ) is non-empty. We could also state this by saying that there exists an covering fUig of X such that the C (Ui) are all non-empty. 3. gerbes are locally connected. This means that for any two objects A; B in C (f : Y ! X ), there exists an surjective local homeomorphism g : Z ! Y such that g, (A) and g, (B ) are isomorphic. In terms of covers: if A; B are objects in C (U ) for some U X , then there exists an open covering fUig of U such that A jUi is isomorphic to B jUi for all i. De nition . A gerbe on X is a locally non-empty and locally connected sheaf of groupoids on X . For any group G let GX be the sheaf of G-valued functions on X . A gerbe is said to have band in G if for any object A 2 C (f : Y ! X ), the sheaf Aut(A) of automorphisms of A on Y is isomorphic to GY , and the isomorphism : Aut(A) ! GY is unique up to an inner automorphism of G. De nition . Quaternionic Gerbe is a gerbe with band in H . 4.1. Neutral Gerbes. A gerbe G is said to be neutral if there exists a global object, A 2 G (X ). Because the automorphism sheaf of Aut(A) is isomorphic to the sheaf of H -valued functions, we can identify the groupoid G (X ) with the groupoid of principal H -bundles, : G (X ) ! Tor(Aut(A)) B 7! Isom(B ! A) Quaternionic gerbes are locally non-empty so we can always nd an object AU 2 G (U ) over the open set U . Using that local object we can identify G (U ) with the groupoid of H -bundles over U . Local non-emptiness implies that the gerbe is locally neutral. In order to understand this local neutrality, it is helpful to consider an analogy with the relation between a principal G-bundle and an associated vector bundle. To any vector bundle we can associate the principal bundle of frames. The local neutralisation associated to a local object AU is sort of \frame" for G over U . The set of all frames for G forms a local groupoid. 1
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Assuming that U is contractable, all the H -bundles di er by an automorphisms valued function : U ! SO(3). We de ne a H bundle associated to AU and by letting A U = AU as a bre bundle. The action of H however is twisted by . Let a 2 AU and let a be the same element considered in A U . Then for any quaternion p 2 H , a p = (a [p]) : If U is not contractable there can be topologically inequivalent H bundles. Then we can replace the function above with an H -bimodule M ! U . If AU and BU are two di erent objects in G (U ), then there is an H -bimodule M such that, A H M = B: The local groupoid H(U ) consists of the \frames" of G (U ). Note that because of the local neutrality axiom, all quaternionic gerbes look the same locally. 4.2. The Local Groupoid. We describe here the local structure of H. An object of the local groupoid H(U ) is a diagram of the form, A ,,,! Aut(H ) ? ? y U where : A ! U is a principle H -bundle and is an H -equivariant map (xp) = (p), (x). As we have seen, this data can also be presented in terms of H bitorsors. An H -bitorsor is a principle right H -bundle that is also a principle left H -bundle for a commuting action of H . For any (A; ) 2 H(U ), the left H action on A is, px = x (x)[p]. The morphisms of H -bitorsors are simply bundle maps that commute with both the left and right actions. 1
4.3. Tensor Product on H(U ). In terms of bitorsors we can present the product structure on H(U ) by using the quaternionic tensor product. For any A; B 2 H(U ), A H B = A R B= where xp y x py.
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Assuming U is contractable and by xing a coordinate basis, we get a canonical trivialisation of the tangent bundle, TU = U H . In this way TU can be considered as an object in H(U ). Relative to this xed object, all the others are given by SO(3) valued functions on U , the morphisms are given by H valued functions. Over U H the local groupoid consists of sections C 1(U; H). However the strength of this approach is in terms of the global structure. A global quaternionic gerbe is given in terms of \transition functions". 4.4. Transition Functions or Bitorsor Cocyle. The transition functions for a quaternionic gerbe are given in terms of H -bitorsors. Maybe we should say \transition bitorsors". Let G be a quaternionic gerbe on X and let fUig be a good cover. Choose Ai 2 G (Ui). Then Aut(Ai) is isomorphic to the sheaf of H valued functions. Using Ai we have the following local neutralisation, i : G (Ui) ! Tor(Aut(Ai )) B 7! Isom(B ! Ai) On any intersection Uij = Ui \ Uj we can de ne, Eij = Isom(Aj jUij ; Ai jUij ); The Eij are H -bitorsors and are the transition functions. The two H -actions are given by the composition of an isomorphism with automorphisms of Ai jUij and Aj jU ij , which are each isomorphic to H -valued functions. Note that the isomorphisms H ' Aut(Ai) are unique up to an automorphism. To be really careful we should take care of those automorphisms as well, however that will work will be presented in a comprehensive way later. The H -bitorsors Eij need to be compared over triple intersections. The natural transformations in the de nition of a stack give us the following morphisms as extra data: ijk : Eij H Ejk ! Eik ; These morphisms live in H(Uijk ) and must satisfy the following coherence condition on four intersections, 4
ijk
Eij H Ejk H Ekl ,,,,! Eik H Ekl Id
?
Id
jkl ? y
Eij H Ejl
ijl ,,,!
4 All intersections Ui \ Uj are contractable.
? ? y
Eil
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The pair (Eij ; ijk ) is called a quaternionic bitorsor cocycle on X. Of course the above description of a particular quaternionic gerbe depends on the choice of Ai 2 G (Ui). We can measure the dependence on those choices with a coboundary. 4.5. Coboundary. Let Bi be a di erent choice of local objects and (Fij ; ijk ) be the associated bitorsor cocyle. Let Mi 2 H(Ui ) be de ned by, Bi = Ai H Mi The pair (Mi; ij ) is a coboundary relating (Fij ; ijk) to (Eij ; ijk ) if ij is a map in H(Uij ), ij : Fij ! Mi H Eij H Mj such that as morphisms in H(Uijk ), ik ijk = ijk (ij jk ) We can present this equation with a commutative diagram, ij jk Fij ?H Fjk ,,,,! Mi H Eij ?H Ejk H Mk ijk ? y
Fik
,,,! ik
? yId
ijk Id
Mi H Eik H Mk
It was demonstrated in [11] that coboundaries de ne an equivalence relation on the set of quaternionic bitorsor cocycles. Moreover, it is possible to construct a quaternionic gerbe from a given cocyle, and that gerbe will be isomorphic to any gerbe constructed from a cocyle from the same equivalence class. Although we have used the terminology of cohomology at present there is no actual theory of H -valued cohomology. We use the terminology because it is convenient, and perhaps to be a little optimistic. 5. Conformal Four Manifolds A conformal structure on a four manifold is a reduction of the frame bundle to R SO(4). As we saw at the beginning, the Euclidean conformal group can be presented using the groupoid H with its tensor product acting as the group structure. We will indicate here brie y how to construct a quaternionic bitorsor cocycle from a given a conformal structure on a four manifold X . The presentation here is very sketchy and a more detailed presentation is being prepared. +
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We can choose charts f i : Ui ! H g that are compatible with the conformal structure, i.e., @ ( i j, ) 2 i(H H ); where @ (f ) is the Jacobian matrix of f considered as an element of H H . Therefore there are H -valued functions xij and yij on Uij such that, @ ( i j, ) = xij yji: Over each chart Ui the tangent bundle has a canonical H -bitorsor structure coming from the coordinate . The tangent gerbe cocycle allows us to relate these various H -bitorsor structures. The xij yij can be used to de ne Eij by twisting the left and right H -actions by (xij ) and (yij ) respectively. In terms of an SO (3) valued function, we can de ne Eij relative to TUi with the function (yij xij ). Over the triple intersections Uijk it is possible to construct isomorphisms ijk : Eij Ejk ! Eik . It can also be shown that the i are coordinate charts compatible with the conformal structure if and only if the (Eij ; ijk ) form a quaternionic gerbe cocyle. 1
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References [1] M. Atiyah, Geometry on Yang-Mills Fields., Scuola Normale Superiore Pisa, Pisa, 1979. [2] Arthur L Besse, Einstein manifolds., Springer-Verlag, Berlin-New York, 1987. [3] L. Breen, On the classi cation of 2-gerbes and 2-stacks., Asterisque, (1994), no. 225. [4] L. Breen, Tannakian Categories., Proceedings of Symposia in Pure Mathematics, Volume 55(1994), part 1. [5] Jean-Luc Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization., Progress in Mathematics 107, Birkhauser, 1993. [6] N. Hitchin, Lectures on Special Lagrangian Submanifolds., School on Di erential Geometry (1999), the Abdus Salam International Centre for Theoretical Physics. [7] M. K. Murray, Bundle gerbes., J. London Math. Soc. (2) 54 (1996), no. 2, 403{416. [8] S. Salamon, Di erential geometry of quaternionic manifolds., Annales Scienti ques de l'Ecole Normale Superieure. Quatrieme Serie, 19 (1986), no. 1, 31{ 55. [9] S. Donaldson and P. Kronheimer, The Geometry of Four Manifolds., Oxford Mathematical Monographs, Oxford 1990. [10] D. Joyce, Hypercomplex Algebraic Geometry., The Quarterly Journal of Mathematics, Vol 49, no. 194, p129 (1998), Oxford Second Series. [11] F. Thompson, New Approachs to Quaternionic Algebra and Geometry., PhD. Thesis, 1999, SISSA, Italia.
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[12] R. Brown, Groupoids and Crossed Objects in Algebraic Topology., Homology, Homotopy and Applications, Vol 1, no. 1, 1-78 (1999), http://www.emis.de/journals/HHA/. Finlay Thompson, Victoria University of Wellington, Wellington, New Zealand
E-mail address :
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