Topics From Spectral Theory of Differential Operators

Contemporary Contemporary Mathematics Mathematics Volume 340, 2004 Topics from spectral theory of diﬀerential operators...

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Contemporary Contemporary Mathematics Mathematics Volume 340, 2004

Topics from spectral theory of diﬀerential operators Andreas M. Hinz Dedicated to Christine

Contents Introduction

2

1.

3

Self-adjointness of Schr¨ odinger operators

1.0.

Solving the Schr¨ odinger equation

3

1.1.

Linear operators in Hilbert space

7

1.2.

Criteria for (essential) self-adjointness

13

1.3.

Application to Schr¨ odinger operators

18

2.

Hardy-Rellich inequalities

21

2.0.

Relative boundedness

21

2.1.

Weighted estimates

24

2.2.

Explicit bounds

26

3.

Spectral properties of radially periodic Schr¨ odinger operators

28

3.0.

Spectra of self-adjoint operators

29

3.1.

Asymptotic behavior of eigensolutions and the spectrum

38

3.2.

Spherical symmetry

40

3.2.1.

Some basic examples

40

3.2.2.

Embedded eigenvalues

42

3.3.

Radial periodicity

44

3.3.1.

Dense point spectrum

44

3.3.2.

Welsh eigenvalues

46

3.4.

Numerical analysis

46

References

48

2000 Mathematics Subject Classiﬁcation. Primary 35J10, 47B25; Secondary 35P20, 35Q40. Key words and phrases. self-adjointness, Schr¨ odinger operator, Rellich inequality, spectrum. My travel to Mexico was supported by IIMAS-UNAM and Deutsche Forschungsgemeinschaft. c 2003 A.M.Hinz c 2004 American Mathematical Society

1

2

ANDREAS M. HINZ

Introduction Spectral methods for (partial) diﬀerential equations emerged as early as the 18th and 19th centuries in the works of D. Bernoulli and J. Fourier ([22]). However, it took almost a hundred years to clarify the notions of function and integrability in order to establish the soundness and applicability of these ideas. This development culminated at the dawn of the 20th century with the integral of H. Lebesgue ([38, 39]), which triggered the accomplishment of the early theory of functional analysis in the works of D. Hilbert, E. Schmidt and F. Riesz, highlighted by the proof of E. Fischer ([21]) of completeness of L2 . The mathematical ground was therefore well prepared, when E. Schr¨ odinger ([58]) came up with his equation (0.1)

i

∂ Ψ(t, x) = (− + V (x)) Ψ(t, x), ∂t

which may well be considered as one of the most outstanding pieces of physics and mathematics of all time. With his spectral theorem, J. v. Neumann ([41, 42]) founded the spectral theory of Schr¨ odinger operators, put into a more general frame by M. H. Stone ([64]). In the ﬁrst part of Chapter 1, we will give a modern and comprehensive presentation of this theory which reduces the problem of existence and uniqueness of solutions for (0.1) with the initial state Ψ0 = Ψ(0, ·) given to the problem of establishing self-adjointness of the corresponding linear operator. The latter question has attracted attention for more than half a century with an ever expanding class of admissible potential functions V and supply of methods evolving. In the second part of Chapter 1, we will provide some abstract criteria for (essential) self-adjointness which can be applied in a very elegant way to Schr¨ odinger operators using only some results from regularity theory of diﬀerential equations. ¿From the late 1930s, originating in the works of F. Rellich and T. Kato, perturbation theory became a mighty tool to investigate both qualitative and quantitative properties of linear operators (cf. [36]). It depends largely on, sometimes ingenious, estimates to show that one part of the operator is subordinate to another one. As an example, we will discuss Hardy-Rellich inequalities in Chapter 2. With the basic properties of diﬀerential operators being established by 1970 (cf. [32, Chapter 3]), the last three decades of the 20th century were marked by a tremendous ﬂow of diverse results about the spectra in a variety of cases like magnetic, random or one-dimensional Schr¨ odinger operators and Dirac operators, a development led by the big promotor in the ﬁeld, B. Simon (cf. [59]). To demonstrate the diversity of spectral phenomena, Chapter 3 will discuss, after a condensed introduction into spectra of self-adjoint operators, the technically rather simple case of spherically symmetric radially periodic Schr¨ odinger operators. Here, we will also indicate that apart from analytical methods, numerical investigations are now viable, given the ever increasing power of electronic computing machinery. The following three chapters want to present a uniﬁed approach from the very beginnings of the theory to topics of current research. We do not attempt to give a comprehensive overview of the theory including appropriate recognition of all contributors, since this would be too enormous a task. .

TOPICS FROM SPECTRAL THEORY OF DIFFERENTIAL OPERATORS

3

1. Self-adjointness of Schr¨ odinger operators In Section 1.0, we will show the necessity and naturalness to reduce the task of solving the initial value problem for the Schr¨ odinger equation to the investigation of self-adjointness of a corresponding linear operator in a Hilbert space. The discussion of basic properties of such operators in Section 1.1 leads to the conclusion that selfadjointness is also suﬃcient for a complete solution. It is therefore imperative to develop criteria for self-adjointness, which will be done in Section 1.2. In particular, we will give a straightforward argument for the fact that essential self-adjointness, i.e. the existence and uniqueness of a self-adjoint extension, is equivalent to selfadjointness of the closure of the operator, avoiding the (explicit) use of the Cayley transform and defect indices. The general theory is then applied in Section 1.3 to Schr¨ odinger operators − + V with the (local) Kato class (named for T. Kato, the “father of the modern theory of Schr¨odinger operators ([59, p. 3523])”) emerging as the most natural and most extensive home for (negative parts of) potential functions V. The Schr¨ odinger equation (0.1) is rooted in the wave model of quantum mechanics, starting from the idea that a free particle, i.e. subject to no outer force ﬁeld, with mass m ∈ ]0, ∞[, total energy E ∈ ]0, ∞[ and velocity v ∈ Rd \{0} (d ∈ N) will behave like a plane wave and can therefore be described by a wave function Ψ of the form ∀ t ∈ R ∀ x ∈ Rd : Ψ(t, x) = A · e2πi(k·x−νt) , which propagates with constant speed ν/|k| ∈ ]0, ∞[ into direction k ∈ Rd \ {0}. (A ∈ C\{0} is a normalization constant.) Using only the most fundamental physical laws of quantum theory, namely Einstein’s equation E = hν (with Planck’s constant h 1 = , one arrives at h) and de Broglie’s relation for the wave length λ := |k| m |v| (0.1) if one wants to determine the time evolution of Ψ starting from some initial state Ψ0 : Ψ has to fulﬁl a diﬀerential equation of ﬁrst order in t, which is linear because of the superposition principle for waves. Therefore it is also evident that wave functions have to be complex valued. The Laplacian −, acting on the space 1 variable x only, represents the kinetic energy m |v|2 , while the so-called potential 2 (function) V in (0.1) embodies the potential energy induced by an external force ﬁeld and has therefore to be real-valued. As an example, one might think of an electron subject to the coulombic force of a charged nucleus. 1.0. Solving the Schr¨ odinger equation. The ﬁrst attempt to ﬁnd a nontrivial solution of equation (0.1) is by separation of variables, i.e. the ansatz Ψ(t, x) = f (t) u(x). Then, for a (t0 , x0 ) ∈ R1+d with Ψ(t0 , x0 ) = 0, we have f (t) =

Ψ(t0 , x) Ψ(t, x0 ) , u(x) = , u(x0 ) f (t0 )

∂Ψ whence f ∈ C1 (R), u ∈ C2 Rd and Ψt = f u, Ψ = f u (we write Ψt for ∂t etc.), such that (1.1)

i f u = f (− + V )u.

4

ANDREAS M. HINZ

Putting x = x0 , we see that f fulﬁls the ordinary diﬀerential equation u(x0 ) − V (x0 ) f (t) =: −i λ f (t), f (t) = i u(x0 ) whose general solution is f (t) = c exp(−i λ t) with some c ∈ C\{0}. If we insert this into (1.1), we ﬁnd that u has to fulﬁl the (time-independent) Schr¨ odinger equation (1.2)

∀ x ∈ Rd : −u(x) + V (x) u(x) = λ u(x).

Since |f (t)| = |c| exp (im(λ) t), the time evolution as given by f is bounded if and only if the eigenvalue λ is real; this is, of course, the physically relevant situation. There are only a few cases, like e.g. the harmonic oscillator (cf. infra, Example 3.26), where there are (suﬃciently many) classical eigensolutions u of (1.2). In particular in view of possible singularities of the potential function V as in the Coulomb case (cf. infra, Example 3.27), we are forced to extend the notion of solution, based on the following observation: let d R := ϕ ∈ C∞ Rd ; supp(ϕ) is bounded , C∞ 0 where the of ϕ is deﬁned by supp(ϕ) = {x ∈ Rd ; ϕ(x) = 0}. Then for every dsupport 2 u ∈ C R for which V u is locally d integrable, i.e. integrable after multiplication R , we get from integration by parts: with any test function ϕ ∈ C∞ 0 d ∀ ϕ ∈ C∞ R : {−u(x) + (V (x) − λ) u(x)} ϕ(x) dx 0 = u(x) {−ϕ(x) + (V (x) − λ) ϕ(x)} dx, such that for these u, equation (1.2) is equivalent to d (1.3) ∀ ϕ ∈ C∞ R : u(x) {−ϕ(x) + (V (x) − λ) ϕ(x)} dx = 0. 0 As there is no regularity requirement on u in (1.3), we call every non-trivial locally integrable u for which V u is locally integrable as well and which fulﬁlls (1.3) a weak eigensolution of the Schr¨ odinger equation for eigenvalue λ. In order to make use of functional analytic methods, we now have to ﬁnd a suitable function space H, in which − + V acts as an operator. For the sake of linearity, H has to have the canonical algebraic structure of a vector space over C. We deﬁne d R : D(S) = u ∈ H; ∃ v ∈ H ∀ ϕ ∈ C∞ 0

u(x) {−ϕ(x) + V (x) ϕ(x)} dx =

v(x) ϕ(x) dx ,

∀ u ∈ D(S) : S(u) = v. This deﬁnes a linear operator in H, i.e. D(S) is a linear subspace of H and ∀ u, v ∈ D(S), α ∈ C : S(u + αv) = Su + αSv. (For linear operators, the brackets for the argument are usually omitted, if no ambiguity is possible.) Let us suppose that we have suﬃciently many weak eigensolutions en ∈ H \ {0} for the eigenvalues λn ∈ R (they are then eigenfunctions of S, i.e. Sen = λn en ), such that any u ∈ H can be written as a Fourier series

5

TOPICS FROM SPECTRAL THEORY OF DIFFERENTIAL OPERATORS

u=

∞

αn en with α ∈ CN0 and accordingly Ψ(t, ·) =

n=0

∞

exp (−i λn t) αn en exists

n=0

in H for all t ∈ R. This presupposes a metric structure on H, compatible with the algebraic one, which can only be achieved by a norm · on H. Then we have (formally) for h = 0 and setting fn (t) = exp (−i λn t): ∞

1 {Ψ(t + h, x) − Ψ(t, x)} − {−i λn exp (−i λn t)} αn en (x) h n=0

=

∞

fn (t + h) − fn (t)

h

n=0

−

fn (t)

αn en (x),

and it would be desirable to have the right hand side tend to 0 as h → 0. Alas, this can not be expected in general! The way out is to assume further that the en are mutually orthogonal. This necessitates the introduction of a compatible geometric structure in the form of an inner product · , ∗ in H, which thus will become a unitary space, where two vectors x and y are called orthogonal, iﬀ x, y = 0. Then there is something like the theorem of Pythagoras, namely u=

∞

αn en ⇒ u2 =

n=0

∞

|αn |2 ,

n=0

where we have assumed that en = 1 for all n. By the properties of an inner product we then have ∞ ∞

αn en , em = αn en , em = αm . ∀ m ∈ N0 : u, em = n=0

We obtain (1.4)

n=0

2

∞

1

{−i λn exp (−i λn t)} u, en en

{Ψ(t + h, ·) − Ψ(t, ·)} −

h n=0 2 ∞

fn (t + h) − fn (t) − fn (t) | u, en |2 . = h n=0

Now the convergence of the right hand side as h → 0 can be investigated with the aid of the following dominated convergence theorem. Lemma 1.1. Let (anm )m∈N0 n∈N be a sequence of null sequences in C, with 0

∀ n ∈ N0 ∀ m ∈ N0 : |anm | ≤ bn and

∞

bn < ∞.

n=0

Then

∞

anm → 0 as m → ∞.

n=0

Proof. For ε > 0 choose N ∈ N0 such that

∞

n=N +1

ε . ∀ m ≥ M ∀ n ∈ {0, . . . , N } : |anm | < 2 (N + 1)

bn <

ε and then M ∈ N0 with 2

6

ANDREAS M. HINZ

fn (t + h) − fn (t) − fn (t) ≤ 2 |λn |, the right hand side of (1.4) tends to 0 As h ∞

|λn |2 | u, en |2 < ∞. This is the case if and only if the series in for every t ∈ R, if n=0

the left hand side of (1.4) converges for every t ∈ R to some Ψt (t, ·) ∈ H, provided that the unitary space (H, · , ∗ ) is complete, i.e. H is a Hilbert space. (Ψ(t, ·) will then be called diﬀerentiable in H with respect to t ∈ R. Rules from classical calculus can be carried over, like e.g. the product rule Φ, Ψ = Φ , Ψ + Φ, Ψ .) d Therefore, by Fischer’s theorem (cf. [21]), H = L2 R is the appropriate home for the Schr¨ odinger operator S. This also leads to the probabilistic interpretation of the wave function (cf. infra, Section 3.0). ∞

λn u, en en ∈ H to evThe operator T in H, which assigns the image T u = n=0 ∞

2 2 λn | v, en | < ∞ , is (formally) symmetric, because ery u ∈ D(T ) = v ∈ H; n=0

∀ u, v ∈ D(T ) : u, T v =

u,

∞

λn v, en en

n=0

=

∞

=

∗

λn v, u, en en =

n=0

∞

λn v, en ∗ u, en

n=0 ∞

λn u, en en , v

= T u, v.

n=0

So Ψ is the unique solution of the initial value problem Ψ(0, ·) = u, ∀ t ∈ R : i Ψt (t, ·) = T Ψ(t, ·), where uniqueness follows from Ψ2 = Ψ, Ψ = Ψ , Ψ + Ψ, Ψ = −i T Ψ, Ψ + Ψ, −i T Ψ = −i { T Ψ, Ψ − Ψ, T Ψ} = 0, because then Ψ = 0, if u = 0.

d Finally, T ⊂ S, because for any u ∈ D(T ) and ϕ ∈ C∞ R , we have 0 T u(x) ϕ(x) dx =

∞

λn u, en en (x)

n=0

= =

∞

u, en

n=0 ∞

ϕ(x) dx

λn en (x)ϕ(x) dx

u, en

en (x) {−ϕ(x) + V (x)ϕ(x)} dx

n=0

=

u(x) {−ϕ(x) + V (x)ϕ(x)} dx.

For u ∈ D(T ), the above Ψ is therefore also a solution of i Φt (t, ·) = SΦ(t, ·). If D(S) = D(T ), there could be other solutions for the corresponding initial value problem. Uniqueness, however, can be guaranteed, if S = T , which means in particular that S has to be (formally) symmetric too. Unfortunately, the domains of both S and T are given implicitly only and depend on properties of V . It is not even

TOPICS FROM SPECTRAL THEORY OF DIFFERENTIAL OPERATORS

7

clear, if they contain a substantial set, namely a dense subspace of H. However, if we d R assume V ϕ ∈ H for all ϕ ∈ C∞ 0 (this is equivalent to local square integrability d R ⊂ D(S) and that S0 ϕ = −ϕ + V ϕ for the of V ), it is obvious that C∞ 0 Rd . Conversely, by the deﬁnition of minimal Schr¨ odinger operator S0 := S C∞ 0 D(S), the maximal Schr¨ odinger operator S is the adjoint operator of S0 : S = S0∗ . The symmetry of S0∗ means that S0 has exactly one self-adjoint extension, namely S = S ∗ . (S0 is then called essentially self-adjoint.) We will now make these notions more precise in Section 1.1 and show that essential self-adjointness of the minimal operator is also suﬃcient for the unique solvability of the initial value problem for the Schr¨ odinger equation, i.e. we will be able to prove the following. Theorem 1.2. Let {en ; n ∈ N0 } be an orthonormal basis of H consisting of eigenfunctions en for eigenvalues λn of S and assume that S0 is essentially selfadjoint. Then for every u ∈ D(S) the unique solution of the initial value problem for the Schr¨ odinger equation Ψ(0, ·) = u, ∀ t ∈ R : i Ψt (t, ·) = SΨ(t, ·), is given by ∀ t ∈ R ∀ x ∈ R : Ψ(t, x) = d

∞

exp(−i λn t) u, en en (x).

n=0

Self-adjointness will also prove to be the key to solve the problem even if there is no orthonormal basis of H consisting of eigenfunctions of S. 1.1. Linear operators in Hilbert space. Properties of operators reﬂect the algebraic, metric and geometric structure of a Hilbert space H. Linearity is associated with the vector space, boundedness, closability and closedness with the norm, symmetry with the inner product and ﬁnally self-adjointness with completeness. Although these properties can be characterized in the corresponding more general settings, we will, for simplicity, concentrate on operators T ⊂ H 2 , i.e. deﬁned on some subset D(T ) of H and with values in H. In view of our application to the solution of the Schr¨ odinger equation, we will also limit our considerations to a nontrivial H over the ﬁeld C. (Many of the results in this and the next section are valid for real Hilbert spaces, but some of the proofs involve subtleties, which we do not want to address here.) We assume familiarity with the basic properties of Hilbert spaces (see, e.g., [65]). Definition 1.3. T is a linear operator in H, iﬀ T is a linear subspace of H 2 and T ∩ ({0} × H) = {(0, 0)}. The domain of T is D(T ) = {u ∈ H; ∃ v ∈ H : (u, v) ∈ T }, and we write T u for v. For λ ∈ C, the subspace (T − λ)−1 ({0}) is called the eigenspace of λ (and T ); if this eigenspace is non-trivial, λ is called an eigenvalue of T and every nontrivial element of the eigenspace is called an eigenvector (or eigenfunction, if H is a function space) for λ (and T ). A linear operator T is a function from D(T ) to H with the property ∀ u, v ∈ D(T ) ∀ κ ∈ C : T (u + κv) = T u + κ T v.

8

ANDREAS M. HINZ

The most handsome operators are those which are bounded. Lemma 1.4. If T is a linear operator in H, then T is continuous, iﬀ T is bounded, i.e. T := sup {T u; u ∈ D(T ), u = 1} < ∞. Proof. If T is continuous, then there is a δ > 0 such that T v < 1 for every v ∈ 1 1 D(T ) with v < 2δ and consequently T u = T (δu) < for every u ∈ D(T ) δ δ with u = 1. Conversely, if T is bounded and u, v ∈ D(T ) with u−v < δ, then T u−T v = T (u − v) ≤ T u − v ≤ T δ. d Unfortunately, diﬀerential operators in L2 R are not bounded in general and therefore we have to resign ourselves to closedness or even closability. Definition 1.5. A linear operator T in H is closable, iﬀ its closure T is a linear operator. D(T ) is then called a core of T . T is called closed, iﬀ T = T . Corollary 1.6. Let T be a bounded linear operator in H. Then T is closable, and T is the only bounded extension of T with domain D(T ). In particular, T is closed if and only if D(T ) is closed. Proof. Let D(T ) ⊃ (un )n∈N → u ∈ D(T ). Then u → lim T un deﬁnes the operator n→∞

T on D(T ) = D(T ): since T un − T uN ≤ T un − uN , (T un )n∈N is a Cauchy sequence; moreover, the limit is independent of the choice of the sequence (un )n∈N approximating u, as can be seen by observing that the images of the mixed sequence built from two such sequences converge as well. Linearity of T is obvious. Furthermore, T u = lim T un ≤ lim T un = T u, n→∞

n→∞

such that T ≤ T ; T ≤ T is trivial since T ⊂ T . If T is a bounded extension of T with D T = D(T ), then for every u ∈ D T , there is a sequence D(T ) ⊃ (un )n∈N → u, such that T un = Tun → Tu by continuity of T, granted by Lemma 1.4. The operator S of Section 1.0 is closed, as can be seen directly or by recourse to the fact that S = S0∗ . Lemma 1.7. Let T be a linear operator in H. Then T ∗ := (u, v) ∈ H 2 ; ∀ ϕ ∈ D(T ) : u, T ϕ = v, ϕ deﬁnes a (closed) linear operator in H, called the adjoint of T , if and only if T is densely deﬁned, i.e. D(T ) = H. Proof. Obviously, T ∗ is a linear subspace of H 2 , and its closedness follows from the continuity of the inner product. T ∗ is a linear operator if and only if D(T )⊥ = {0}. Corollary 1.8. Let T be a densely deﬁned linear operator in H. Then T = ∗∗ T ; T is closable if and only if T ∗ is densely deﬁned, in which case ∗ in particular, ∗ T =T .

TOPICS FROM SPECTRAL THEORY OF DIFFERENTIAL OPERATORS

9

Proof. We have T

⊥ = T ⊥⊥ = (T ∗ ϕ, −ϕ) ∈ H 2 ; ϕ ∈ D (T ∗ ) = (u, v) ∈ H 2 ; ∀ ϕ ∈ D (T ∗ ) : u, T ∗ ϕ = v, ϕ = T ∗∗ ,

∗ ∗ and the equivalence ∗ follows from Lemma 1.7. If we apply this to T , we get T = ∗∗∗ ∗ T =T = T .

Another type of closable operators are symmetric operators. Definition 1.9. A linear operator T in H is called symmetric, iﬀ T is densely deﬁned and T ⊂ T ∗ . Corollary 1.10. Let T be a symmetric operator in H. Then T is closable, and T is symmetric. ∗ Proof. By Corollary 1.8, T is closable and T = T ∗ . As T ⊂ T ∗ , and T ∗ is closed ∗ by Lemma 1.7, we get T ⊂ T . Symmetric operators have other nice features. Lemma 1.11. Let T be a symmetric operator in H. Then a) ∀ u ∈ D(T ) : T u, u ∈ R; in particular all eigenvalues of T are real. b) Eigenspaces for diﬀerent eigenvalues of T are orthogonal. c) ∀ λ ∈ C ∀ u ∈ D(T ) : (T − λ)u ≥ |im(λ)| u.

Proof. a) T u, u∗ = u, T u = T u, u. If u is an eigenvector for the eigenvalue λ, then λ u2 = λu, u = T u, u ∈ R and consequently λ ∈ R. b) Let λe and λf be eigenvalues with eigenvectors e and f , respectively. Then (λe − λ∗f ) e, f = T e, f − e, T f = 0, whence from λe = λf = λ∗f , we obtain e, f = 0. c) We may assume u = 1. Then (T − λ)u ≥ | (T − λ)u, u| = | T − re(λ) u, u − i im(λ)| ≥ |im(λ)|, the latter since T u, u ∈ R from (a).

We are now ready for the decisive step to prove Theorem 1.2. Theorem 1.12. Let M be an orthonormal basis of H and λ ∈ RM . Then

D(T ) = u ∈ H; λ2e | u, e|2 < ∞ , T u = λe u, ee, e∈M

e∈M

deﬁnes a self-adjoint operator, i.e. T = T ∗ .

Proof. Obviously, D(T ) is a subspace of H, and D(T ) = H since span(M ) ⊂ D(T ). The existence of T u is guaranteed by Fourier expansion in H, which also yields linearity of T . Moreover, by Parseval’s identity,

u, e λe v, e∗ = λe u, e v, e∗ = T u, v, ∀ u, v ∈ D(T ) : u, T v = e∈M

such that T is symmetric.

e∈M

10

ANDREAS M. HINZ

If u ∈ D (T ∗ ), then λe u, e = u, T e = T ∗ u, e for every e ∈ M ⊂ D(T ) and consequently u ∈ D(T ), whence T = T ∗ . This theorem now completes the proof of Theorem 1.2, because the self-adjoint operator T cannot have a strict extension S which is symmetric, by virtue of T ∗ = T ⊂ S ⊂ S∗ ⊂ T ∗. We now try to liberate ourselves from the assumption of the existence of an orthonormal basis consisting of eigenfunctions. We observe that the operator T in Theorem 1.12 can be rewritten as

2 2 λ Pλ u < ∞ , T u = λPλ u, D(T ) = u ∈ H; λ∈R

λ∈R

where Pλ u is the projection of u to the eigenspace of λ. (A projector P is a

2 2 symmetric operator with D(P ) = H and P = P .) Here, the sum λ Pλ u2 λ∈R

is built up in a monoton increasing way in countably many positive steps, if we let λ grow from −∞ to ∞. This suggests a generalization by replacing the sum with an integral. To this end, we observe further that the family (Pλ )λ∈R , being orthogonal, i.e. Pλ Pµ = 0 for λ = µ, by Lemma 1.11b, generates, by putting

Pµ , a spectral family (Eλ )λ∈R . Eλ = µ≤λ

Definition 1.13. A family (Eλ )λ∈R of projectors in H is called a spectral family, iﬀ it is • non-decreasing, i.e. ∀ µ ≤ λ ∀ u ∈ H : Eµ u, u ≤ Eλ u, u, (Note that this implies ∀ µ ≤ λ : Eµ Eλ = Eµ = Eλ Eµ .) • right-continuous, i.e. ∀ λ ∈ R : Eλ = lim Eλ+ n1 =: Eλ+ , n→∞

and E−∞ := lim E−n = 0, E∞ := lim En = 1. n→∞

n→∞

Then, with Eλ− := lim Eλ− n1 , n→∞

λ2 Eλ u2 − Eλ− u2 < ∞ , T u = λ (Eλ − Eλ− ) u, D(T ) = u ∈ H; λ∈R

λ∈R

or, using the Cauchy-Stieltjes integral, 2 2 D(T ) = u ∈ H; λ dEλ u < ∞ , ∀ v ∈ H : T u, v = λ d Eλ u, v. The famous spectral theorem of von Neumann [41, Satz 3.6] states that the connection between spectral families and self-adjoint operators in H is not restricted to those with a complete set of eigenvectors. Theorem 1.14. Let (Eλ )λ∈R be a spectral family in H and f ∈ C(R). Then 2 2 D = u ∈ H; |f (λ)| dEλ u < ∞ is dense in H and

∀ v ∈ H : f (T )u, v =

f (λ) d Eλ u, v

TOPICS FROM SPECTRAL THEORY OF DIFFERENTIAL OPERATORS

11 11

deﬁnes a linear operator f (T ) in H with D (f (T )) = D = D (f (T )∗ ), ∗ ∀ v ∈ H : f (T ) u, v = f (λ)∗ d Eλ u, v, and f (T )u2 =

|f (λ)|2 dEλ u2 = f (T )∗ u2 .

For f (λ) = λ, the corresponding map from the set of spectral families to the set of self-adjoint operators in H is bijective. An immediate consequence is the following result. Lemma 1.15. If T is a self-adjoint operator in H and (Eλ )λ∈R its spectral family, then for any λ ∈ R, the operator Eλ − Eλ− is the projector to the eigenspace of λ. Proof. If u = (Eλ − Eλ− ) u, then ∀ µ ∈ R : Eµ u = (µ ≥ λ) u. (We use the Iverson convention: for a statement A, (A) is 1, if A is true and (A) is 0, if A is false.) Hence for all ε > 0 and f ∈ C(R):

λ 2

f (µ) dEµ (u)2

f (µ) dEµ (u) = λ−ε

and consequently

µ2 dEµ (u)2 < ∞, i.e. u ∈ D(T ) and

(T − λ) u2

λ (µ − λ)2 dEµ (u)2 =

=

≤

2

ε

(µ − λ)2 dEµ (u)2 λ−ε

dEµ (u) = ε u2 , 2

2

whence T u = λu. On the other hand, if u ∈ D(T ) and T u = λu, then λ−ε

∀ −∞
(µ − λ)2 dEµ (u)2 ≤ (T − λ) u2 = 0, a

from which (Eλ−ε − Ea ) u = 0 follows by the mean value property of the CauchyStieltjes integral. For ε → 0 and a → −∞ we get Eλ− u = 0. In the same way one can show that Eλ u = u. We will not attempt to give a proof of the spectral theorem here; for a convenient approach, see [40] (cf. also [2, Section 26.3] and [65, Theorems 7.14,7.17]). In view of equation (1.4), we put f (λ) = exp(−i λ t), which yields the unitary operators U (t) := exp(−i T t). (A unitary operator U is a Hilbert space homomorphism, i.e. U : H → H is bijective and linear and ∀ u, v ∈ H : U u, U v = u, v; by the polarization identity, the latter property is equivalent to U being isometric, i.e. ∀ u ∈ H : U u = u). As we will see, the U (t), t ∈ R, are forming an evolution group. Definition 1.16. A family of operators (U (t))t∈R is called an evolution group, iﬀ

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ANDREAS M. HINZ

• • • •

∀ t ∈ R : U (t) is a unitary operator on H, U (0) = 1, ∀ s, t ∈ R : U (s + t) = U (s)U (t), ∀ u ∈ H : s → t ⇒ U (s)u → U (t)u.

This notion is the basis for Stone’s theorem (cf. [64]), which we will not prove either (cf. [65, Theorem 7.38]). Theorem 1.17. Let (U (t))t∈R be an evolution group in H and deﬁne the operator A by D(A) = {u ∈ H; Ψ := U (·)u is diﬀerentiable at 0} , Au = Ψ (0).

Then the operator i A is self-adjoint. We are now able to generalize Theorem 1.2 to every self-adjoint operator.

Theorem 1.18. Let T be a self-adjoint operator in H. Then for every u ∈ D(T ) the unique solution of the initial value problem Ψ(0) = u, ∀ t ∈ R : i Ψ (t) = T Ψ(t),

is given by Ψ(t) = U (t)u = exp(−i T t)u.

Proof. By virtue of Theorem 1.14 it is clear that U (t) = exp(−i T t) is deﬁned on all of H and isometric. Obviously, U (0) = 1. Furthermore, for u, v ∈ H and t ∈ R, we have Eλ U (t)u, v = U (t)u, Eλ v = exp(−i µ t) d Eµ u, Eλ v λ = exp(−i µ t) d Eµ u, v, −∞

whence, for s ∈ R,

U (s)U (t)u, v =

= =

exp(−i λ s) d Eλ U (t)u, v λ exp(−i λ s) d exp(−i µ t) d Eµ u, v −∞

exp(−i λ (s + t)) d Eλ u, v = U (s + t)u, v.

Surjectivity of U (t) now follows from U (t)U (−t) = 1. Next we show that −i T u is the derivative of U (·)u at 0 for any u ∈ D(T ). For h > 0 we have

2 2

1

{U (h)u − u} + i T u = exp(−i λ h) − 1 + i λ dEλ u2 .

h h For ε > 0, R > 0 can be chosen independently of h, such that 2 2 −R ∞ exp(−i λ h) − 1 exp(−i λ h) − 1 dEλ u2 + dEλ u2 + i λ + iλ h h −∞ R ∞ −R ε 2 2 2 2 ≤4 |λ| dEλ u + |λ| dEλ u < . 2 −∞ R

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13 13

With this R ﬁxed, we proceed further with 2 R exp(−i λ h) − 1 h2 4 R h2 4 2 2 + i λ R R u2 , dE u ≤ dE u ≤ λ λ h 4 4 −R −R ε for suﬃciently small h. 2 To prove continuity of U (·)u for any u ∈ H, we choose an approximating sequence D(T ) un → u and observe that

which is smaller than

U (s)u − U (t)u ≤ U (s)u − U (s)un + U (s)un − U (t)un + U (t)un − U (t)u = 2 u − un + U (s)un − U (t)un , which can be made small by ﬁrst choosing n large and then using the fact already established that U (·)un is diﬀerentiable and consequently continuous. So we have proved that (U (t))t∈R is an evolution group and T ⊂ i A for the operator A of Theorem 1.17. By that theorem, iA is self-adjoint, such that T = i A. Now let Ψ := U (·)u for u ∈ D(T ). Then Ψ(0) = u is obvious. For t, h ∈ R we have Ψ(h) − u 1 {Ψ(t + h) − Ψ(t)} = U (t) , h h and therefore, by continuity of the operator U (t), i Ψ (t) = i U (t)Au = i AU (t)u = T Ψ(t). Uniqueness of the solution can be shown based on symmetry of T as above for Theorem 1.2. Solving the Schr¨ odinger equation (0.1) has now been entirely reduced to estabishing self-adjointness of the operator S of Section 1.0 or any property of the handier operator S0 which in turn will guarantee self-adjointness of S. We will address the issue of suitable criteria in the following section. 1.2. Criteria for (essential) self-adjointness. The equivalence of self-adjointness of the operator T and the existence of an evolution group (exp(−i T t))t∈R , as expressed by Theorems 1.17 and 1.18, suggests the following idea: represent −n iT t ing exp(−i T t) formally by the compound interest formula lim 1 + , we n→∞ n should have n −1 n H=D T− i = T − i D(T ). t t This, in fact, leads to the fundamental criterion for self-adjointness. Theorem 1.19. Let T be a symmetric operator in H. Then the following are equivalent. o) T is self-adjoint, i) ∀ λ ∈ C \ R : (T − λ)D(T ) = H, ii) ∃ λ ∈ C : (T − λ)D(T ) = H = (T − λ∗ )D(T ).

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Proof. Let T be self-adjoint and λ ∈ C \ R. Then λ is not an eigenvalue of T by Lemma 1.11a, whence T − λ is injective. The operator (T − λ)−1 is bounded by virtue of Lemma 1.11c, and closed, since T is closed. By Corollary 1.6, the domain (T − λ)D(T ) of (T − λ)−1 is closed as well. Furthermore, (T − λ)D(T )⊥ = ((T − λ)∗ )

−1

−1

({0}) = (T ∗ − λ∗ )

({0})

∗ −1

= (T − λ )

({0}) = {0},

such that (T − λ)D(T ) = (T − λ)D(T ) = H. The implication from (i) to (ii) is trivial. Let λ ∈ C with (T − λ)D(T ) = H = (T − λ∗ )D(T ). Since T is symmetric, we only have to show that D (T ∗ ) ⊂ D(T ) to obtain self-adjointness of T . For u ∈ D (T ∗ ), there is a v ∈ D(T ) ⊂ D (T ∗ ) with (T − λ)v = (T ∗ − λ) u, since T − λ maps D(T ) onto H. From T ∗ v = T v it follows that (T ∗ − λ) (u − v) = (T ∗ − λ) u − (T − λ)v = 0 and consequently u − v ∈ (T ∗ − λ)−1 ({0}) = (T − λ∗ )D(T )⊥ = {0}, that is u = v ∈ D(T ).

In order to make use of Theorem 1.19, it is necessary to establish symmetry of the operator ﬁrst. This is, in general, not easy for an operator with maximal domain like S in Section 1.0. On the other hand, for a small, symmetric operator, like S0 , it is not evident that it has a self-adjoint extension at all. We will now introduce an important class of operators which do have self-adjoint extensions. Definition 1.20. Let T be a symmetric operator in H. Then µ := inf { T u, u; u ∈ D(T ), u = 1} is called the lower bound of T , and T is called semi-bounded (from below), iﬀ µ > −∞. For semi-bounded operators, the existence of a distinguished self-adjoint extension, the Friedrichs extension can be proved. Theorem 1.21. In H let T be a semi-bounded operator with lower bound µ. Deﬁne HT := u ∈ H; ∃ (un )n∈N ⊂ D(T ) : un → u and ∀ ε > 0 ∃ N ∈ N ∀ n ≥ N : | T (un − uN ), un − uN | < ε} . Then TF := T bound µ.

∗

(HT ∩ D (T ∗ )) deﬁnes a self-adjoint extension of T with lower

Sketch of proof. We may assume µ = 1. Then u, v = T u, v deﬁnes an inner product in D(T ) and every Cauchy sequence in (D(T ), · , ∗ ) is also a Cauchy sequence in (H, · , ∗ ), such that the completion of (D(T ), · , ∗ ) can be identiﬁed with (HT , · , ∗ ). Since · , ∗ is continuous, we have ∀ u ∈ HT ∩ D (T ∗ ) ∀ v ∈ HT : u, v = T ∗ u, v.

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15 15

As T ⊂ TF , it follows that TF is symmetric and again by continuity of · , ∗ that it has the same lower bound as T . To establish self-adjointness of TF by Theorem 1.19, it suﬃces to show that TF D (TF ) = H. For u ∈ H, ϕ → ϕ, u deﬁnes a bounded linear functional on (HT , · , ∗ ), and the representation theorem of Fr´echet and Riesz guarantees the existence of v ∈ HT with ϕ, u = ϕ, v for every ϕ ∈ HT . In particular, for ϕ ∈ D(T ) we have v, T ϕ = v, T ∗ ϕ = v, ϕ = u, ϕ, whence v ∈ HT ∩D (T ∗ ) = D (TF ) and TF v = T ∗ v = u. If a symmetric operator has a self-adjoint extension, it is not clear whether the latter, and consequently the corresponding spectral family, is unique. Definition 1.22. A densely deﬁned operator T in H is called essentially self-adjoint, iﬀ it has exactly one self-adjoint extension. An obvious suﬃcient condition for essential self-adjointness of T is the self, because then for every self-adjoint extension T of T we adjointness of its closure T ∗ ∗ ⊂ T = T , that is T = T . For symmetric T we can have T ⊂ T ⊂ T = T therefore employ the criteria of Theorem 1.19 as applied to T . In practice, this is not too easy, but it is unavoidable, because self-adjointness of T turns out to be also necessary for essential self-adjointness of T . The proof of the latter fact can be based on the following lemma. Lemma 1.23. Let T be a symmetric operator in H. Then U → T with D T = (U − 1)H, T = −i (U + 1)(U − 1)−1 , deﬁnes a bijection between the set of all unitary extensions U of (T − i)(T + i)−1 and the set of all self-adjoint extensions T of T . Proof. The domain of the mapping under consideration makes sense, because (T +i) is injective by Lemma 1.11a. Let U be a unitary extension of (T − i)(T + i)−1 . Then 1 (1.5) ∀ ϕ ∈ D(T ) : ϕ = (U − 1) i (T + i)ϕ , 2 such that D(T ) ⊂ D(T). U − 1 is injective: let u ∈ H with U u = u; then from (1.5) we get 1 ∀ ϕ ∈ D(T ) : u, ϕ = − i u, (U − 1)(T + i)ϕ 2 1 = − i { U u, U (T + i)ϕ − u, (T + i)ϕ} = 0, 2 the latter because U is unitary. Since D(T ) is dense in H, we get u = 0. Again from (1.5) we deduce 1 ∀ ϕ ∈ D(T ) : Tϕ = (U + 1)(T + i)ϕ = T ϕ, 2 such that T ⊃ T . T is symmetric, because for all u, v ∈ H we have

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ANDREAS M. HINZ

(U − 1)u, T(U − 1)v = = =

i (U − 1)u, (U + 1)v = i ( U u, v − u, U v) −i ( U u, U v + u, U v − U u, v − u, v) −i (U + 1)u, (U − 1)v T(U − 1)u, (U − 1)v . =

Finally,

1 (1.6) ∀ u ∈ H : u = T + i (U − 1) iu , 2 such that T + i D T = H and (1.7) ∀ u ∈ H : U T + i (U − 1)u = T − i (U − 1)u, whence also T − i D T = H. By Theorem 1.19, T is self-adjoint. Formula (1.7) implies that the inverse mapping must be given by −1 U = T − i T + i for any self-adjoint extension T of T . In fact, the mappings T ± i : D T → H are surjective by Theorem 1.19 and injective by Lemma 1.11a. U is isometric, since

2 2

∀ ϕ ∈ D T : T ± i ϕ = Tϕ + ϕ2

and therefore

−1 −1

2

2 T + i

+ T + i

= U u2 . T ∀ u ∈ H : u2 = u u

Finally, since T is an extension of T , we have ∀ ϕ ∈ D(T ) : U (T + i)ϕ = U T + i ϕ = T − i ϕ = (T − i)ϕ. The mapping in question is surjective: since −1 −1 ∀ u ∈ H : (U − 1)u = T − i T + i u − u = −2 i T + i u, we have (U − 1)H = D T and ∀ u ∈ H : T + i (U − 1)u = −2 i u = −i (U + 1)u + i (U − 1)u, whence T = −i (U + 1)(U − 1)−1 . The mapping in question is injective, because by (1.6) and (1.7) we have −1 1 ∀ u ∈ H : T − i T + i u = T − i (U − 1) i u = U u. 2 We will now collect criteria for essential self-adjointness of symmetric operators.

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17 17

Theorem 1.24. Let T be a symmetric operator in H. Then the following are equivalent. o) T is essentially self-adjoint, i) T is self-adjoint, ii) T ∗ is symmetric, iii) ∀ λ ∈ C \ R : (T − λ)D(T ) = H, iv) ∃ λ ∈ C \ R : (T − λ)D(T ) = H = (T − λ∗ )D(T ), and if T is semi-bounded with lower bound µ, v) ∀ λ < µ : (T − λ)D(T ) = H, vi) ∃ λ < µ : (T − λ)D(T ) = H.

∗ ∗ Proof. Let T be self-adjoint. ∗T is densely deﬁned, ∗because T ⊂ T , and from ∗ Corollary 1.8 we get T = T = T ; in particular, T is symmetric. Conversely, let assume that T ∗ is symmetric. Then, by Corollaries 1.8 and 1.10, we have ∗ us ∗ T = T ∗ and T is symmetric. Therefore, T ⊂ T = T ∗ ⊂ T ∗∗ = T , whence ∗ T = T . So we have established the equivalence of (i) and (ii).

To prove the mutual equivalence of (i), (iii) and (iv), we only have to show, in view of Theorem 1.19, that T − λ D T = (T − λ)D(T ) (1.8) for any symmetric operator T and λ ∈ C \ R. This follows from u∈ T −λ D T ⇔ ∃ (ϕn )n∈N ⊂ D(T ) : ϕn → ϕ ∈ D T , (T − λ)ϕn → T − λ ϕ = u ⇔ u ∈ (T − λ)D(T ), where we have made use of the continuity of (T − λ)−1 from Lemma 1.11c. The same argument also applies in the case of a semi-bounded operator for λ < µ, because for any normalized ϕ ∈ D(T ) we have (T − λ)ϕ ≥ | (T − λ)ϕ, ϕ| ≥ µ − λ > 0. So the implications from (v) through (vi) to (i) are proved. For the implication from (i) to (v), we remark that for any self-adjoint operator T and λ which is not an eigenvalue of T (and neither is λ∗ by Lemma 1.11 a) we have ⊥ ∗ −1 −1 T − λ D T = T − λ∗ ({0}) = T − λ∗ ({0}) = {0} and therefore T − λ D T = H. In particular, this is true for any λ less than the lower bound of T, which clearly cannot be an eigenvalue. In our situation, T −1 is densely is self-adjoint and semi-bounded with lower bound µ and so T − λ deﬁned, closed and, as above, bounded for any λ < µ. But then, by Corollary 1.6, its domain T − λ D T = (T − λ)D(T ) is H. We have already seen earlier (after Deﬁnition 1.22) that (i) implies (o). Now let T have the unique self-adjoint extension T and assume that T is not self-adjoint.

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ANDREAS M. HINZ

Then by Theorem 1.19 and (1.8) there is a v ∈ (T +i)D(T )⊥ with v = 1 (one may have to replace T by −T ). Since the operator Φ deﬁned on H by Φu = u − 2 u, vv ◦ Φ, where U is the unitary operator corresponding to T is unitary, so is U := U (T + i)D(T ) = according to Lemma 1.23. Furthermore, U (T + i)D(T ) = U −1 (T − i)(T + i) . As U v = −U v = 0, U = U , and Lemma 1.23 guarantees the existence of a self-adjoint extension of T diﬀerent from T. So (o) implies (i). One consequence of Theorem 1.24 is that a closed operator T is self-adjoint iﬀ T restricted to some (any) core of T is essentially self-adjoint. This allows for some choice of a suitable domain where the operator can easily be deﬁned. We are now prepared to apply these general results to the situation of the Schr¨ odinger operator of Section 1.0. 1.3. Application to Schr¨ odinger operators. Throughout this section, we are concerned with the operators dof Section 1.0, namely the minimal d Schr¨odinger operator S0 = − + V on C∞ 0 R with the potential V ∈ L2,loc R , whence S0 is a linear operator in H = L2 Rd . Furthermore, V is assumed to be real-valued, such that S0 is symmetric. As before, the adjoint operator S0∗ of S0 will be denoted by S. Note that for all u ∈ D(S) both u and V u are locally integrable. As S0 represents an energy operator with (1.9)

S0 ϕ, ϕ = ∇ϕ2 +

V |ϕ|2

interpreted as the expected value of the total energy of the (normalized) state ϕ, the operator S0 will be semi-bounded in most physical applications. The Friedrichs extension S0 F is then distinguished among all self-adjoint extensions of S0 by the fact that every element of its domain can be approximated in L2 Rd by a sequence of test functions which is convergent in the energy norm given by (1.9). (If the lower bound µ of S0 is not positive, one has to add (1 + µ)ϕ2 to produce a norm.) We illustrate the situation withthekinetic energy operator, i.e. the case where V = 0. 1 d Then obviously HS0 = W 2 R . On the other hand, u ∈ D(S) means that both d u and u are in L2 R . By an interpolation argument (cf. [31, Lemma 3a]), u ∈ W21 Rd , such that S = S0F and S0 is essentially self-adjoint. The methods used in literature to prove essential self-adjointness of S0 if V = 0 were mainly perturbational (cf. [36]; for an historical outline, see [32, Chapter 3]). We want to present a more direct approach, based on Theorem 1.24, namely we will prove that (S0 − λ)D(S0 )⊥ = {0} for (some) λ < µ, that is, we have to show that λ is not an eigenvalue of S. What one can achieve in that direction is the following. d R be real-valued and S0 = − + V on Proposition 1.25. Let V ∈ L 2,loc d ∞ C0 R be semi-bounded with lower bound µ. Then there are no locally bounded eigenfunctions for S = S0∗ and λ < µ. d R , Su = 0. We will Proof. We may assume λ = 0. Let u ∈ L2 Rd ∩L∞,loc 1 Rd , and we may also assume show that u = 0. By [31, Lemma 3a], u ∈ W2,loc that u is real-valued.

19 19

TOPICS FROM SPECTRAL THEORY OF DIFFERENTIAL OPERATORS

For ε > 0 and k ∈ N consider ψ := uε ηk2 , where uε is the regularized u (cf., |·| , with η a mesa function e.g., [32, Deﬁnition and Lemma 2.2]) and ηk = η k 1 (smooth, with values in [0, 1], 1 in B 0; 2 , 0 outside B (0; 1)). Then 0 = u, S0 (ψε ) = u, −ψε + u, V ψε ∇u · ∇ψε + V u, ψε = ∇uε · ∇ψ + (V u)ε , ψ 2 = ηk |∇uε | + 2 uε ηk ∇uε · ∇ηk + V uε , uε ηk2 + (V u)ε − V uε , uε ηk2 2 = uε ηk , S0 (uε ηk ) − |∇ηk | uε + (V u)ε − V uε , uε ηk2 2 max |η | 2 2 uε L2 (B(0;k)) ≥ µ uε ηk − k2 − (V u)ε − V uε L1 (B(0;k)) uL∞ (B(0;k+ε)) . =

As ε → 0, we get 2

u ηk ≤

max |η |2 u2 , µ k2

since both (V u)ε and V uε tend to V u in L1 (B (0; k)). For k → ∞, we arrive at u2 ≤ 0, whence u = 0.

So we are left with the problem of ﬁnding suﬃcient conditions for V which guarantee local boundedness of weak eigensolutions, i.e. solutions of u = V u in L1,loc Rd (again we assume λ = 0). We apply Green’s representation formula d R : ϕ = − s(· − y)ϕ(y) dy, ∀ ϕ ∈ C∞ 0 1 where s(z) = ωd

|z|−1

ρd−3 dρ, with ωd the area of the unit sphere in Rd , to ϕ = |·| ηr0 (x − ·)uε ; here ηr = η with a mesa function η, and uε is the regularized r u as before. Then, for ε → 0, we get u(x) = u (r0 ; x) − s(x − y)ηr0 (x − y)u(y) dy, 1

(r; ·) at r = r0 for where u (r0 ; ·) ∈ C∞ Rd . So it suﬃces to investigate vr := u − u s(x − y)|u(y)| dy, local regularity. Since vr0 − vr ∈ C∞ Rd and |vr (x)| ≤ B(x;r)

this suggests the following deﬁnitions. Definition 1.26. (Ω Rd means Ω ⊂ Rd is open and bounded.) d 0 Kloc R := f : Rd → C measurable; ∀ Ω R ∃ r ≤ 1 : sup d

x∈Ω B(x;r)

s(x − y)|f (y)| dy < ∞ ,

20

ANDREAS M. HINZ

Kloc Rd :=

f : Rd → C measurable;

∀ Ω R : lim sup d

r→0 x∈Ω B(x;r)

s(x − y)|f (y)| dy = 0 .

Then u ∈ K0loc Rd implies u ∈ L∞,loc Rd and u ∈ Kloc Rd guarantees continuity of u. The problem with assumption on this is the question: under which V do we have u = Vu ∈ K0loc Rd ? Of course, V ∈ K0loc Rd would be suﬃcient, if we knew u ∈ L∞,loc Rd — but that is what we are about to prove! The way out is an iteration process necessitating the local Kato condition V ∈ Kloc Rd . Moreover, with the interpretation of |u(x)|2 as the density of the probability to encounter the particle described by u at x in mind, it is clear that a local singularity of u at x can only occur, if the potential is strongly attractive at x, i.e. if the negative part V− := max{0, −V } of V is very singular at x. So it seems reasonable that only This is substantiated by Kato’s inequality a local assumption on V− is necessary. |u| ≥ sign(u)u in L1,loc Rd (cf. [19, p. 357–359]) with the consequence that |u| ≥ V |u| ≥ −V− |u|. So we can make use of the following regularity statement, which can be found in [32, Theorem 2.1]. Proposition 1.27. Let f ∈ Kloc Rd be real-valued, v ∈ L1,loc Rd non negative with f v ∈ L1,loc Rd and ( + f ) v ≥ 0. Then v ∈ L∞,loc Rd . Together with Proposition 1.25 and Theorem 1.24 we arrive at Theorem 1.28. Let V ∈ L2,loc Rd be real-valued, V− ∈ Kloc Rd and such Rd is semi-bounded. Then S0 is essentially self-adjoint that S0 =− + V on C∞ 0 d in L2 R . For d ≤ 3, L2,loc Rd ⊂ Kloc Rd (cf. [32, Proposition 1.9(ii)]), and therefore the assumption on V− is always fulﬁlled. Although some assumption on V− is needed for d ≥ 5, the local Kato condition on V− is not necessary for essential selfadjointness of theoperator S0 , as can be seen from the example V (x) = −α|x|−2 , d / Kloc R (cf. [32, Example 1.7]), but S0 is bounded from below iﬀ where V− ∈ 2 2 d−2 d−2 and essentially self-adjoint iﬀ α ≤ − 1 (cf. [19, VII Propoα≤ 2 2 sition 4.1]). The case d = 4 seems to be open. Another open problem for d ≥ 4 is J¨ orgens’s conjecture (1972) (cf. [8]). Conjecture 1.29. Let V, W ∈ L2,loc Rd be real-valued and V ≤ W . Rd is bounded from below and essentially in L2 Rd , If − + V C∞ 0 self-adjoint ∞ d d then − + W C0 R is essentially self-adjoint in L2 R . d If S0 = −+V on C∞ R is semi-bounded with bound µ, then, as mentioned 0 before, the property (P)

all eigenfunctions for S0∗ and λ are locally bounded

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21 21

for some λ < µ is suﬃcient for essential self-adjointness of S0 (Proposition 1.25 and Theorem 1.24, vi ⇒ o). If one can prove that it is also necessary, then it would suﬃce to prove, e.g. by subharmonic comparison (cf. [30]), that property (P) is preserved under a positive perturbation of S0 . Let us ﬁnally mention that boundedness from below of S0 can be guaranteed by the assumption of a global Kato condition V− ∈ K Rd (where Ω = Rd in Deﬁnition 1.26) due to relative form boundedness with respect to − (cf. [32, Corollary 3.3]). Moreover, by truncating the negative part of the potential, we may even use Theorem 1.28 to obtain essential self-adjointness of Schr¨ odinger operators which are not bounded from below, namely allowing for a behavior of the potential like −O |x|2 at inﬁnity (cf. [32, Theorem 3.4]). Theorem 1.30. Let V ∈ L2,loc Rd be real-valued and V− ∈ K Rd + O |x|2 . Rd is essentially self-adjoint in L2 Rd . Then − + V C∞ 0 Under the same assumptions on V , this approach allows d to treat magnetic 2 R , as long as b is Schr¨ odinger operators as well, i.e. − (∇ − i b) + V C∞ 0 continuously diﬀerentiable as a function from Rd to Rd (cf. [33, Theorem 2.1]); if one employs a method of H. Leinfelder and C. G. Simader, one can even cover d and ∇ · b ∈ L2,loc Rd (cf. [34, the most general case, where b ∈ L4,loc Rd Theorem 2.5]).

2. Hardy-Rellich inequalities The statement of J¨orgens’s conjecture is an example of an argument from perturbation theory, which for a long time had been predominant in the spectral theory of diﬀerential operators, in particular for questions of self-adjointness. In this chapter, we want to present such a perturbational argument which is based on estimates of types ﬁrst studied by G. H. Hardy and F. Rellich. After some general background about relative boundedness in Section 2.0 we will give a short account of how to arrive at Hardy and Rellich type weighted estimates (Section 2.1) and produce sharp constants for Hardy-Rellich inequalities (Section 2.2) in the spirit of [14]. For more details, including the early literature, we refer to that paper. 2.0. Relative boundedness. In view of Theorems 1.21 and 1.28, d the ﬁrst R is semiproperty we have to establish for the operator S0 = − + V C∞ 0 α boundedness. For instance, if d = 3 and the potential is given by V (x) = − |x| for some constant α > 0 (hydrogen atom with Coulomb interaction), we have for R3 and ε > 0: ϕ ∈ C∞ 0 |ϕ(x)|2 2 S0 ϕ, ϕ = ∇ϕ − α dx |x| 2 |ϕ(x)| 1 ε 2 ≥ ∇ϕ − dx − α2 ϕ2 , 2ε |x|2 2 such that S0 is semi-bounded, if

22

ANDREAS M. HINZ

3 R : ∀ ϕ ∈ C∞ 0

(2.1)

|ϕ(x)|2 dx ≤ 2 ε ∇ϕ2 |x|2

1 for some ε > 0. We have employed here the useful inequality 2ab ≤ ε a2 + b2 , ε 2 √ 1 which follows immediately from ε a − √ b ≥ 0, but is also a consequence of ε the following even more useful inequality, which we reproduce for convenience. Lemma 2.1. ∀ γ ≥ 0 ∀ ε > 0 ∀ a, b ≥ 0: a

1+γ

1+γ

+b

≤ (a + b)

1+γ

≤ (1 + ε) a γ

1+γ

γ 1 + 1+ b1+γ . ε

Proof. We may assume γ > 0 and b = 1. The ﬁrst inequality follows readily from Taylor’s formula. For the second inequality we deﬁne f (a) = (a + 1)1+γ − (1 + γ γ γ = (1+γ) ε)γ a1+γ . Then f (0) = 1, f (∞) = − ∞ and f(a) γ((a + 1) − (1 + ε) a ), 1 1 1 which is 0 iﬀ a = , such that f (a) ≤ f . = 1+ ε ε ε Inequality (2.1) is similar to Hardy’s original inequality for d = 1: b b |ϕ(x)|2 (2.2) dx ≤ 4 |ϕ (x)|2 dx, 2 x a a (cf. [26, p. 110]), which will turn out to hold for d = 3 as well (cf. infra, Example 2.14), i.e. (2.1) is fulﬁlled with ε = 2. (For an overview of Hardy-type inequalities, see [13].) Furthermore, we then have, using the same trick as before, |ϕ(x)|2 2 2 V ϕ = α dx 2 |x| ≤ 4 α2 |∇ϕ(x)|2 dx = − 4 α2 ϕ(x) ϕ(x)∗ dx ≤ 2 α2 ε ϕ2 +

2 α2 ϕ2 . ε

1 This means that V is relatively bounded with respect to − and putting ε < 2 α2 even relatively small w.r.t. −. Definition 2.2. Let T and τ be linear operators in a normed space H with D(T ) ⊂ D(τ ). Then ι := inf {a ≥ 0; ∃ A ≥ 0 ∀ u ∈ D(T ) : τ u ≤ a T u + A u} is called the relative bound of τ with respect to T in H. If ι < ∞ (< 1), then τ is called relatively bounded (small) w.r.t. T in H. Remark 2.3. The estimate in the deﬁnition of ι can be replaced by τ up ≤ ap T up + Ap up

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for any p ∈ [1, ∞[.

23 23

This is an easy consequence of Lemma 2.1. The famous perturbation theorem of Rellich and Kato (cf. [36, p. 287]) reads as follows. Theorem 2.4. Let T be a self-adjoint operator in a Hilbert space H. If τ is a symmetric operator in H and relatively small w.r.t. T , then T + τ D(T ) is self-adjoint. Proof. Obviously, T + τ D(T ) is symmetric. We have for some ι < a < 1, A > 0 A and every λ ∈ C with re(λ) = 0 and |λ| ≥ : a 2 ∀ u ∈ D(T ) : τ u2 ≤ a2 T u + A2 u2 ≤ a2 T u2 + |λ|2 u = a2 (T − λ)u2 . By Theorem 1.11c, (T − λ) : D(T ) → H is bijective, such that

1.19 and Lemma ∀ v ∈ H : τ (T − λ)−1 v ≤ av. Now, if an operator B : H → H is bounded with B < 1 and u ∈ H, then f : H → H given by f (v) = u−Bv is a contraction, such that by Banach’s ﬁxed point theorem there is a v with u = (1 + B)v, i.e. 1 + B is surjective. It follows that 1 + τ (T − λ)−1 : H → H and consequently (T + τ − λ) : D(T ) → H is surjective. Again by Theorem 1.19, T + τ is self-adjoint. Essential self-adjointness is also stable under relatively small perturbations. Corollary 2.5. Let T be an essentially self-adjoint operator in a Hilbert space H. If τ is a symmetric operator in H and relatively small w.r.t. T , then T +τ D(T ) is essentially self-adjoint with D T + τ = D T , T + τ = T + τ . Proof. By Corollary 1.10, τ is symmetric. Obviously, τ is T -small and D T ⊂ D T + τ . By Theorem 2.4, T + τ is self-adjoint on D T . Because this is a closed operator, we have T + τ ⊂ T + τ . α This establishes essential self-adjointness of our example S0 = − − on |·| 3 3 3 ∞ 1 C0 R in L2 R with D(S) = W2 R (cf. Section 1.3). To guarantee relative smallness w.r.t. − for a potential with a quadratic local α singularity, V (x) = − 2 , we need a stronger estimate, namely |x| (2.3)

|ϕ(x)|2 dx ≤ c2 ϕ2 , |x|4

such that V ϕ2 ≤ α2 c2 ϕ2 , and V is relatively small w.r.t. − provided that 1 α < . This kind of inequality has been established by Rellich [46, p. 247–249] for c 4 . dimensions d ≥ 5 and with c = d(d − 4) A further motivation to study inequalities of Hardy and Rellich type or, more general, Hardy-Rellich inequalities (2.4)

∀ u ∈ Wpk : V uLp ≤ c ∇k uLp

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ANDREAS M. HINZ

for potential functions V with local singularities, k ∈ N and p ∈ [1, ∞[, and to establish optimal constants c (cf. [44, 14]) comes from the fact that relative boundedness with respect to the operator − in Lp (Ω) with Dirichlet boundary conditions for a bounded region Ω with smooth boundary ∂Ω can be reduced to such a local for these operators [14, Lemma 7]. (For problem. In fact, C∞ 0 Ω forms d a core ∞ ∩ C (Ξ) := ϕ ∈ C R (Ξ); supp(ϕ) ⊂ Ω is bounded .) Moreover, Ξ ⊂ R d , C∞ 0 if for x ∈ Ω we deﬁne c(x) := lim ι (B(x; r) ∩ Ω) with ι(Ψ) denoting the relative r→0 Ψ , then it can be shown using an appropribound of V with respect to − C∞ 0 ate partition of unity [14, Lemma 8] and a compactness argument [14, Lemma 9] that c is upper semi-continuous and that max c(x); x ∈ Ω is the relative bound of V w.r.t. − in Lp (Ω) [14, Theorem 10]. Another application of (2.4) is in investigations about the domain of higher order operators (cf. [15, 45, 12]; a general overview of the Lp -theory of higherorder operators is given in [11]). Optimal constants have also been obtained for p = 2 and non-integer k in [68, 20]. 2.1. Weighted estimates. How to obtain an inequality of the type (2.4)? Let us start with the easiest case k = 1, i.e. we want to establish ∞ p p X|ϕ| ≤ c Y |∇ϕ|p ∀ ϕ ∈ C0 (Ω) : for some (positive) weights X and Y on the open set Ω ⊂ Rd and with explicit constants c. In order to make use of information about ∇ϕ, X has to be of the 2 (Ω), such that formally we get by integration by form −V for some V ∈ W1,loc parts and using H¨ older’s inequality: X |ϕ|p = ∇V · ∇ (|ϕ|p ) = p |ϕ|p−1 ∇V · ∇ϕ p−1 1−p ≤ p |ϕ|p−1 X p X p |∇V | |∇ϕ| p−1 p1 p |∇V |p p p ≤ p X |ϕ| |∇ϕ| . X p−1 By cancelling and raising both sides to the power p, we arrive at 2 (Ω) with V < 0. Then Theorem 2.6. Let V ∈ W1,loc |∇V |p |∇ϕ|p |V | |ϕ|p ≤ pp |V |p−1

for all ϕ ∈ C∞ 0 (Ω).

In this formal derivation we used ϕ ∈ C∞ 0 (Ω) and ϕ > 0 simultaneously, which is, of course, absurd. However, we may approximate |ϕ|p by p/2 − εp ∈ C∞ 0 ≤ ϕε := |ϕ|2 + ε2 0 (Ω), which has the same support as ϕ, and apply dominated convergence.

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25 25

For the more ambitious case k = 2 of (2.4), we proceed in the same manner: X |ϕ|p = − V |ϕ|p = − V (|ϕ|p ) p−2 2 = −p (p − 1) V |ϕ| |∇ϕ| − p V |ϕ|p−1 ϕ, and we arrive at 2 Lemma 2.7. Let V ∈ W1,loc (Ω) with V ≥ 0 and V < 0, and assume that there is a c ≥ 0 such that c |V | |ϕ|p ≤ p (p − 1) V |ϕ|p−2 |∇ϕ|2 {x∈Ω; ϕ(x)=0}

for all ϕ ∈

C∞ 0 (Ω).

Then

(1 + c)p

|V | |ϕ|p ≤ pp

Vp |ϕ|p |V |p−1

for all ϕ ∈ C∞ 0 (Ω).

A suﬃcient condition for V to fulﬁl the assumptions of Lemma 2.7 is given in the following statement. 2 Lemma 2.8. Let p > 1. If 0 < V ∈ C(Ω) ∩ W1,loc (Ω) with V < 0 and δ (V ) ≤ 0 for some δ > 1, then V |ϕ|p−2 |∇ϕ|2 < ∞ (δ − 1) |V | |ϕ|p ≤ p2 {x∈Ω; ϕ(x)=0}

for all ϕ ∈

C∞ 0 (Ω).

1 Proof. The assumption on V δ makes sense, because V ∈ W2,loc (Ω) by an interpolation argument (cf. [31, Lemma 3]) and therefore V δ = δ V δ−2 (δ − 1) |∇V |2 + V V

by regularization. The case p = 2 can now be based on Theorem 2.6: |∇V |2 |∇ϕ|2 (δ − 1) |V | |ϕ|2 ≤ 4 (δ − 1) |V | 2 ≤ 4 V |∇ϕ| = 4

V |∇ϕ|2 ,

{x∈Ω; ϕ(x)=0}

the latter since |{x ∈ Ω; ϕ(x) = 0 = |∇ϕ(x)|}| = 0 (cf. [17, VII. Corollary 20.3]). The general case for p is reduced to this by approximating |ϕ|p/2 as before. p−1 Combining Lemma 2.8 and Lemma 2.7 (with c = (δ − 1)), we arrive at p the fundamental weighted estimate. 2 Theorem 2.9. If 0 < V ∈ C(Ω) ∩ W1,loc (Ω) with V < 0 and (V δ ) ≤ 0 for some δ > 1, then p2p Vp p |ϕ|p |V | |ϕ| ≤ {(p − 1) δ + 1}p |V |p−1

26

ANDREAS M. HINZ

for all ϕ ∈ C∞ 0 (Ω). Remark 2.10. The inequality in Theorem 2.9 remains valid for all ϕ ∈ if Ω is bounded and V ∈ W12 (Ω) (cf. [14, Theorem 11]).

C∞ 0

Ω ,

2.2. Explicit bounds. Our main goal in this section is to obtain sharp constants c for bounds of the type |u(x)|p |u(x)|p dx ≤ c dx |x|β |x|α for functions u which lie in appropriate Sobolev spaces. By iteration, these estimates yield similar bounds in which u on the right-hand side is replaced by m u or by ∇(m u). We will be able to show that these iterated bounds also have sharp constants. We start by putting Ω = Rd \{0} in the considerations of the preceding section. If V (x) = |x|−κ for some κ > 0, then (V δ )(x) = κ δ (κ δ − (d − 2)) |x|−(2+κδ) for any δ ≥ 1, so we may apply Theorem 2.9 whenever κ < d − 2 and, to get the d−2 best constant, for δ = , yielding κ |ϕ(x)|p |ϕ(x)|p p dx ≤ c dx (2.5) 2+κ |x| |x|2+κ−2p d R \ {0} , where for all ϕ ∈ C∞ 0 c=

p2 . {d − 2 − κ}{(p − 1)(d − 2) + κ}

For p = 2 = κ and d > 4, this includes Rellich’s original inequality (2.3). Formula (2.5) is the case m = 1 of the following result (with β = 2 + κ). Theorem 2.11. Let m ∈ N and 2 (1 + (m − 1) p) < β < d. Then |m ϕ(x)|p |ϕ(x)|p p dx ≤ c(d, m, p, β) dx β |x| |x|β−2mp d for all ϕ ∈ C∞ R \ {0} , where 0 m−1 d−2 c(d, m, p, β) = γ d, β − 2 (1 + k p), ,p β − 2 (1 + k p) k=0

and γ(d, β, δ, p) =

p2 . β{d − 2 − β}{(p − 1) δ + 1}

The proof of Theorem 2.11 is by induction on m, where the induction step is carried out by applying (2.5) with ϕ and κ replaced by m ϕ and β − 2 (1 + m p), respectively. The same procedure which led to (2.5) can be used in Theorem 2.6 with the result |∇ϕ(x)|p |ϕ(x)|p p dx ≤ c dx, (2.6) |x|2+κ |x|2+κ−p

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27 27

d R \ {0} and all 0 < κ < d − 2, where for all ϕ ∈ C∞ 0 p c= . d−2−κ Since there is no restriction on the sign of V in Theorem 2.6, however, we may put V (x) = sign(κ) |x|−κ to prove the validity of (2.6) for all −∞ < κ < d − 2 (in fact any κ = d − 2 replacing c by | c|), the case κ = 0 following from a limiting process. Here p = 2 and κ = 0 leads to the Hardy inequality for d = 2 (cf. (2.1)): 4 |ϕ(x)|2 dx ≤ |∇ϕ(x)|2 dx. |x|2 (d − 2)2 Again this can be extended to a generalized Hardy-Rellich inequality: Theorem 2.12. Let m ∈ N0 and 2 (1 + (m − 1) p) < β < d. Then |∇(m ϕ)(x)|p |ϕ(x)|p p dx ≤ c (d, m, p, β) dx, β |x| |x|β−(2m+1) p d for all ϕ ∈ C∞ R \ {0} , where 0 p c(d, m, p, β) = c(d, m, p, β). d + 2mp− β The proof is based on Theorem 2.11, combined with formula (2.6), where ϕ and κ are replaced by m ϕ and β − 2 (1 + m p), respectively, as before. d−β As long as 0 < κ < , one may consider functions u with u(x) = O (|x|−κ ) p at the origin, such that by induction: m−1 m m u ≈ (−1) (κ + 2 k) (d − 2 − (κ + 2 k)) |x|−(κ+2m) k=0

for every m ∈ N0 , whence m−1 m | u| ≈ (κ + 2 k) (d − 2 − (κ + 2 k)) |x|−(κ+2m) k=0

and

|∇( u)| ≈ m

(κ + 2 m)

m−1

(κ + 2 k) (d − 2 − (κ + 2 k)) |x|−(κ+2m+1)

k=0

near the origin.

The products in curly brackets are equal to c(d, m, p, β) and d−β , which shows that these constants are c(d, m, p, β), respectively, for κ = p sharp. We ﬁnally obtain Hardy-Rellich inequalities of the form (2.4), if the weights in the right-hand integrals of Theorems 2.11 and 2.12 are bounded, which is the case Rd \ {0} is dense for β = 2 m p and β = (2 m + 1) p, respectively. Noting that C∞ 0 k d in Wp R as long as d > k p, we arrive at Corollary 2.13. Let m ∈ N0 , d ∈ N.

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ANDREAS M. HINZ

i) If 1 < p < ∞ and d > 2 m p, then |u(x)|p p dx ≤ K(m, p, d) |m u(x)|p dx |x|2 m p for all u ∈ Wp2m Rd , where −1 m 2m K(m, p, d) = p (d − 2 k p) (2 (k − 1) p + (p − 1) d) ; k=1

ii) if 1 ≤ p < ∞ and d > (2 m + 1) p, then |u(x)|p p dx ≤ K(m, p, d) |∇(m u)(x)|p dx |x|(2 m+1) p for all u ∈ Wp2m+1 (Rd ), where m −1 2m+1 p K(m, p, d) = (d − (2 k + 1) p) ((2 k − 1) p + (p − 1) d) . d−p k=1

We end this chapter by recovering the classical case p = 2 of Corollary 2.13. Example 2.14. Let m ∈ N0 , n ∈ N. Then K(m, 2, 4 m + n) =

Γ( n ) Γ( n ) n 4 , K(m, n 4 2, 4 m + 2 + n) = 4m Γ 4 + 2 m 2 · 4m Γ 4 + 2 m + 1

in Corollary 2.13. This follows from the formula ∀ x > 0 ∀ M ∈ N0 : Γ(x + M ) = Γ(x)

M −1

(x + k).

k=0

3. Spectral properties of radially periodic Schr¨ odinger operators The mathematical interest associated with Schr¨odinger operators − + V can be characterized either by topical issues or by considering variants of the operator itself. Central in most investigations is the spectrum of the operator (cf. Section 3.0). ` E. ` Shnol’, the interest in the connecStarting in the early 1950s in the work of E. tions between the spectrum of Schr¨ odinger operators and the asymptotic behavior of eigensolutions, to which we will refer in Section 3.1, reached its peak in the 1970s and early 1980s (cf. [32, Chapter 4], [59, Chapter IV]). As in the question of self-adjointness, addressed above in Section 1.3, the (local) Kato class appears again as a natural setting for potential functions V . This was accompanied by the development of semigroup techniques and scattering theory, which led to a more thorough investigation into the ﬁne structure of the spectrum these last twenty years. The discovery of peculiar spectral phenomena, like e.g. embedded eigenvalues, dense point spectrum and singular continuous spectrum, formed a motivation to consider magnetic (cf. [59, Chapter X]), random (cf. [59, Chapter VII]) and in particular one-dimensional (cf. [59, Chapter V]) Schr¨ odinger operators as well as Dirac operators. Originally just a mathematical curiosity, there is now evidence

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29 29

that embedded eigenvalues can actually be observed experimentally (cf. [5]), thus adding to the revived interest in these questions. In this chapter we will concentrate on a class of operators which may not have received the attention it deserves, namely spherically symmetric Schr¨ odinger operators (Section 3.2). It turns out that they live a life in between their one-dimensional and general higher dimensional brethren. On one hand, their spectra are qualitatively diﬀerent from the spectra of the corresponding Sturm-Liouville operators; on the other hand the emergence of phenomena like embedded eigenvalues and dense point spectrum can be obtained by recourse to the one-dimensional theory. The spectral properties especially of radially periodic Schr¨ odinger operators (Section 3.3) are therefore extra-ordinary in every sense of the word. For the same reason, spherically symmetric Schr¨odinger operators can be approached by numerical methods too, and we will report on an example of a numerical investigation into the distribution of eigenvalues in intervals of dense point spectrum in Section 3.4. We hope that the elementary character of the cases treated in this chapter will contribute to the understanding of non-orthodox patterns in the spectral theory of diﬀerential operators. 3.0. Spectra of self-adjoint operators. Throughout this chapter, T will be a self-adjoint operator in a non-trivial complex Hilbert space H. Its spectral family according to Theorem 1.14 will be denoted by (Eλ )λ∈R . The spectrum σ of T and its parts are well-deﬁned. We will consider the decompositions σe (T ) ∪· σd (T ) = σ (T ) = σc (T ) ∪ σp (T ), where σp is the point spectrum, i.e. the set of eigenvalues, σd is the discrete spectrum, i.e. the set of eigenvalues of ﬁnite multiplicity which are isolated from other elements of the spectrum, with the essential spectrum σe being its complement in σ, and σc denotes the continuous spectrum. A further decomposition of the latter into an absolutely continuous and a singular continuous part, useful in scattering theory, will not be pursued here. Operators T are used in applications to represent measurable quantities like for instance the energy of a system of particles in a certain (normalized) state u ∈ D(T ). The expected value of that quantity is then eT (u) = T u, u, which is real if T is symmetric (Lemma 1.11a). The precision of the measurement is given by the variance sT , i.e. the square root of the expected value of (T − eT )2 : 1/2 sT (u) = (T − eT (u))2 u, u = T u − eT (u)u . Therefore, the physical quantity can be measured precisely for the state u if and only if u is an eigenvector (eigenfunction) of T and eT (u) is an eigenvalue (cf. Definition 1.3). The point spectrum σp (T ) := {λ ∈ C; λ is an eigenvalue of T } of T can be characterized in several ways. Theorem 3.1. For λ ∈ C, the following are equivalent. o) λ ∈ σp (T ), i) ∃ Cauchy sequence (un )n∈N ⊂ D(T ), un = 1 : eT (un ) → λ ∧ sT (un ) → 0,

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ANDREAS M. HINZ

ii) iii) iv) v)

∃ Cauchy sequence (un )n∈N ⊂ D(T ), un = 1 : (T − λ) un → 0, T − λ : D(T ) → H is not injective, (T − λ) D(T ) = H, λ ∈ R and Eλ− = Eλ+ .

Proof. “(o) ⇔ (ii)” and “(o) ⇔ (iii)” are immediate consequences of the deﬁnition of an eigenvalue. “(o) ⇔ (v)” follows from Lemma 1.11a and Lemma 1.15. For “(o) ⇒ (iv)”, assume that (T − λ) D(T ) = H. Then (T − λ∗ ) D(T ) = H as well, since by Theorem 1.19, (T − λ) D(T ) = H for any λ ∈ C \ R. Therefore −1

(T − λ)

−1

({0}) = (T ∗ − λ)

({0}) = (T − λ∗ ) D(T )⊥ = {0},

i.e. λ ∈ / σp (T ). The implication “(iv) ⇒ (o)” has been shown earlier, in the proof of Theorem 1.24. The equivalence of (i) and (ii) is an immediate consequence of the following lemma. Lemma 3.2. For λ ∈ C and any sequence (un )n∈N ⊂ D(T ) with un = 1, we have eT (un ) → λ ∧ sT (un ) → 0 ⇔ (T − λ) un → 0. Proof. “⇒”: (T − λ) un ≤ T un − eT (un ) un + eT (un ) un − λ un = sT (un ) + |eT (un ) − λ| → 0. “⇐”: | (T − λ) un , un | ≤ (T − λ) un → 0 and (T − eT (un )) un ≤ (T − λ) un + |λ − eT (un )| → 0. for Remark 3.3. In the situation of Lemma 3.2, eT (un ) → λ is not suﬃcient 1 1 1 0 2 (T − λ) un → 0; example [55]: H = C , T u = u, un = √ , √ , 0 −1 2 2 1 1 such that eT (un ) = 0, sT (un ) = 1 and T un = √ , − √ . 2 2 The point spectrum of a symmetric operator in a separable Hilbert space (like L2 Rd ) is at most countable (cf. infra, Remark 3.19). On the other hand, many observable quantities allow for continuous values. In view of (i) in Theorem 3.1, it is therefore reasonable to consider approximate eigenvalues and deﬁne the spectrum of T by σ(T ) := {λ ∈ C; ∃ (un )n∈N ⊂ D(T ), un = 1 : eT (un ) → λ ∧ sT (un ) → 0}. (Such a sequence is then called a characteristic sequence for λ ∈ σ(T ).) Again there are a couple of characterizations of the spectrum of T .

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31 31

Theorem 3.4. For λ ∈ C, the following are equivalent. o) i) ii) iii) iv) v)

λ ∈ σ(T ), ∃ (un )n∈N ⊂ D(T ), un = 1 : eT (un ) → λ ∧ sT (un ) → 0, ∃ (un )n∈N ⊂ D(T ), un = 1 : (T − λ) un → 0, T − λ : D(T ) → H is not bijective, (T − λ) D(T ) = H, λ ∈ R and ∀ ε > 0 : Eλ−ε = Eλ+ε .

Proof. The equivalence of (o), (i) and (ii) is clear from the deﬁnition of the spectrum and Lemma 3.2. To prove the equivalence of (ii), (iii) and (iv), we show for T := T − λ : D(T ) → H the chain of implications T is surjective ⇒ T is bijective ⇒ T is injective and T−1 is bounded ⇒ T is surjective. For the ﬁrst implication, we know from Theorem 3.1 that T is injective for non-real λ; for real λ, we have T−1 ({0}) = (T ∗ − λ∗ )−1 ({0}) = T D(T )⊥ = {0}. Now let T be bijective. Since T is closed, so is T−1 . The latter operator is deﬁned on the whole space H and therefore bounded by virtue of the so-called closed graph theorem (cf. [17, VI. Theorem 8.4]). For the last implication, Theorem 3.1 tells us that D T−1 is dense in H. Moreover, T−1 is closed and bounded and therefore its domain closed (cf. Corollary 1.6). To prove “(ii) ⇒ (v)”, we ﬁrst remark that T−1 is surjective for any λ ∈ C \ R by Theorem 1.19, such that σ(T ) ⊂ R. Now suppose there is an ε > 0 such that Eλ−ε = Eλ+ε . Then Eκ = Eµ for all λ − ε ≤ κ ≤ µ ≤ λ + ε, such that λ+ε f (µ) dEµ u2 = 0 for any f ∈ C(R) and u ∈ D(T ), whence λ−ε

2

(T − λ) u =

(µ − λ)2 dEµ u2 ≥ ε2

dEµ u2 = ε2 u2 .

Finally, for “(v) ⇒ (ii)”, we ﬁnd for every n ∈ N a un = Eλ+ n1 − Eλ− n1 un with un = 1, and we have 1 λ+ n

(T − λ) un 2

=

(µ − λ)2 d Eµ un 2 =

≤ n−2

(µ − λ)2 d Eµ un 2 1 λ− n

d Eµ un = n−2 → 0. 2

An easy consequence of Theorem 3.4 is the closedness of the spectrum.

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R.

ANDREAS M. HINZ

Corollary 3.5. The spectrum of a self-adjoint operator T is a closed set in

Proof. Let (λn )n∈N ⊂ σ(T ) with λn → λ ∈ R. Then for every n ∈ N there is a 1 un ∈ D(T ) with un = 1 and (T − λn ) un < . Hence n 1 (T − λ) un ≤ (T − λn ) un + |λn − λ| un < + |λn − λ| → 0. n The last criterion in Theorem 3.4 provides a tool to detect spectrum in an interval. Corollary 3.6. For −∞ < a < b < ∞, we have σ(T ) ∩ ]a, b[ = ∅ ⇔ Ea = Eb− . 1 1 Proof. “⇒”: For n ∈ N with a + ≤ b − we have from Theorem 3.4: n n 1 1 ∀λ ∈ a + ,b − ∃ ε > 0 : Eλ−ε = Eλ+ε . n n 1 1 The intervals ]λ − ε, λ + ε[ cover a + , b − , such that by compactness this is n n already the case for ﬁnitely many of them ]λj − εj , λj + εj [, j ∈ {1, . . . , J}, where we may assume that λj+1 − εj+1 ≤ λj + εj ≤ λj+1 + εj+1 for j < J. We then 1 1 have Eλ1 −ε1 = Eλj +εj for all j and since λ1 − ε1 < a + and λJ + εJ > b − , we n n arrive at Ea+ n1 = Eb− n1 , whence Ea = Eb− . “⇐”: For λ ∈ ]a, b[ there is an ε > 0 with [λ − ε, λ + ε] ⊂ ]a, b[ and consequently Eλ−ε = Eλ+ε . By Theorem 3.4, λ ∈ / σ(T ). Letting a → − ∞ and b → ∞ in Corollary 3.6, we immediately get that the spectrum of a self-adjoint operator in a non-trivial Hilbert space is not empty, because E−∞ = 0 = 1 = E∞ . (The spectrum of the operator in H = {0} is obviously empty.) In many applications there is a lowest possible value for the physical quantity (e.g. energy) expressed by T . This should be reﬂected as the minimum of the spectrum of T , which exists if and only if T is semi-bounded from below. Lemma 3.7. With µ the lower bound of T , µ = inf σ(T ).

Proof. From Corollary 3.6 we learn that ∀ λ < inf σ(T ) : Eλ = 0. Therefore ∀ u ∈ D(T ) : T u, u = λ dEλ u2 ≥ inf σ(T ) dEλ u2 = inf σ(T ) u2 , whence µ ≥ inf σ(T ). On the other hand, if λ < µ, then ∀ u ∈ D(T ), u = 1 : 0 < µ − λ ≤ (T − λ) u, u ≤ (T − λ) u , such that λ ∈ / σ(T ) by Theorem 3.4.

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Another consequence of Corollary 3.6 is the following. Corollary 3.8. An isolated element λ of the spectrum of a self-adjoint operator T is an eigenvalue of T . Proof. By Corollary 3.6, there is an ε > 0 with Eλ− = Eλ−ε and Eλ = Eλ+ε , whence Eλ− = Eλ+ , since otherwise λ ∈ / σ(T ) by Theorem 3.4. But then λ ∈ σp (T ) by Theorem 3.1. In view of Theorem 3.4 v one may collect those points of the spectrum, where something “essential” happens into the essential spectrum σe (T ) := {λ ∈ R; ∀ ε > 0 : dim ((Eλ+ε − Eλ−ε ) H) = ∞} of T . An accumulation point λ of the spectrum belongs to the essential spectrum. To see this, let ε > 0 and take a sequence (λn )n∈N ⊂ σ(T ) converging to λ and with the property that for some 0 < εn → 0 the intervals ]λn − εn , λn + εn [ are mutually disjoint. Choose un ∈ (Eλn +εn − Eλn −εn ) H with un = 1. They are mutually orthogonal, and un ∈ (Eλ+ε − Eλ−ε ) H for suﬃciently large n, whence dim ((Eλ+ε − Eλ−ε ) H) = ∞. This also shows that σe (T ) is a closed set in R. By Corollary 3.8, the complement σd (T ) = σ(T ) \ σe (T ) of the essential spectrum inside the spectrum, called the discrete spectrum of T , consists of all isolated eigenvalues of ﬁnite multiplicity. A fundamental tool for the analysis of the essential spectrum is the following. |T − λ| , un = 1 and un 0. Lemma 3.9. Let λ ∈ R; (un )n∈N ⊂ D Then

2

dist (λ, σe (T )) ≤ lim inf |T − λ| un . n→∞

2

Proof. Let a := lim inf |T − λ| un < ∞ and assume that σe (T ) ∩ [λ−a, λ+a] = n→∞

∅. Then for every µ ∈ [λ−a, λ+a ] there is an ε > 0 with dim ((Eµ+ε − Eµ−ε ) H) < ∞. A compactness argument yields dim ((Eλ+a+ε − Eλ−a−ε ) H) < ∞ for some ε > 0. For E := Eλ+a+ε − Eλ−a−ε let (ek )k=1,...,K be an orthonormal basis of E H. Then K K

E un = E un , ek ek = un , E ek ek → 0, k=1

k=1

because un 0. Moreover,

2

|T − λ| un

= ≥ =

|µ − λ| dEµ un 2 |a + ε| {1 − (λ − a − ε < µ ≤ λ + a + ε)} dEµ un 2 2 |a + ε| un 2 − E un .

For n → ∞, the right-hand side tends to |a + ε|, in contradiction to the deﬁnition of a. The essential spectrum can be characterized as follows.

34

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Theorem 3.10. For λ ∈ C, the following are equivalent. λ ∈ σe (T ), ∃ (un )n∈N ⊂ D(T ), un = 1, un 0 : eT (un ) → λ ∧ sT (un ) → 0, ∃ (un )n∈N ⊂ D(T ), un = 1, un 0 : (T − λ) un → 0, λ ∈ R and ∃ (un )n∈N ⊂ D |T − λ| , un = 1, un 0 : |T − λ| un → 0, iv) λ ∈ R and ∀ ε > 0 : dim ((Eλ+ε − Eλ−ε ) H) = ∞.

o) i) ii) iii)

Proof. The equivalence of (o) and (iv) is by deﬁnition. Statements (i) and (ii) are equivalent by Lemma 3.2. If λ ∈ σe (T ), then either λ is an (isolated) eigenvalue of inﬁnite multiplicity or it is an accumulation point of the spectrum. In the ﬁrst case, the corresponding eigenspace contains an (at least) countably inﬁnite orthonormal system (un )n∈N , which converges weakly to 0 by Pythagoras’s theorem and therefore has the desired properties in (ii). For an accumulation point λ of the spectrum, we take the orthonormal sequence un = (Eλn +εn − Eλn −εn ) vn , vn ∈ H, as before and have 2 2 (T − λ) un = (µ − λ)2 d Eµ (Eλn +εn − Eλn −εn ) vn ≤ (εn + |λn − λ|)2 → 0. By Lemma 3.2 inconjunction with Lemma 1.11a, we have λ ∈ R in the situation of (ii), (un )n∈N ⊂ D |T − λ| and

2

|T − λ| un = |T − λ| un , un 1/2 |µ − λ|2 dEµ un 2 ≤ |T − λ| un = = (T − λ) un → 0. Therefore, (un )n∈N is admissible in (iii). The ﬁnal implication “(iii) ⇒ (o)” follows from Lemma 3.9, because the essential spectrum is a closed set. A characteristic sequence (un )n∈N ⊂ D(T ) with un 0 is called a singular (or Weyl) sequence for λ ∈ σe (T ). Remark 3.11. In Theorems 3.1, 3.4 and 3.10, one can replace, in (i) and (ii), respectively, D(T ) by any core D of T , as can be seen by approximation and normalization. A corresponding statement holds for Theorem 3.10 iii. Remark 3.12. Although (ii) and (iii) in Theorem 3.10 are equivalent, (iii) is a more general criterion to guarantee λ ∈ σe (T ), as can be seen from the following examples [55] in H = 2 with orthonormal basis (en )n∈N given by en = ((k = n))k∈N , λ = 0 and ∞ ∞

k2 ϕ22k < ∞ : T ϕ = k ϕ2k e2k . ∀ ϕ ∈ 2 , k=1

k=1

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35 35

!

1 1 e2n−1 + e2n . Then un = 1, un n2 n 1 0, T un = e2n , such that T un = 1, T un , un = → 0. This sequence fulﬁlls (iii), n but not (ii) of Theorem 3.10. The ﬁrst example is with un =

1−

For a sequence which fulﬁlls (iii), but does not even lie in the domain of T , take 1/2 ∞ ∞

−3 un = 1 − k e2n−1 + k−3/2 e2k ; k=n

k=n

∞

2

k−2 → 0. then un = 1, un 0 and |T | un = k=n

If we allow for λ = ±∞ in criterion (iii) of Theorem 3.10, then by Lemma 3.7 the spectrum is not bounded, such that ±∞ is an accumulation point of the spectrum. If we deﬁne the accumulation spectrum of T by λ is an accumulation point of σ(T ) λ ∈ R; σ e (T ) := ∪ λ ∈ R; dim (T − λ)−1 ({0}) = ∞ = R ∪ {−∞, ∞}), we have the following. (R we have for any core D of T : Theorem 3.13. For λ ∈ R, λ∈σ e (T ) ⇔ ∃ (un )n∈N ⊂ D, un = 1, un 0 : eT (un ) → λ ∧ sT (un ) → 0. Proof. We only have to show “⇒” for |λ| = ∞ and D = D(T ). For λn → λ we choose as before a normalized sequence un = (Eλn +εn − Eλn −εn ) vn , vn ∈ H. Then eT (un ) = µ dEµ (Eλn +εn − Eλn −εn ) vn 2 ∈ [λn − εn , λn + εn ] → ∞ and

2

sT (un ) =

2

(µ − eT (un )) dEµ (Eλn +εn − Eλn −εn ) vn 2 ≤ 4 ε2n → 0.

We collect some facts about the accumulation spectrum. Proposition 3.14. o) i) ii) iii) iv)

σ e (T ) \ {−∞, ∞} = σe (T ); if T is semi-bounded from below, then −∞ < min σ e (T ) = min σe (T ); if inf σ e (T ) = −∞, then T is not semi-bounded from below; if dim(H) ∈ / N, then σ e (T ) = ∅; if (un )n∈N ⊂ D(T ) with un = 1 and un 0, then inf σ e (T ) ≤ lim inf eT (un ). n→∞

36

ANDREAS M. HINZ

Proof. Part (o) is obvious from Corollary 3.8. Statements (i) and (ii) follow from Lemma 3.7. For (iii), assume that σe (T ) = ∅. Then σ(T ) = σd (T ), and from the spectral theorem there have to be inﬁnitely many eigenvalues which accumulate at an inﬁnite point. For (iv), we may assume T to be semi-bounded from below. Then for λ = min σ(T ) we get from Lemma 3.9, since T − λ ≥ 0: min σe (T ) − λ = dist (λ, σe (T )) ≤ lim inf (T − λ) un , un = lim inf eT (un ) − λ. n→∞

n→∞

Note that criterion (iv) in Proposition 3.14 cannot decide whether σe (T ) is bounded from below or not. Before we analyze the essential and discrete spectra of Schr¨odinger operators in the next section, let us make some remarks about another decomposition of the spectrum of a self-adjoint operator. The basic tool is provided in the following statement. Lemma 3.15. Let G be a closed subspace of H, and assume that PG T ⊂ T PG (PG is the projection onto G). Then the restrictions TG and TG⊥ of T on G ∩ D(T ) and G⊥ ∩ D(T ) are self-adjoint operators in G and G⊥ , respectively. Furthermore, σ(T ) = σ(TG ) ∪ σ (TG⊥ ). The proof is a straightforward application of deﬁnitions and therefore not carried out here. We use Lemma 3.15 to show the following. Theorem 3.16. Let M denote the set of eigenvectors of T . Then for Tp := Tspan(M ) and Tc := Tspan(M )⊥ , we have: i) σp (Tc ) = ∅ = σc (Tp ); ii) σ(Tp ) = σp (T ); iii) σ(T ) = σp (T ) ∪ σc (T ), where σc (T ) := σ(Tc ) denotes the continuous spectrum of T .

Warning. The decomposition of the spectrum in (iii) is not disjoint! In fact, (i) is even compatible with σ(Tc ) to contain eigenvalues of T . While the deﬁnition of the point spectrum is essentially uniform throughout literature (some authors use this term for the closure of the set of eigenvalues, which is not totally unreasonable in view of (ii) and (iii)), there are (at least) two deﬁnitions of the continuous spectrum of a self-adjoint operator which are not compatible with ours, namely as σ(T ) \ σp (T ) (with the advantage of being disjoint with σp (T ) and somehow justifying the word “continuous” with respect to the spectral family, but otherwise useless) or as what we called the essential spectrum (which is rather audacious, since then certain (projection) operators will have the “continuous” spectrum {0, 1}). Proof of Theorem 3.16. (o) In order to prove the correctness of the deﬁnition of Tp and Tc according to Lemma 3.15, we have to show that PG T ⊂ T PG for G := span(M ). We chose for every eigenvalue of T an orthonormal basis of the corresponding eigenspace; uniting these orthonormal bases, we get an orthonormal basis m ⊂ M of G. Further let n be an orthonormal basis if G⊥ . Then N := m ∪ n

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37 37

is an orthonormal basis of H. Let u ∈ D(T ); then for every v ∈ D(T ) (with uG := PG u, u⊥ G := u − uG ):

T v, uG = T v, e uG , e∗ = v, T e u, e∗ e∈m e∈N

= λe v, e u, e∗ = v, e u, T e∗ . Since

| T u, e|2 ≤

e∈m

e∈m

e∈m

| T u, e|2 < ∞, we have

e∈N

T u, e e ∈ G, and

e∈m

T v, uG =

v,

T u, e e ,

e∈m

that is uG ∈ D (T ∗ ) = D(T ) and T uG = T ∗ uG ∈ G. Furthermore, ∀ e ∈ m : ⊥ T uG , e = T u, e and therefore T u⊥ G ∈ G , i.e. PG T u = T PG u. ⊥

(i) For u ∈ span(M ) with Tc u = λ u we have u ∈ H and T u = λ u, such that u ∈ span(M ) as well and consequently u = 0. The second identity in (i) holds since T and Tp have the same eigenvectors. (ii) It is clear that σp (T ) ⊂ σ(Tp ) and therefore σp (T ) ⊂ σ(Tp ) by Corollary 3.5. For λ ∈ / σp (T ) there is

a c > 0 such that |µ − λ| ≥ c for every µ ∈ σ(Tp ). (λe − λ)−1 u, e e. Then, as before, for u ∈ G and Deﬁne T : G → G, u → e∈m

any v ∈ D(T ) we have T v, Tu = v, λe (λe − λ)−1 u, e e , whence Tu ∈ e∈m

D (T ∗ ) ∩ G = D(T ) ∩ G = D(Tp ) and (Tp − λ)Tu = (T − λ)Tu = (T ∗ − λ) Tu = u. / σ(Tp ) by Theorem 3.4. This shows (Tp − λ) D(Tp ) = G, such that λ ∈ Finally, (iii) is an immediate consequence of (ii) together with Lemma 3.15. The continuous spectrum is part of the essential spectrum. Corollary 3.17. σc (T ) ⊂ σe (T ).

Proof. We have λ ∈ σc (T ) ⇔ λ ∈ σ(Tc ) ⇔ λ ∈ σe (Tc ) ⇒ λ ∈ σe (T ), the latter by taking a singular sequence for λ in D(Tc ) ⊂ D(T ).

For an application of Theorem 3.16, we reconsider the operator of Theorem 1.12. Corollary 3.18. For the operator T of Theorem 1.12 we have σp (T ) = λ(M ), σc (T ) = ∅ and σ(T ) = λ(M ). Proof. Since span(M ) = H, we only have to show that σp (T ) = λ(M ). It is clear that λ(M ) ⊂ σp (T ). Assume that λ ∈ σp (T ) \ λ(M ); then M ∪ {eλ } represents an orthonormal system for any corresponding normalized eigenvector eλ . This is impossible because M is maximal.

38

ANDREAS M. HINZ

Remark 3.19. A consequence is |σp (T )| ≤ dim(H), since σp (T ) = σp (Tp ) and |λ(M )| ≤ |M |. In a ﬁnite dimensional Hilbert space H we know by virtue of Theorems 3.1 and 3.4 that σ(T ) = σp (T ) for every self-adjoint (= symmetric) operator T . Corollary 3.18 shows that in the case of inﬁnite dimension σ(T ) = σp (T ) = σp (T ) can occur even for bounded self-adjoint operators T ; take, for instance, an orthonormal 1 basis (en )n∈N in a separable Hilbert space and put λ(en ) = . n Operators T with σc (T ) = ∅ are often called operators with pure point spectrum. This is a rather unfortunate notion, because, as we have just seen, σ(T ) = σp (T ) does not only contain eigenvalues. Moreover, it may happen that σ(T ) = σp (T ) and σc (T ) = ∅. What is actually meant by “pure point spectrum” is that the spectrum is not contaminated by continuous spectrum. Therefore the following characterization of these operators is preferable. Theorem 3.20. T has a pure point spectrum if and only if H has an orthonormal basis consisting of eigenvectors of T . (T is then of the form of the operator in Theorem 1.12.) ⊥

Proof. By (the remark after) Corollary 3.6, σc (T ) = ∅ is equivalent to span(M ) = {0}, i.e. span(M ) = H. As in the proof of Theorem 3.16 we get an orthonormal basis of H consisting of eigenvectors of T . The other direction follows from Corollary 3.18.

3.1. Asymptotic behavior of eigensolutions and the spectrum. There are a couple of tools to investigate the essential spectrumwhich are speciﬁc for d d Schr¨ odinger operators S = − + V C∞ and show that it 0 (R ) in H = L2 R depends mainly on the behavior of V at ∞. The most important are the following, ` E. ` Shnol’ and A. Persson. which go back to ideas of I. M. Glazman, E. d d Theorem 3.21. Let S := − + V C∞ 0 (R ) be self-adjoint and V− ∈ Kloc R , Then λ ∈ R. d R \ Bn , ϕn = 1 : eS (ϕn ) → λ ∧ sS (ϕn ) → 0. a) λ ∈ σ e (S) ⇔ ∃ ϕn ∈ C∞ 0 d R \ Bn , ϕ = 1 . b) inf σ e (S) = sup inf eS (ϕ); ϕ ∈ C∞ 0 n∈N

Here Bn := B(0; n).

Proof. For (a), it suﬃces to prove “⇒” for |λ| = ∞, the ﬁnite case being covered by [28, Lemma 1] in conjunction with [32, Lemma 3.2] and the implication “⇐” following from Theorem 3.10 and Proposition 3.14 o. There exists a sequence with λn → λ and (λn )n∈N ⊂ σe (S) or (λn )n∈N ⊂ σd (S). In the former case we may select an appropriate mixed sequence from the singular sequences of the λn s. For λn ∈ σd (S) we chose a normalized eigenfunction vn ∈ D(T ). From Lemma 1.11 and Pythagoras’s theorem they converge weakly to 0 and Rd vn ) = λn and sS ( vn ) = 0. We approximate vn by vn ∈ C∞ obviously fulﬁl eS ( 0 and proceed as in the proof of [28, Lemma 1] to obtain the desired sequence by cutting out inside balls of increasing radii.

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39 39

d R with For “≤” in (b), ﬁx ε > 0 and choose a sequence (ϕn )n∈N ⊂ C∞ 0 d R , ϕ ϕ n ∈ C∞ \ B = 1 and n n 0 d ∞ eS (ϕn ) < inf eS (ϕ); = 1 + ε ϕ ∈ C0 R∞ \ Bdn , ϕ ≤ sup inf eS (ϕ); ϕ ∈ C0 R \ Bn , ϕ = 1 + ε. n∈N

Then Proposition 3.14 iv yields d R \ Bn , ϕ = 1 + ε. inf σ e (S) ≤ sup inf eS (ϕ); ϕ ∈ C∞ 0 n∈N

Finally we remark that inf σ e (S) ∈ σ e (S), such that “≥” in (b) follows easily from part (a) of this theorem. We can now analyze the close relations between the diﬀerent parts of the spectrum and the asymptotic behavior of eigensolutions at inﬁnity, the obvious one being λ ∈ σp (S) ⇔ ∃ eigensolution u ∈ L2 Rd . The general view is that λ ∈ σ(S) is associated with bounded eigensolutions and that eigenfunctions for discrete eigenvalues do decay exponentially. The proofs can be based on Lemma 3.9, the precise statements depending on the behavior of the potential V at inﬁnity. They read as follows. Proposition 3.22. If V− ∈ K Rd + o |x|2 , then −γ σ(S) = λ ∈ R; ∃ γ > 0 ∃ eigensolution u for S and λ : (1 + | · |) u ∈ L∞ (Rd ) . This follows from [34, Main Theorem], where a magnetic potential b is allowed for too. There are two obvious questions, namely if it is utterly necessary to take the closure on the right-hand side and if γ = 0 would suﬃce for the implication “⊂”. of S Proposition 3.23. If V− ∈ K Rd + o | · |2 , then every eigenfunction d for λ ∈ σd (S) decays faster than any inverse polynomial; if V− ∈ K R , then the decay is faster than exp(−µ|x|) for some µ > 0. This can be found in [32, Corollary 4.5]. In the last statement, it may actually be possible to allow for any µ < dist (λ, σe (S)). The case where V− (x) behaves like O |x|2 at inﬁnity is open for dimensions d ≥ 2. An example of G. Halvorsen for d = 1, where there is a λ ∈ R \ σ(S) with a bounded eigensolution and an eigenfunction for a discrete eigenvalue which decays only polynomially, indicates that both Propositions 3.22 and 3.23 may fail for these potentials in any dimension (cf. the discussion in [32, Chapter 5]). More explicit bounds on eigenfunctions are known for the standard case of a potential which tends to some constant at inﬁnity (cf. [32, Proposition 4.1]). Proposition 3.24. Let V ∈ Kloc Rd with V (x) → V0 ∈ R ∪ {∞}, as |x| → ∞. Then for every eigenfunction u for λ ∈ σp (S) and any µ < dist (λ, σe (S)): ln(u(x)) u(x) = O (exp(−µ|x|)); if u > 0, then → − dist (λ, σe (S)), as |x| → ∞. |x| Moreover, σe (S) = [V0 , ∞[.

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ANDREAS M. HINZ

The last statement is based on Theorem 3.21. For short range potentials with V (x) ≥ V0 − c |x|−1−ε outside a compact set for some c > 0 and ε > 0, we even get, by subharmonic comparison, u(x) = − d−1 2 exp − dist (λ, σe (S)) |x| (cf. [6, Theorem 2] or [30, Theorem 2] in O |x| conjunction with [6, Lemma 10]). The lower bound for positive u suggests that the exponential decay rate given in Proposition 3.24 is optimal in general. Two cases are of interest: eigenvalues below the essential spectrum, as in Proposition 3.24, and eigenvalues in gaps of the essential spectrum, whose existence has been proved in [37, Theorem (2.2)] and [16] (cf. also [29]). We will approach this question in the following section. 3.2. Spherical symmetry. We now come to the investigation of the spectrum of spherically symmetric Schr¨ odinger oparators, where d ∈ N \ {1}, and the potential is of the form V (x) = q(r) with some q : [0, ∞[ → R, r := |x|. They have been used to demonstrate statements (cf., e.g., [60, Problem 8]) or to provide (counter-)examples in spectral theory and also in scattering theory (cf. [1, Chapter 11]). We will start with some basic examples leading to classical types of spectra and then turn to the phenomenon of embedded eigenvalues, i.e. λ ∈ σp (S) ∩ σe (S). All examples will be in R3 . 3.2.1. Some basic examples. The ﬁrst few examples are provided to set the stage for more unusual spectral behavior to come and to mention some of the basic kinds of argument to prove statements about the spectrum and its components. Example 3.25 (free particle). Let q = 0. Then σe (S) = σ (S) = σc (S) = [0, ∞[, σp (S) = ∅. Proof. Because S is bounded from below by 0, we have σ (S) ⊂ [0, ∞[ by Lemma 3.7. By Proposition 3.24, σe (S) = [0, ∞[. Finally, σp (S) ∩ [0, ∞[ = ∅ follows from the virial theorem (cf. [65, p. 324]). The operator of the free particle provides an example of a purely continuous spectrum. Example 3.26 (harmonic oscillator). Let q(r) = α2 r 2 for some α > 0. Then σ(S) = σd (S) = {α (3 + 2k); k ∈ N0 }.

The proof is by constructing an orthonormal basis consisting of eigenfunctions. The harmonic oscillator has a purely discrete spectrum. α Example 3.27 (hydrogen atom). Let q(r) = − for some α > 0. Then r α2 σe (S) = σc (S) = [0, ∞[, σp (S) = σd (S) = − 2 ; k ∈ N0 . 4 (k + 1)

The proof is as for Examples 3.25 and 3.26, though somewhat more involved. (The α2 proof that σd (S) ⊂ − seems not to be readily available in 2 ; k ∈ N0 4 (k + 1) literature.)

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41 41

The hydrogen operator has a combined discrete/continuous r spectrum, which is the is a ground state eigenstandard type of spectrum. The function u(x) = exp − 2 1 function, i.e. for the lowest point λ = − of the spectrum (α = 1). Eigenfunctions 4 which do not change sign are always associated with the lowest point of the spectrum. More generally, we have the following. d Theorem 3.28. Let S0 = − + V C∞ R be bounded from below and 0 d V ∈ Kloc R . If there exists a positive eigensolution u for λ ∈ R and S = S0 , then λ ≤ min σ(S). d d 1 R be real-valued. Since u ∈ C Rd ∩ W2,loc R (cf. [32, Proof. Let ϕ ∈ C∞ 0 d Corollaries 2.8 and 2.9]), we may replace ψ ∈ C∞ R in 0 0 = − u ψ + u (V − λ)ψ = ∇u · ∇ψ + u (V − λ)ψ by

ϕ2 , whence after some calculation we get u ϕ 2 ϕ2 2 2 0 = |∇ϕ| − u ∇ + u (V − λ) , u u

that is

ϕ 2 u2 ∇ ≥ 0. u Rd : λ ϕ2 ≤ Sϕ, ϕ, whence As S and λ are real, we arrive at ∀ ϕ ∈ C∞ 0 λ ≤ min σ(S) by Lemma 3.7, because S is self-adjoint by Theorem 1.28. (S − λ)ϕ, ϕ =

Ground state eigenfunctions can have a faster decay rate than the one expected in view of Proposition 3.24. Here is an example. 1 Example 3.29. With α(r) = r + sin(3 r) let 6 2 2 2 q(r) = α (r) 1− − α (r) + α (r) tanh (α(r)). 2 r cosh (α(r)) Then 0 ∈ σd (S) with eigenfunction u(x) = µ0 < 1 (µ0 ≈ 0.9466) (cf. Figure 1).

1 and σe (S) = [µ0 , ∞[, where cosh (α(r))

The proof for Example 3.29 depends on the following theorem, where s=−

d2 + q(r) C∞ 0 (R) dr 2

in L2 (R) (q(−r) = q(r)). Theorem 3.30. Let q ∈ C ([0, ∞[), with |q(r) − q(ρ)| 1 sup ; 0 < |r − ρ| < 1 → 0, as r → ∞, r |r − ρ| and such that s and S are self-adjoint. Then ]inf σe (s), ∞[ ⊂ σe (S). If q− is bounded, then [inf σe (s), ∞[ = σe (S).

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ANDREAS M. HINZ

4

4

2

2

0

0

-2

-2

-4

-4

-6

-6

-8

-8 0

2

4

6

8

10

12

0

2

4

6

8

10

12

Figure 1. Potential function and eigenfunction (dashed) for the ground state eigenvalue λ = 0 in Examples 3.29 (left) and 3.31 (right) The proof of the ﬁrst statement (cf. [28, Corollary 1]) is based on rectangular separation and Theorem 3.21(a). (The same fundamental ideas have been employed in [49] to study the corresponding question for three-dimensional spherically symmetric Dirac operators.) If q− ∈ K(R) (and consequently V− ∈ K(Rd ); cf. [32, Lemma 1.6]), we can prove [inf σe (s), ∞[ ⊂ σe (S) without any further local assumption on q, based on spherical separation and the construction of singular sequences by cutting oﬀ eigensolutions (cf. [28, Proposition 2]). (For an alternative proof of this implication, see [67].) Under the same assumption on q and making use of Theorem 3.21(b), it is possible to show that inf σe (s) ≤ inf σe (S) (cf. [28, Proposition 1]), with the second statement of Theorem 3.30 as an immediate consequence. We refer to [28], where details on the determination of µ0 in Example 3.29 can be found as well. Theorem 3.30 limits the possibilities to construct an example of an isolated eigenvalue λ of S of inﬁnite multiplicity, i. e. ∃ ε > 0 : Eλ−ε = Eλ− , Eλ = Eλ+ε , dim (Eλ − Eλ− ) L2 Rd = ∞. On the other hand, the results of this section yield many other eigenvalues in the essential spectrum. 3.2.2. Embedded eigenvalues. Theorem 3.28 allows us to construct an example of an eigenvalue at the bottom of the (essential) spectrum. Example 3.31. Let q(r) = 2 [0, ∞[.

r2 − 3

2.

(1 + r 2 )

Then 0 ∈ σp (S), σd (S) = ∅, σe (S) =

1 is an eigenfunction for λ = 0 (cf. Figure 1). The rest follows 1 + r2 from Theorem 3.28 and Proposition 3.24. Proof. u(x) =

As mentioned in the introduction, there is now a revived interest in eigenvalues which are strictly embedded in the essential spectrum (cf. also [62]). The example

43 43

TOPICS FROM SPECTRAL THEORY OF DIFFERENTIAL OPERATORS

which produced the ﬁrst scandal is due to J. von Neumann and E. Wigner (cf. [43]; note that the source of this reference is often cited inaccurately); here is a slightly corrected and simpliﬁed version. Example 3.32 (von Neumann and Wigner). With α(r) = 2 r − sin(2 r) let q(r) = − 1 −

8 sin(2r) 32 sin(r)4 + . 1 + α(r) (1 + α(r))2

Then 0 ∈ σp (S), σe (S) = [−1, ∞[.

sin(r) is an eigenfunction for λ = 0 (cf. Figure 2). The rest r (1 + α(r)) follows from Proposition 3.24. Proof. u(x) =

4

4

2

2

0

0

-2

-2

-4

-4

-6

-6

-8

-8 0

2

4

6

8

10

12

0

2

4

6

8

10

12

Figure 2. Potential function and eigenfunction (dashed) for the embedded eigenvalue λ = 0 in Examples 3.32 (left) and 3.33 (right) As in Example 3.31, the eigenfunction in the von Neumann/Wigner example decays only polynomially. With Theorem 3.30 on hand, we are now able to produce an embedded eigenvalue with an exponentially decaying eigenfunction. 2 1 sin(r)4 − sin(2r). Example 3.33. Let q(r) = −1 + 25 5 14 Then σe (S) ⊃ − , ∞ , and 0 ∈ σp (S), with eigenfunction 25 sin(r) 1 1 u(x) = exp − r − sin(2r) r 10 2 (cf. Figure 2). For the proof we only have to observe that min σe (s) ≤ max q(r) ≤ − rem 3.21(b) and use Theorem 3.30.

14 by Theo25

Example 3.33 seems to be the only existing example where both an embedded eigenvalue and its exponentially decaying eigenfunction are known explicitly. It puts an end to eﬀorts to provide lower bounds for eigenfunctions, even for spherical

44

ANDREAS M. HINZ

means, for large classes of Schr¨odinger operators and opens one path to the phenomenon which has now become known as localization, i.e. the existence of dense point spectrum associated with exponentially decaying eigenfunctions (cf. [63]). 3.3. Radial periodicity. Examples 3.29 and 3.33 suggest a further investigation into spherically symmetric Schr¨ odinger operators which are radially periodic, i.e. where q is a periodic (even) function; we also assume q to be bounded, throughout. To ﬁx notation, we remark the following: for f ∈ L1,loc (R), let Πf := {τ ∈ R; f = fτ }, where fτ ∈ L1,loc (R) is given by ∀ r ∈ R : fτ (r) = f (r + τ ), and deﬁne τf := inf {τ ∈ Πf ; τ > 0}. Lemma 3.34. If f ∈ L1,loc (R), then 0 < τf < ∞ ⇔ Πf = Zτf τf = 0 ⇔ f is constant.

Proof. Πf is a subgroup of (R, +), such that it is either trivial, non-trivial and discrete or dense in R, with τf = ∞, Πf = Zτf , τf = 0, respectively. In the latter case there is a sequence (τn )n∈N ⊂ Πf with 0 < τn → 0 as n → ∞. Let ϕ ∈ C∞ 0 (R) be real-valued. Then ϕ − ϕ−τn f − fτn f ϕ = lim = lim ϕ = 0, f n→∞ n→∞ τn τn whence f is constant by the Lemma of DuBois-Reymond.

Now the following deﬁnition makes sense. Definition 3.35. f ∈ L1,loc (R) is called periodic with (principal) period τf , iﬀ 0 < τf < ∞. Remark. It is easy to extend this notion of periodicity and period to distributions. Typically, the spectrum of a one-dimensional periodic Schr¨ odinger operator has band structure (cf., e.g., [18, Section 5.3], [66, Section 12]), as for instance in the prototype case of the Mathieu operator, where q = cos; here σ(s) = σe (s) =

∞ "

[µk−1 , Mk ] , σp (s) = ∅,

k=1

with µk−1 < Mk < µk → ∞, as k → ∞ (µ0 ≈ −0.3785, M1 ≈ −0.3477, µ1 ≈ 0.5948, M2 ≈ 0.9181, µ2 ≈ 1.293, . . .). By Theorem 3.30, this is not so for radially periodic Schr¨ odinger operators, their essential spectrum being a half-line. 3.3.1. Dense point spectrum. A spherically symmteric extension S of s could possibly produce any kind of spectrum of S in spectral gaps of s: (absolutely or singular) continuous or dense point spectrum or even a combination of these. In order to characterize the quality of the spectrum of S in the gaps of the spectrum of a periodic s, we note the following. By spherical separation, (3.1)

σc (S) =

∞ " l=0

σc (tαl ),

TOPICS FROM SPECTRAL THEORY OF DIFFERENTIAL OPERATORS

45 45

1 where for α ≥ − , sα is the Friedrichs extension (cf. Theorem 1.21) in L2 (]0, ∞[) 4 d2 α of − 2 + q(r) + 2 on C∞ 0 (]0, ∞[) (this operator being semi-bounded by Hardy’s dr r 1 inequality (2.2)); αl = l(l + d − 2) + (d − 1)(d − 3), l ∈ N0 . Since the diﬀerence 4 −1 of resolvents s−1 − s (assuming q ≥ 1) is compact (cf. [27, Lemma 1]) and the α 0 essential spectra of s0 and s are the same by virtue of Glazman’s decomposition principle (cf. [24, Chapter I, Theorem 23]), we have σe (sα ) = σe (s) (cf. [2, Theorem 27.2.6]). Combining this with (3.1) and using Corollary 3.17, we arrive at σc (S) ⊂ σe (s). Together with Theorem 3.30, this yields the following result. Theorem 3.36. Let q be bounded. If min σe (s) ≤ λ1 < λ2 with ]λ1 , λ2 [ ∩ σe (s) = ∅, then ]λ1 , λ2 [ ∩ σc (S) = ∅, [λ1 , λ2 ] ⊂ σp (S).

For more details of the proof, see [27, Section 1].

For every such q with a gap in the essential spectrum of the corresponding onedimensional operator s we therefore have an interval of dense point spectrum for S. Example 3.37. Let q = cos, then σc (S) =

∞ "

[µk−1 , Mk ], {µ0 } ∪ σp (S) = σd (S) ∪ {µ0 } ∪

k=1

∞ "

[Mk , µk ].

k=1

If d ≥ 3, then σd (S) = ∅ and µ0 ∈ / σp (S).

Proof. The ﬁrst two results follow from Theorem 3.36, together with the fact that σp (sα ) ∩ ]µk−1 , Mk [ = ∅ (cf. [27, Corollary 1]). For the last result we note that αl ≥ 0 for d ≥ 3. We have thus obtained a very elementary example of a Schr¨ odinger operator with a spectrum consisting of alternating intervals of (absolutely, cf. [27, Theorem 2]) continuous and dense point spectrum. The presence of intervals of dense point spectrum had been known for magnetic Schr¨ odinger operators since the example of K. Miller and B. Simon (cf. [9, Section 6.2]). Their construction, together with the ideas presented above formed also the basis for a more general investigation into the spectrum of two-dimensional magnetic Schr¨odinger operators with radial periodicity of both, the electric potential V (x) = q(r) and the magnetic ∂b1 ∂b2 (x) − (x) = B(r) with B and q periodic with period τ (cf. [35]). ﬁeld, i.e. ∂x1 ∂x2 It turned out that here too there are alternating intervals of absolutely continuous τ

spectrum and dense point spectrum, provided that

B(r) dr = 0, and that oth0

erwise the essential spectrum consists entirely of dense point spectrum. Moreover, intervals ﬁlled with dense point spectrum can also be observed for spherically symmetric Dirac operators; cf. [50]. For localization in random Schr¨ odinger operators we refer to [63] and literature cited there. We also do not want to go into the onedimensional case, for which we point to [47]. The construction of one-dimensional Dirac operators with a prescribed dense set of eigenvalues can be found in [51].

46

ANDREAS M. HINZ

An interesting question is the persistence of dense point spectrum in our radially periodic examples under a compact support perturbation, say (cf. [25]; for a result on a perturbation of the von Neumann/Wigner example, see [7]). 3.3.2. Welsh eigenvalues. In Example 3.37 the question of existence of discrete eigenvalues and the status of the lowest point µ0 of the essential spectrum of S remained open for d = 2. As in connection with Example 3.29, where we constructed an admissible function for which the value of a quadratic form associated with s is strictly less than 1, thus showing that min σe (S) = min σe (s) < 1, we now produced a function in the form domain of s0 with a value of the form strictly less than µ0 , such that for q = cos and d = 2, we have σd (S) = ∅ (cf. [3]). Numerical calculations, based on the SLEIGN2 code to ﬁnd eigenvalues of s0 as restricted to functions deﬁned on ]a, b[ ⊂ ]0, ∞[ with 0 < a < b < ∞, revealed the ground state, which was baptized the Welsh eigenvalue and denoted by λλ for its place of discovery, at about −0.4016. The question if the lower spectrum, i.e. the discrete spectrum below the essential spectrum, is ﬁnite or not is a delicate one, because 1 the perturbation − 2 represents a borderline case which had not been studied be4r fore with suﬃcient thoroughness. The following can be shown by oscillation theory (cf. [52, Theorem 2]). Proposition 3.38. Let q ∈ L2,loc (R) be periodic with q− ∈ L∞ (R); d = 2. Then |σd (S)| = ∞ and min σe (S) ∈ σp (S). 3.4. Numerical analysis. The preceding results suggest to look at the contribution of sαl for l ∈ N0 to σp (S) ∩ ]Mk , µk [ for k ∈ N as well, if this gap in the spectrum of s is not empty, as in Example 3.37. For l ﬁxed and k large enough, |σp (sαl ) ∩ ]Mk , µk [ | < ∞ (cf. [48, Corollary 3]). (It seems to be open, however, if this number is 0 (or 1) for even larger k; cf. the analysis in [23], the assumptions of which do not cover our case.) On the other hand, if we look into a ﬁxed interval ]Mk , µk [, we get a similar result as in Proposition 3.38, at least for suﬃciently large l. Proposition 3.39. Let q ∈ L2,loc (R) be periodic with period τ and let D(µ) denote the discriminant of −u + q u = µ u. τ2 , with eigenvalues accuThen |σp (sα ) ∩ ]Mk , µk [ | = ∞ for α > αcrit = 4|D| (Mk ) # α αcrit − 1 mulating at Mk like | ln (λ − Mk ) | and no accumulation of eigenvalues 4π at µk . (The discriminant is the trace of the canonical fundamental system after one period; cf. [66, p. 180].) For the proof we refer to the article [53], an extension of which to periodic Dirac systems is given in [54]. Note that αcrit > 0 in Proposition 3.39, so it applies to positive α only. In the sole case where αl < 0, namely d = 2 and l = 0, there is no accumulation at the left end of a gap, but may be at the right end, as in Proposition 3.38. To obtain further insight inside the gaps of σ(s), we employed a numerical analysis to count eigenvalues of sα in a closed subinterval of ]Mk , µk [. The analytic foundation for our method to calculate N (λ1 , λ2 ; α) := |σp (sα ) ∩ [λ1 , λ2 ] | is the

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47 47

following result based on relative oscillation theory of Sturm-Liouville operators (cf. [4, Proposition 1]). 3 , 4 [λ1 , λ2] ⊂ ]Mk , µk [, k ∈ N. Choose constants a > 0, m , m ∈ N such that 1 2 α c q(r) + 2 > λ2 , and 2 ≤ dist (λj , σ(s)) for r ≥ mj τ and j ∈ {1, 2}. inf r r r∈]0,a[ Denote by nj the number of zeros in ]a, mj α] of a non-trivial real-valued solution uj of α −u (r) + q(r) − λj + 2 u(r) = 0 r Proposition 3.40. Let q ∈ L2,loc (R) be periodic with period τ . Let α ≥

satisfying the boundary condition uj (a) = 0. Then N (λ1 , λ2 ; α) − (n2 − n1 + (m1 − m2 ) k) ∈ {−4, . . . , 3}.

3 Remark. The restriction to α ≥ has been made for technical reasons only. It 4 does not eﬀect but the case d = 2 and l = 0. With Proposition 3.40 at hand, the problem is therefore reduced to count zeros of solutions in ﬁnite intervals. This counting is particularly simple, if the solutions are piecewise trigonometric or hyperbolic functions, which is the case for piecewise constant coeﬃcients in the equations. Such calculations have been performed in [4, Section 2]. The results suggest a formula

(3.2)

N (λ1 , λ2 ; α) ≈

√

λ2 α

f (λ) dλ λ1

with some density function f depending on q and k only. This compares to an asymptotic formula for α → ∞ (cf. [61, (1.8) and Theorem 3.8]; for the corresponding question for Dirac systems, see [57]) √ α (3.3) N (λ1 , λ2 ; α) ∼ χλ1 ,λ2 (λ, r) dκ(λ) dr; πτ here χλ1 ,λ2 is the characteristic function of the set 1 (λ, r) ∈ R × ]0, ∞[ ; λ + 2 ∈ [λ1 , λ2 ] , r and κ is related to the discriminant D of s by D(λ) = 2 cos (κ(λ)) for λ inside the spectral bands, and it is constant in the spectral gaps of s. For instance, if k = 1, we have µ0 ≤ λ(κ) = D−1 (2 cos(κ)) ≤ M1 < λ1 < λ2 < µ1 , whence (3.3) can be written as √ π α 1 1 − dκ, N (λ1 , λ2 ; α) ∼ πτ λ1 − λ(κ) λ2 − λ(κ) 0

48

ANDREAS M. HINZ

that is we have f = F in formula (3.2) with F (λ) = −

1 πτ

π

dκ , which λ − λ(κ)

0 |D (M1 )| ln(λ − M1 ) as λ → M1 , in perfect accordance with Propobehaves like 2πτ sition 3.39 for large α. Together with the numerical attestation (cf. [4, Section 3]) this provides strong evidence for formula (3.3) to hold already for small values of the coupling constant α. (A similar numerical investigation into the corresponding question for perturbed periodic Dirac operators can be found in [56].)

Such an inference is an example of the great potential which lies in numerical spectral analysis to obtain insight into the unexpected spectral behavior of diﬀerential operators where non-asymptotic analytical methods seem to fail. A possible ﬁeld of investigation would be the decay of eigenfunctions for embedded eigenvalues or in examples like Halvorsen’s. Spherically symmetric (radially periodic) Schr¨ odinger operators with their neither typically higher-dimensional nor simply one-dimensional spectral characteristics are a promising ﬁeld for further research. Acknowledgements. I thank R. R. Del R´ıo Castillo (Mexico City), C. Villegas Blas (Cuernavaca) and R. Weder (Mexico City) for their kind hospitality. I am grateful to E. B. Davies (London), H. Kalf (Munich) and K. M. Schmidt (Cardiﬀ) for valuable discussions.

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[44] Okazawa, N., Lp -theory of Schr¨ odinger operators with strongly singular potentials, Japan. J. Math. (N.S.) 22(1996), 199–239. [45] Owen, M. P., The Hardy-Rellich inequality for polyharmonic operators, Proc. Roy. Soc. Edinburgh Sect. A 129(1999), 825–839. [46] Rellich, F., Halbbeschr¨ ankte Diﬀerentialoperatoren h¨ oherer Ordnung, in: Gerretsen, J. C. H., de Groot, J. (Eds.), Proceedings of the International Congress of Mathematicians 1954, Volume III, Noordhoﬀ, Groningen, 1956, 243–250. [47] Remling, C., Some Schr¨ odinger Operators with Power-Decaying Potentials and Pure Point Spectrum, Comm. Math. Phys. 186(1997), 481–493. [48] Rofe-Beketov, F. S., Kneser constants and eﬀective masses for band potentials, Soviet Phys. Dokl. 29(5)(1985), 391–393. [49] Schmidt, K. M., On the essential spectrum of Dirac operators with spherically symmetric potentials, Math. Ann. 297(1993), 117–131. [50] Schmidt, K. M., Dense point spectrum and absolutely continuous spectrum in spherically symmetric Dirac operators, Forum Math. 7(1995), 459–475. [51] Schmidt, K. M., Dense point spectrum for the one-dimensional Dirac operator with an electrostatic potential, Proc. Roy. Soc. Edinburgh Sect. A 126(1996), 1087–1096. [52] Schmidt, K. M., Oscillation of the perturbed Hill equation and the lower spectrum of radially periodic Schr¨ odinger operators in the plane, Proc. Amer. Math. Soc. 127(1999), 2367–2374. [53] Schmidt, K. M., Critical Coupling Constants and Eigenvalue Asymptotics of Perturbed Periodic Sturm-Liouville Operators, Comm. Math. Phys. 211(2000), 465–485. [54] Schmidt, K. M., Relative Oscillation–Non-Oscillation Criteria for Perturbed Periodic Dirac Systems, J. Math. Anal. Appl. 246(2000), 591–607. [55] Schmidt, K. M., private communication, 2002. [56] Schmidt, K. M., Eigenvalues in gaps of perturbed periodic Dirac operators: numerical evidence, J. Comput. Appl. Math. 148(2002), 169–181. [57] Schmidt, K. M., Eigenvalue asymptotics of perturbed periodic Dirac systems in the slowdecay limit, Proc. Amer. Math. Soc. 131(2003), 1205–1214. [58] Schr¨ odinger, E., Abhandlungen zur Wellenmechanik, Johann Ambrosius Barth, Leipzig, 1927. [59] Simon, B., Schr¨ odinger operators in the twentieth century, J. Math. Phys. 41(2000), 3523– 3555. [60] Simon, B., Schr¨ odinger operators in the twenty-ﬁrst century, in: Fokas, A., Grigoryan, A., Kibble, T., Zegarlinski, B. (Eds.), Mathematical Physics 2000, Imperial College Press, London, 2000, 283–288. [61] Sobolev, A., Weyl Asymptotics for the Discrete Spectrum of the Perturbed Hill Operator, in: Birman, M. Sh. (Ed.), Estimates and Asymptotics for Discrete Spectra of Integral and Diﬀerential Equations, American Mathematical Society, Providence, 1991, 159–178. [62] Stahlhofen, A. A., A remark on von Neumann–Wigner type potentials, J. Phys. A 29(1996), L581–L584. [63] Stolz, G., Localization for Schr¨ odinger Operators with Eﬀective Barriers, J. Funct. Anal. 146(1997), 416–429. [64] Stone, M. H., Linear Transformations in Hilbert Space and Their Applications to Analysis, American Mathematical Society, New York, 1932. [65] Weidmann, J., Linear Operators in Hilbert Spaces, Springer, New York, 1980. [66] Weidmann, J., Spectral Theory of Ordinary Diﬀerential Operators, Springer, Berlin, 1987. [67] Weidmann, J., Note to the paper by R. Hempel, A. M. Hinz, and H. Kalf: On the Essential Spectrum of Schr¨ odinger Operators with Spherically Symmetric Potentials, Math. Ann. 277(1987), 209–211. [68] Yafaev, D., Sharp constants in the Hardy-Rellich inequalities, J. Funct. Anal. 168(1999), 121–144. Center for Mathematics, Technical University Munich, Germany E-mail address: [email protected] Mathematical Institute, University of Munich, Germany E-mail address: [email protected] URL: http://www.mathematik.uni-muenchen.de/∼hinz/