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(20)
be fulfilled where the rnj are integers. These conditions are called the Maslov quantization conditions. They are independent of the choice of basis in the homology group Hi (AN (cx), 2). Th e quantization rules are a system of q equations in q+l unknowns err,. . . , aQ, h. We set M = (ml,. . . , m,), a vector with integer components. In the general position case, for every fixed M and for h E (0, ho) with ho < 1, the system (20) has a unique solution (Y = aM(h). The integer vectors iV form a cubic lattice with edge of order h’ for small h. In this manner, the family of lagrangian manifolds {AN(~)} quantizes: the MC0 exists for cy = aM (h). We construct the approximate eigenfunctions of the operator A(h). The commutation formula (for a = ok (h))
n(h)Kip=K[(L(x,p)+~~)~+O(h’)] = K(E(a)cp
is satisfied where K = KIN. A(h)Kp
+ f$
+ O(h2)),
We set cp= 1 on AN(a), = E(m&))Kv
then
+ O(h2),
therefore u = K( 1) is an approximate eigenfunction of the operator A(h). follows that the operator A(h) has a series of eigenvalues of the form EM(h) = E(wv(h))
It
+ O(h2),
where A4 varies along some integer lattice. We return to (18). Let x(E) < x+(E) be points such that U(x*(E)) with E > 0. Then
= E
M. V. Fedoryuk
40
f
pdx=2
Al(E)
“+ (El s z+(E) s
Z=indAr(E)=2,
dmdx,
z (El
and the quantization rule (20) takes the form dmdx
= ~h(m
+ a).
z (El
(21)
This is the classical BohrSommerfeld quantization rule in quantum mechanics. For fixed m > 0, (21) has a unique solution E = E,(h). Equation (21) determines the asymptotics of the eigenvalues lying in the interval of the form 0 < El < E < Ez where Ei is independent of h. For partial differential equations, the asymptotics of the eigenvalues is investigated in Maslov (1965a, 1965b) and in Leray (1981). In the case where the operator A(h) has a continuous spectrum, the scattering problem arises, which is examined for the Schrodinger equation in the article by B. R. Vainberg in this book. We mention the work Kucherenko (1969) also, in which the MC0 method is developed for the asymptotics as Ic + co of the Green’s function G(x) for the Helmholtz equation, i.e., the solution of the equation (A + k2n2(x))G(x)
= S(x  x0)
with radiation conditions at infinity.
$6. The 6.1.
Secondorder
WKB
Method
Ordinary
for Nonlinear Differential
Equations Equations.
The first ap
proximation. We consider the equation 22 + f(x, t) = 0)
(1)
where E > 0 is a small parameter and f(t, x) E Cm(1w2). We are interested in the asymptotics of rapidly oscillating solutions of (1) as E+ +O. First we consider the autonomous equation e2, + f(x)
= 0.
(2)
The potential energy is U(X) = s: f(u) dy and the total energy of the particle is equal to E = ic2i2 + U(x) and is conserved in time. Let the potential have a well with x = 0 being the floor of the well, i.e., x = 0 is the point were the potential energy is minimum. Then U(Z) is strictly increasing on some interval 0 < x < a with a > 0, and strictly decreasing on b < x < 0 with b > 0. We take a and b such that U(a) = U(b) = Eo. If the energy of the particle E < EO then the particle will oscillate periodically between the points x+(E) > 0 and x(E) < 0 where 17(x+(E)) = E. They are called turning points.
I. Equations with Rapidly Oscillating Solutions
41
Equation (2) can be integrated, but we will not use this fact. Our goal is of to determine, with the help of (2), in what form to seek the asymptotics solutions of (1). We make the change of variables ti = t/c in (2) and obtain
$+f(x)=0, 1
which does not contain the parameter e. Every solution of the last equation has the form x(t, e) = y(tl, E). If we specify initial data in the form Ei(0) = Xl(E) )
40) = X0(E),
where the functions Q(E) and XI(E) are expanded in asymptotic series in powers of E and the energy of the particle E(E) < Ee, then it is not difficult to seethat the function y may also be expanded in an asymptotic series in E:
k=O
By analogy with this expansion, we will seek a formal asymptotic solution of (1) in the form x(6
6) = Y(tl,t,
E),
(3)
t1 = S(t)/E,
and y is an asymptotic series
n=O
The unknown function S(t) (the phase) is normalized by the conditions S(0) = 0 and S(0) > 0. On the function f we place the fundamental Condition
1. f(t, 0) z 0, f(t,x)
0 < z < A, and
fL(t,
0) > 0. Here
t
< 0 for A < z < 0, f(t,z) E I = [0, to] where to > 0.
> 0 for
We introduce the potential energy of the particle U(t, X) = sl f(t, y) dy. Under condition 1, for every fixed t E I the point 2 = 0 is a minimum point of U and there is a potential well with floor x = 0. We obtain equations for the functions S, ye, yi, . . We have E2, = s2L1=
d2Y at:
d2Y
+ cL1y + E2at2 ’
2s&
++ 1
1
We expand the function f(t, X) = f(t, ye + eyi + . . .) in a power series in E, substitute the series for y into (l), and set the coefficients of the powers of t equal to zero. We then obtain a recurrent system of equations, the first two of which have the form 32!$ 1 LOYl
= hYo,
+ f(h Lo =
(6)
Y/o> = 0 1 sg
+ &(t, 1
Y/o).
(7)
M. V. Fedoryuk
42
Equation (6) is an ordinary differential equation (in ti): the variable t plays the role of a parameter. In view of condition l., equation (6) has a solution which is periodic in tl with period T(t). We assume that the solution yc(ti, t) is periodic in ti with period T, independent of t. The point is that if the period T depends on t then the derivative dyc/dt is unbounded for ti E R, t E I. In fact, differentiating the identity yo(tl + mT, t) = yo(tl, t) we obtain gl/o(tl
+ mT, t) = mayoFt’
t, g
+ 1
dYo(t1,
a
q ’
and as m is any integer and aT/at $ 0, then the derivative dyo/dt is unbounded. There are two unknown functions appearing in (6), yc and S. A situation typical of nonlinear equations arises: in order to find the first approximation (in this case the function yo(tl, t)), i t is necessary to consider the second approximation. We will examine (7). It is a linear ordinary differential equation (in the variable tl), where the coefficients and the forcing term are Tperiodic. The corresponding homogeneous equation LOW = 0 has the solution wl(tl,t)
=
$Yo(tl,t). 1
To show this, one merely differentiates both sides of (6) with respect to tl. We assume that (7) has a Tperiodic solution. If this is not the case, then one can easily see that any solution will grow linearly in tl, that is, yi will be of order l/e. We introduce some necessary material concerning equations with periodic coefficients of the form G + q(t)w = 0. (8) Let q(t + T) = q(t) with T > 0, and let (8) h ave a Tperiodic solution WI(~). Then there exists a solution 202, linearly independent of WI, such that wz(t+T)=wz(t)+Aw~(t). We consider the inhomogeneous
(9)
equation ti + q(t)w
= f(t)
(10)
with a Tperiodic forcing term. Then the following statement holds. Let A # 0 (i.e., not all solutions of (8) are Tperiodic). Then in order for (10) to have a Tperiodic solution, it is necessary and sufficient that the following orthogonality relation hold: T
J
f(t)w1(t)
dt
=
0.
0
We introduce Condition 2. Not all solutions of the equation LOW = 0 are Tperiodic.
01)
I. Equations with Rapidly Oscillating Solutions Then we must satisfy o=
the orthogonality
relation
43
(11) which has the form
T 2 JTdlJo ($2 oT t,)"dtl=co, SC d2Yo
0
at1
dt2
+ut,yo))a
dY0
= $J
(x>
0
1
1
dtl,
or, equivalently,
%t)
&Yo(tl,
(12)
where Cc is a constant independent oft. Equation (12) is called the HamiltonJacobi equation. From the two equations (6) and (12) we obtain a system for the two unknown functions ys and S. 6.2. Analysis of the HamiltonJacobi Equation. From (l), it follows that ye is a function of ti, t., and S such that (12) has the form @(t, S) = 0. If it is possible to express S as a smooth function of t, S = p(t), from this equation, then we will have found S(t). Our problem reduces to the testing of the condition &D/a,!? # 0, after which we can apply the implicit function theorem. From (6)) follows
;s2(~>’
+ U(t, yo) = E(t). 1
(13)
We set y/0(0,0) = 20, and $O)&YO(O,O) = xi, which corresponds to Cauchy data in the form xjtEO = x0, E?/,=~ = x1 + O(E) for (l), and obtain E(0) = ix: + U(0, x0). Let Ee = min(U(0, A), U(0, A)),
E(t) < EO ,
then the equation U(t,x)
=
E(t)
has exactly two roots x*(t) on the interval [A, A], both simple with x+(t) > 0 and x(t) < 0 (turning points). The solution ye is periodic in tl with period T = d%‘(t)
z+(t) J z(t)
JE(t)
dx  U(t,x)
(14)
We fix T > 0 and obtain the three equations (6), (12) and (13) in the three unknown functions yo(t, tl), S(t) and E(t), which include the unknown constant CO.
We calculate the quantities mentioned above in the following order: S(O), dyo(tl,O)/dtl, CO. For this, it is sufficient to set t = 0 succesively in (14), (13) and (12). We continue with these steps: find E = E(t, S) and ye = yo(tl, t, S) from (13) and (14) and substitute them in (12). We denote the right hand side of (14) by @(S, E, t), then
44
M. V. Fedoryuk
If the numerator of the integrand is identically equal to zero, then U(X) = a(t)z2, that is, equation (1) is linear. We will not consider this case; note that condition 2 does not hold here. Therefore, we may assume that a@/aE(S(O), E(O), 0) # 0. Further, aE/a,!i = T(>&D/aE)‘. Bearing (13) in mind, we write (12) in the form
Let $(S,t)
be the left hand side of this equation,
then
If the numerator of this fraction is different from zero at t = 0, then, by the implicit function theorem, equation (12) determines S as a smooth function of t: s = cp(t) on some interval [0, To]. We will suppose that this interval coincides with I. Thus, the existence of a smooth solution to the HamiltonJacobi equation (12) is proved. If the function S(t) is known, then the solution ye(ti,t) is calculated by quadratures. We discuss condtion 2 further. If q(t) E const # 0, then all solutions of (8) are Tperiodic. However, there exist infinitely many Tperiodic functions q(t) $ const such that all solutions of (8) are Tperiodic. Condition 2 is satisfied in the case of the “general position”, but the verification of the condition for a given concrete equation is not an easy matter. For Mathieu’s equation
Ince showed that if all solutions are periodic, The constant A in (9) is equal to
A=
J
0
T
 dt
wT(t)
then b = 0.
.
We will consider the case where the function ye (tl , t) is oscillatory and the function wi = dyo/dt, therefore has zeros. The integral A therefore diverges and it is necessary to regularize it. We restrict to the case where the function q(t) is analytic in a neighborhood of the interval [O,T] and WI(O) # 0. Let 0 < t1 < t2 < . . . < t, < T be all of the succesive zeros of the solution WI(~) on [0, T]. We replace the interval [0, T] with a contour in the complex plane with the same endpoints but which avoids the zeros tl, . . , t,. For this, it is irrelevant whether we pass above or below a given zero since resl,+ wc2(t) = 0. The resulting integral converges and gives a regularized value for the integral A. We note that with this method of regularization
J
2T
0
dt ~ =o. COG t
I. Equations with Rapidly Oscillating Example
Solutions
45
1. We consider the linear equation
e2z + a(t)x = 0)
(15)
where a(t) > 0. Equation (6) takes the form d2Yo
52
@
+
4QYo
=
0
1
and has the solution yo(t1,t) = A(t)cos($tl). If this soution is Tperiodic, then &T/s and so
= 21rn, where n is a positive integer,
Thus, the function S(t) is found from the first approximation. This is specific to the linear case. We have y. = A(t)
We set S(t) = & md7,
cos {;l”&?+}.
then T = 27r. Equation (7) has the form . d2Yo
a(t)
(%
+ Yl > = 2sjtl
&h 
dtl
)
and the HamiltonJacobi equation (12) takes the form 27r
S(t)A2(t)
Setting f. a. s.
t
sin2 tl dtl = C.
s0
= 0 we find CO, so that A(t) 51(t,e) = a “4(t)
= Cu1/4(t).
Equation (15) has the
cos { 5 Lt JuT;idT}
.
In just the same way, we can show that (15) has the f. a. s. X2(&E)
= a ‘l”(t)
sin { f .I’ mdr}
.
Thus, we have obtained the classical WKB approximations. For (15), condition 2. is not satisfied since all solutions of the equation + y = 0 are 2rrperiodic. However, by direct verification it is easy to see that the functions zi and 22 are formal asymptotic solutions. d2y/@
Example
2. We consider the equation e2, + u(t)x + b(t)x3 = 0.
(16)
M. V. Fedoryuk
46
If the functions a and b don’t depend on t, then (16) is the classical Dufing ‘s equation which integrates via elliptic functions. Duffing’s equation arises, for example, in the approximation of the equation for the oscillations of a pendulum 2 + sinz = 0. The linear approximation is 2 + II: = 0. If we approximate sine by two terms of its Taylor series, then we obtain Duffing’s equation 2 + x  x3/6 = 0, which already exhibits nonlinear effects. Equation (6) takes the form
$2 d2Y0 dt2
1
+ a(t)y0 + b(t)yi
(17)
= 0.
Let a(t) > 0 and b(t) < 0 for definiteness. We will make use of the fact that every solution of the equation & dz12 + ay + by3 = 0
has the form y = Asn[B(u + C), k] , where sn(u, AY)is the Jacobi elliptic function, and B2(1 + k2) = a, and C is an arbitrary period
2B2k2
= A2
(18)
constant. The function sn(u, k) is periodic in u with 1
T=4K,
K=
We set C = 0 and B = K(k)
s o V(l
dt
 t2)(1  k2t2) .
and take the solution of (17) in the form
yo(tl, t) = A(t) sn(K(k)tl,
k)tl = S(t)/e ,
(19)
where k = k(t), then ye will be periodic in ti with period T = 4, independent of t. From the relations (18) we find
WI k(t) A(t)= d lb(t)1 qTcF@ ’ SE 4m K(k)dm
(20)
.
We are left with finding the unknown function k(t); it is determined from relation (12), which in this case has the form K(k)SA2(t)L(k)
where C is a constant and L(k) obtain the equation
= C,
= Ji J( 1  r2)( 1  k2T2) dr.
Finally we
I. Equations
with
Rapidly
Oscillating
2a3’2w2@) Q(Q) b(t)(l + k2(t))3’2 from which the function k(t) is determined. The equations for higher approximations
where Under
F, is a polynomial
47
= C)
(21)
have the form
in the functions
the condition
Solutions
ye,. . . , yni
and their
derivatives.
T
F*& =O “at, l ’ J0 the equation for the nth approximation has the solution tl with period T. For n > 2, the additional condition
(22) yn(ti,
t), periodic
in
T
J
f(k
Yoig$
dtl
#
0.
0
arises. If it is fulfilled, then the orthogonality relation (22) may be satisfied. We note that if (1) is autonomous, then condition (23) is not fulfilled since ye is independent of t, so that in this problem, the autonomous equation is a singular case. However, for the autonomous equation S(t) = t, y = y(ti, E), we have &ye E 0 and the given methods are simply not needed. Let us formulate the exact result. Consider a truncation of the series (4) XN(h
and the Cauchy
E) = 2 EnY7L(tl, n=O
t)
problem Zlt=0
=
xN(o,
61,
+,
=
6)
kN(o,
Let the condition o(t,z) 5 0 hold for t E I and 1x1 < A (that is, U(t,x) Lyapunov function of equation (1)). Then there exists tl > 0 (independent e) such that f)  zN(t,
E)l 5 cfNl
,
Ik(t, E)  k,(t,
6)I 5 cfN2
.
Iz(t,
The method
examined
here may be generalized
to an equation
is a of
of the form
fz2ii + f(t, 2, Ek, t) = 0. 6.3. Partial equation
Differential t2(utt
Equations.  c2(t, z)Au)
We consider + f(t, 2, u) = 0 )
the
nonlinear
wave
(24) where c and f are functions in C”, c > 0 and conditions on the function f will be formulated below. We will seek a f. a. s. of (24) in the same form as for (l), i.e.,
M. V. Fedoryuk
48
21 = y(t1, t, 2, E), Y=
t1 = dqt,
x) ) (25)
&Yn(W,2), n=O
and the series for y is asymptotic. Substituting these expressions into (24) and equating coefficients of differing powers of Eto zero, we obtain a recurrent system of equations. The first two have the form [(~)zc~(~)2]~+~(t.r,Yo)=o,
[(fg)”
(26)
cqg)‘]gl
+.f~(t,x,Yo)
= bYo,
where L1 is defined by
L1= [( 2
?++EL
j=ldXj axJ+(S c2,,,]&.
As before, the equation for the first approximation contains two unknown functions ys and 5’. Again we assumethat (26) has a solution ye(tr, t, X) which is Tperiodic in the variable tl where T is independent of t. The conditions on the function f are analogous to condition 1. in 2.1. Assume further that condition 2. in 2.1 holds. Then for (27) to have a Tperiodic solution, it is necessary and sufficient to satisfy the orthogonality relation
T .I o
dY0 at,LlyOdtl
= 0.
This relation is equivalent to the following:
(28)
Equation (28) is the HamiltonJacobi equation for the function S(t,z). The solvability of the system (26), (27) is proven in Dobrokhotov and Maslov (1980), however rigorous results on the closenessof the f. a. s. to the true solution are not available. Example
3. We consider the Heisenberg e2(utt
 Au)
equation
+ u  2y(t)u3
= 0,
(29)
where y(t) E C(R) and y(t) 2 7s > 0. We seek a f. a. s. in the form (25), and obtain the following equation for yc
I. Equations
with Rapidly
Oscillating
Solutions
49
(30)
This is an ordinary easily into
differential
equations
in the variable
+ Yo”  rvo4 = qt, where E is an arbitrary function expressible in elliptic functions. y/o=*
tr which
transforms
x) 3
oft and x. The solutions of this equation We consider the special case E E 0, then
( &cos
are
t1 +c 1 >
fi
’ where C = C(t,z) is an arbitary function. We assume that the solution ye is periodic in tl with period 27r, so m = 1 must hold. This leads to the HamiltonJacobi equation for the function S: s,”  (v,s)2 and the first approximation
= 1)
of the solution
of (29) has the form
uo(t,x)= [ficos(F +c(t,x))]l. We note that
(30) is the HamiltonJacobi
equation
for the hamiltonian
H(x,p) = &I;  p2  1). The function tion. As shown
C is determined from the equation for the second approximain Maslov (1983), the function cp, related to C by the identity C = i ln((pJ;;)
must
satisfy
i
the equation Stcpt  (Sz, (~5) + f (stt  AS)p
This form
(32)
is the transport (32).
equation
corresponding
Comments
= 0.
to the hamiltonian
H
in the
on the Literature
The asymptotic expansion (1.2) was first proposed in 1911 by Debye in his remarks on an article by Sommerfeld for the Helmholtz equation. Since then it has been repeatedly applied to problems in quantum mechanics, acoustics, electrodynamics, optics, and elsewhere. For the higher order equations, the method was developed by Birkhoff. The canonical operator method was developed by V. P. Maslov (1965a, 1965b). The method is also explained in the monographs Maslov and Fedoryuk (1981), Mishchenko, Sternin, and Shatalov (1978), Guillemin and Sternberg (1977), and Leray (1981). For a different definition of the KellerMaslov index see Arnol’d (1967) and
50
M. V. Fedoryuk
Leray (1981). The asymptotics of solutions to the Dirac equation are investigated in Maslov (1965a), Maslov and Fedoryuk (1981) and Leray (1981). The Maslov canonical operator, its applications and the KellerMaslov index are described in an extensive literature. Detailed bibliograhies can be found in Maslov and Fedoryuk (1981), Guillemin and Sternberg (1977) and Leray (1981). The extension to the complex case is developed in Kucherenko (1977), Maslov (1973, 1977) and Mishchenko, Sternin, and Shatalov (1978). The WKB method for secondorder nonlinear ordinary differential equations was first presented in an article by G. E. Kuzmak (1959) (see Dobrokhotov and Maslov 1980, Kevorkian and Cole 1981 and Lion and Vergne 1980, also). For partial differential equations the method was developed by Whitham (1974) and is often referred to as Whitham’s method. For further extensions of the method, see Dobrokhotov and Maslov (1980), Maslov (1977) and Nayfeh (1973); these works also contain detailed bibliographies.
References* Arnol’d, V. I. (1967): On the characteristic class entering quantization conditions. Funkts. Anal. Prilozh. 1, no. 1, 114, Zbl. 175.203. English transl.: Funct. Anal. Appl. 1, no. 1, 113 (1967) Arnol’d, V. I., Varchenko, A. N., and GuseinZade, C. M. (1982): Singularities of Differentiable Maps. Moscow: Nauka, 304 pp. English transl.: Monogr. Math., ~01s. 82 and 83, Birkhluser 1985, Zbl. 513.58001, Zbl. 545.58001 Dobrokhotov, S. Yu., and Maslov, V. P. (1980): Almost periodic finitegap solutions in the WKB approximation. Itogi Nauki i Tekhn., Ser. Sovrem. Probl. Mat. 15, 394 (Russian), Zbl. 446.35008. English transl.: J. Sov. Math. 16, 143331487 (1981) Fedoryuk, M. V. (1977): Singularities of the kernels of Fourier integral operators and the asymptotics of solutions to boundary value problems. Uspekh. Mat. Nauk 32, no. 6, 67115 (Russian), Zbl. 376.35001 Guillemin, V., and Sternberg, S. (1977): Geometric Asymptotics. Providence, R. I.: American Mathematical Society, 474 pp. Kevorkian, J., and Cole, J. D. (1981): Perturbation Methods in Applied Mathematics. New York: SpringerVerlag, 558 pp. Kucherenko, V. V. (1969): Quasiclassical asymptotics of pointsource functions for the stationary Schrodinger equation. Teor. Mat. Fiz. 1, no. 3, 384406 (Russian) Kucherenko, V. V. (1977): Asymptotic of the solution of the Cauchy problem for equations with complex characteristics. Itogi Nauki i Tekhn. Ser. Sovrem. Probl. Mat. 8, 41136 (Russian), Zbl. 446.35091 Kuzmak, G. E. (1959): Asymptotic solutions of nonlinear secondorder differential equations with variable coefficients. Prikl. Mat. Mekh. 23, no. 3, 515526, Zbl. 89,289. English transl.: J. Appl. Math. Mech. 23, 730744 (1959) Leray, J. (1981): Lagrangian Analysis and Quantum Mechanics: a Mathematical Structure Related to Asymptotic Expansions and the Maslov Index. Cambridge, Mass.: MIT Press, 271 pp., Zbl. 483.35002 * For the convenience of the reader, references to reviews in Zentralblatt fur Mathematik (Zbl.), compiled using the MATH database, and Jahrbuch fiber die Fortschritte der Mathematik (Jbuch) have, as far as possible, been included in this bibliography
I. Equations
with
Rapidly
Oscillating
Solutions
51
Lion, G., and Vergne, M. (1980): The Weil Representation, Maslov Index, and Theta Series. Progress in Math. 6, Boston: Birkhauser, 337 pp., Zbl. 444.22005 Lukin, D. S., and Palkin, E. A. (1982): A N umerical Canonical Method in Problems of Diffraction and Propagation of Electromagnetic Waves in Inhomogeneous Media Moscow: MFTI Publ., 160 pp., (Russian), Zbl. 527.65083 Maslov, V. P. (1965a): Perturbation Theory and Asymptotic Methods. Moscow: Moscow State Univ. Publ., 554 pp. (Russian) Maslov, V. P. (1965b): The WKB method in many dimensions. Supplement to the book of G. Heding. Introduction to the Method of Phase Integrals. Moscow: Mir, pp. 177237 (Russian) Maslov, V. P. (1973): Operational Methods. Moscow: Nauka, Zbl. 288.47042. English transl.: Moscow : Mir, 1976, 559 pp., Zbl. 449.47002 Maslov, V. P. (1977): The Complex WKB Method for Nonlinear Equations. Moscow: Nauka, 384 pp., Zbl. 449.58001. English transl. of Part I: Progress in Phys. 16, Basel: Birkhauser, 300 pp. (1994), Zbl. 811.35088 Maslov, V. P. (1983): Nonstandard characteristics in asymptotic problems. Uspekh. Mat. Nauk 38, no. 6, 336. English transl.: Russ. Math. Surv. 38, l42 (1983), Zbl. 562.35007 Maslov, V. P., and Fedoryuk, M. V. (1981): SemiClassical Approximation in Quantum Mechanics. Moscow: Nauka 1976, Zbl. 364.53011. English transl.: Dordrecht: Reidel 301 pp., Zbl. 458.58001 Mishchenko, A. S., Sternin, B. Yu., and Shatalov, V. E. (1978): Lagrangian Manifolds and the Canonical Operator Method. Moscow: Nauka, 352 pp. English transl.: Berlin Heidelberg New York: SpringerVerlag, 395 pp., 1990, Zbl. 727.58001 Nayfeh, A. H. (1973): Perturbation Methods. New York: Wiley, 425 pp., Zbl. 265.35002 Povzner, A. Ya. (1974): Linear methods in problems of nonlinear differential equations with a small parameter. Int. J. Nonlinear Mech. 9, 2799323, Zbl. 302.34075 Vainberg, B. R. (1984): Asymptotic Methods in the Equations of Mathematical Physics. Moscow: Moscow State Univ. Publ., 296 pp. English transl.: Gordon and Breach Publ., 1989, 498 pp., Zbl. 518.35002 Whitham, G. B. (1974): Linear and Nonlinear Waves. New York: Wiley, 636 pp., Zbl. 373.76001 Supplementaq
Literature
Bogaevskii, V. N., and Povzner, A. Ya. (1991): Algebraic Methods in Nonlinear Perturbation Theory. Moscow: Nauka 1987, Zbl. 611.34002. English transl.: New York: SpringerVerlag, 265 pp., Zbl. 727.34049 Maslov, V. P., and Nasaikinskii, V. E. (1987): Asymptotics of Operator and Pseudo Differential Equations. Moscow: Nauka, Zbl. 525.35001. English transl.: New York: Consultants Bureau, 313 pp., 1988, Zbl. 702.35002
II. Asymptotic Expansion as t ) oo of the Solutions of Exterior Boundary Value Problems for Hyperbolic Equations and Quasiclassical Approximations B. R. Vainberg Translated from the Russian by S. A. Wolf
Contents Part I. The Asymptotic Expansion of Solutions to Exterior Mixed Boundary Value Problems as t ) 00 . . . . . . . . . . . . . . $1. Analytic Continuation of the Resolvent for Exterior Elliptic Problems and the Short Wave Approximation 1.1. Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Description of the Class of Problems . . . . . . . . . . . . . . . . . . . 1.3. Analytic Continuation of the Resolvent through the Continuous Spectrum . . . . . . . . . . . . . . . . . . . . 1.4. The Behavior of the Resolvent at High Frequencies . . . . . . . $2. The Longwave Approximation and the Asymptotic Expansion as t + co of Solutions to Mixed Boundary Value Problems .. . . . 2.1. Oddn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Longwave Asymptotic Expansion . . . . . . . . . . . . . . . . . . . . . 2.3. Mixed Problems, Even n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Problems with Bounded Energy . . . . . . . . . . . . . . . . . . . . . . . 2.5. The KleinGordon Equation . . . . . . . . . . . . . , . . . . . . . . . . . . .
55
. ..
55 55 56
.. ..
58 60
.. .. .. .. .. ..
62 62 64 66 67 68
B. R. Vainberg
54
Part II. The Scattering
Problem
..........................
51. Quasiclassical Asymptotic Expansion of the Solution to the Scattering Problem ......................... 1.1. Introduction, the Geometry of Rays ........... 1.2. Maslov Canonical Operator (MCO) ............ 1.3. Asymptotic Expansion of the Solution to the Scattering Problem .................... 52. Asymptotics of the Scattering Amplitude. ........... Part III. The Parametrix and the Full Asymptotic of the Spectral Function of Differential Operators
. . ..
69
.. .. ..
69 69 70 72 74
Expansion in KY. . . . . .
$1. The Parametrix for Hyperbolic Equations and Systems 1.1. Introduction ................................ 1.2. Second Order Equations ..................... 1.3. Higher Order Equations and Systems .......... $2. Asymptotics of the Spectral Function ............... 2.1. Second Order Equations ..................... 2.2. Systems of First Order Equations ............. References ..........................................
.. .. .. .. .. .. ..
76 .. .. .. .. .. .. .. .. . .
76 76 76 79 81 81 84 86
Part I The Asymptotic Expansion of Solutions to Exterior Mixed Boundary Value Problems as t + 00
Exterior
$1. Analytic Continuation of the Resolvent for Elliptic Problems and the Short Wave Approximation
1.1. Statement of the Problem. Let 0 be an unbounded connected domain in iRn with an infinitely smooth compact boundary r and v be a solution of the (exterior) mixed boundary value problem
(1)
where L = L(z, i&,i&) is a strictly hyperbolic I x 1 matrix of differential operators of order m, v and cpare vector functions with 1 components (they will be called simply functions from here on), and B is a (mZ/2) x 1 matrix of differential operators of order no higher than m  1. From the conditions B formulated below, it follows that ml is even. As a special case, (1) may be a Cauchy problem (this means 0 = R?). Whenever the reverse is not expressly stipulated, we will assumethat the medium under consideration is homogeneous in a neighborhood of infinity. More precisely, we assume that the following condition holds Condition A.
L= LcI(~~,i&)
+Q(z,
ig,
i&),
where the elements of the matrix Lo are either homogeneous operators of order m with constant coefficients or zeros, and the elements of the matrix Q are operators of order no higher than m with coefficients in C”, equal to zero for 1x1>> 1. One of the fundamental questions, whose answer will be obtained in this part and which is closely connected to the other problems discussed in this article, is contained in the following. What can one say about the asymptotic behavior of solutions of the problem (1) as t + 00 with 1x15 b < 00 under the condition that the function ‘p has compact support? In particular, we wish to describe the asymptotic behavior of that part of the energy of the
56
B. R. Vainberg
solution (the local energy) which remains in a bounded region of space for all t > 0 (i.e., does not dissipate to infinity) under the condition that the initial disturbance is localized in a bounded domain in space. In the case of an unbounded medium, the situation described above is typical from a physical point of view. The initial disturbance is localized in space and the state of the medium also is of interest only in a bounded part of space. Many physical processes give rise to this type of problem: acoustical, optical, electromagnetic wave propagation, etc. Closely related questions in the stationary case arise in several problems in quantum mechanics. If the full energy of the solution is of interest, the answer in the most important problems in terms of application is exceedingly simple: the full energy is independent of time. However, even in this conservative case, the important and difficult question of the behavior of the local energy as t + 00 or of the asymptotic expansion of the solution of (1) for 1x1 < b < 00 as t + 03 remains. This question is nontrivial even in the most simple case when (1) has the form: I,
Utt
 a(x)Au
{ vJt,o = 0,
= 0, ?&o
x E R”,
t>o;
SEEP,
= cp>
where a E Cffi(EP), a(x) > 0, and a(z)1 = (P(Z) = 0 for 1~) > CY.If a(x) G 1, then the answer is obtained from explicit formulas for the solution: for odd n, by virtue of Huygens’ principle, v = 0 when t > CY+ 1x1; for the same cp in the case of even n, v = O(t“12) as t + co for 1x1 < b. In the latter case, it is also possible to find the full asymptotic expansion for the function v as t + co for (5) < b. The results obtained below on the asymptotic behavior as t + M of solutions of (1) may be viewed as a generalization of Huygens’ principle to equations with variable coefficients. 1.2. Description of the Class of Problems. We denote by H the hamiltonian corresponding to (1): H(z, X,p) = det a(L), where a(L) is the principal homogeneous part of the matrix L(z, X,p). We recall that the operator L is called hyperbolic if the equation H(x, X,p) = 0 has, for every x and p # 0, exactly ml roots in X that are real and mutually distinct. Without loss of generality, we may suppose that the coefficients of d”/dt” in the matrix L form the identity matrix. Condition
B’.
For any point H(z’,O,p)
x0 E R # 0
for
This condition ensures that plane waves v equation a(L) ( x0, i a/& i a/ax)v = 0 have a Sometimes this condition is called the condition mental frequency in the matrix g(L). Clearly, to an ellipticity condition for the operator LI, not depend on the value of k).
pf
0.
= v(ct + (p, z)) satisfying the nonzero propagation velocity. of absence of a zero fundathe condition !3’ is equivalent ‘3 L(x, k, i&) (the latter does
II. Asymptotics Condition
t3. There
exists
of Solutions
to Exterior
BVP
a cpo, 0 < cpo < X, such that
57 the problem
is an elliptic problem with paramater Ic = peivo, p > 0, (see Agranovich and Vishik 1964 and Vainberg 1982). In other words, the problem which results after t,he substitution of eipo aGa for k in (2) is an elliptic problem in the variables (20,~) on the cylinder lR x R. The latter is equivalent to the ellipticity of the equation and the fulfillment of an algebraic condition relating the coefficients of the operator L and the boundary operators B (a covering, or ShapiroLopatinskij condition, see for example Lions and Magenes 1968). Ellipticity in (~0, Z) of the corresponding equation is by virtue of the hyperbolicity of the system (1) equivalent to condition B’. Hence condition D is stronger, in comparison with condition B’, and coincides with the latter in the case of the Cauchy problem. It is often necessary for the wellposedness of (1). We will denote the (vector) Sobolew space of functions on 0 by HS = H”(Q) where s E lR and the norm on it by 11. IIs. W e will denote the Sobolev space HS(IR+ x 0) of functions on the halfcylinder t > 0, x E R by BS, in order not to confuse it with H”(R), and the norm on it by III . IIIs. Let BS>Y be the space B” with weight eyt and III IIIs,y be its norm, i.e., Ill~lll~,~ = lll~eYtllls. Condition C. (wellposedness of (1)). W e suppose the standard existence and uniqueness theorem with a priori estimates for mixed boundary value problems is satisfied for (l), that is, there exists a y < 0~) such that for any v > y, (1) has a unique solution z1 in the space B”+’ for any ‘p E C?(Q) and
lll4llm1,r e CIIPIIO.
(3)
The conditions B, C are fulfilled for example for all three fundamental boundary value problems for the wave equation. Condition C is fulfilled for first order positive symmetric systems, and for conservative or dissipative problems. We will also consider a weaker variant of condition C (condition C’) in which we have, instead of the estimate (3), that for any function p = p(t) E C$‘(E%) with support in the interval (0,l)
111~ * Plllm1,Y I clllPucplll0~ Here the convolution symbol indicates convolution in the variable t. Clearly, v * p is a solution of (1) with zero initial data and the righthand side of Th e conditions the equation equal to p(t)cp(x). B, C’ are fulfilled for general coercive mixed boundary value problems for arbitrary hyperbolic systems for which condition 23’ holds. The Schwartz kernel E = E(t, x, x0) of the operator P, which transforms a function ‘p E Cr into a solution of (1) belonging to Brn,y, is called the Green matrix of (l), i. e., the matrix E is a solution of problem (1) for cp = IS(z  x0), where I is the identity matrix and S is the Dirac delta function.
B. R. Vainberg
58
Condition V. The discontinuities of the Green matrix go to infinity as t + co, that is, for any c > 0 there exists a T = T(c) such that the matrix E is infinitely differentiable for Iz/,[x’/ 2 c and t > T.
The following hamiltonian
system corresponds to problem (1).
dH t:=,x>
dH X:=%7
xl,=o,
p:=g.
(4)
The projections into lKg of the phase curves of the system (4), for which
(~J,~P)I~=~ = (O,Y,~~,P~), H(y,X”,po),
lpol = 1,
(5)
are called rays. The rays reflect off the boundary of the domain R according to the laws of geometric optics. For the Cauchy problem (see the construction of E in Maslov and Fedoryuk (1981) or the theorems on propagation of singularities in Hormander (1971))) and also for some boundary value problems, in particular, for the exterior problems for the wave equation (see Morawetz and Ludwig 1969, Babich and Grigor’ev 1974, Majda and Osher 1975, Taylor 1976, Ivrij 1978, Melrose and Sjijstrand 1978) condition D is equivalent to the following condition. Nontrapping condition. For any c, there exists an s = s, such that rays for which [y/15 c lie outside the ball 1x1< c for IsI > s,. Let a and b be arbitrary fixed constants such that Q E 0 for 1x1 > a, I’ c {x : 1x1 < a}, cp = 0 for 1x1 > a, and b > a. We will not follow the dependence of any quantities on the constants a and b. 1.3. Analytic Continuation of the Resolvent through the Continuous Spectrum. We will regard the parameter Ic, on which problem (2) depends polynomially, as a spectral parameter. We will denote by Rk the operator (the resolvent) which transforms functions cpE Ho(Q) into solutions of (2) belonging to H”(f2): R&D = U. From the conditions d, B it follows that the real axis belongs to the continuous spectrum of the operator corresponding to (2)) and for Im k # 0 the operator & : Ho(R) + Hm( fi) is defined and depends meromorphically on I?. Let ,. RI, = JIRkJZ : H,” + H& O<argIc<~. Here H,” is a subspace of Ho(O) consisting of functions that are zero for 1x1> a, Jz : H,” + H’(O) is the inclusion operator, H$) is the Sobolev space of functions in fib = L?n{z : 1x1< b}, and JI is the restriction operator to the domain fib. We will denote the norms on the spacesHz and H&, by II . Ils,a and II /Is,(b), resp. The operator & is distinct from & in that it has a smaller domain of definition and the result of the action of the operator is considered in a larger space. Also, the parameter Ic varies over one of the halfplanes, for definiteness, the upper halfplane.
II. Asymptotics
of Solutions
to Exterior
BVP
59
1. Let the conditions A, l3 be fulfilled. Then I. The operator Rk depends meromorphically on k for Im k > 0. The operator Rk admits a meromorphic continuation (for which we retain the same notation) to the lower halfplane if n is odd, or to the Riemann surface of the function lnx if n is even. II. The operator Rk has poles at exactly those points of the halfplane 0 < &g k < n at which the homogeneousproblem (2) has a nontrivial solution in H”(R). The operator Rk has poles at exactly those points of the rays arg k = 0 and arg k = rr for which (2) h as a nontrivial solution satisfying the radiation condition at infinity. Theorem
The radiation conditions for (2) (see Vainberg 1982) are analogous to the Sommerfeld conditions (seeTikhonov and Samarskij 1953, Vainberg 1982) for the Helmholtz equation. They guarantee finite dimensionality of the kernel and solvability of (2) for any cp E H,“, satisfying a finite number of orthogonality conditions. In order to illustrate the first (fundamental) assertion of Theorem 1, we will consider the simplest case of (2): x E R3,
 [A + k2 + q(x)]u = cp,
q E c,(R3).
(6)
In the case q E 0 we will denote the operator Rk by Ri. Clearly, for Im k > 0 &P = Ek * P,
Ek = (&‘rr)leikr.
(7)
The function Ek decays exponentially as r + 00 if Im k > 0 and grows exponentially if Im k < 0. It follows that the operator (7) is a bounded operator on La(Iw3) for Im k > 0 and has no analytic continuation (in k) to the lower halfplane as an operator on the space L2(R3). If, however, the function cphas compact support, then the convolution (7) is defined for all k E Q1.But, the function Ricp grows exponentially as r + 00 for Im k < 0. Its restriction to the ball 1x1< b belongs to H$, and depends analytically on k, that is, the operator Ri is an entire function of k. If q $0 and cpE H,” then from (6) we obtain Rk~ = Ro,(I  qR;)‘p
= R;(I
 qR:)‘P,
Imk>O.
In the last equality, we made use of the fact that q = 0 for 1x12 a. It is easy to show that the operator (Iq@)’ : H,” + H$,, is a meromorphic function of k E Cc(cf. Theorem 2.). It follows therfore that the operator Rk depends meromorphically on k for Im k > 0. Furthermore,
and this operator depends meromorphically on k E @, as was to be shown. In the twodimensional case the operator RE can be expressed in terms of Hankel functions and RE has a logarithmic branch point at k = 0.
60
B. R. Vainberg
For (2), the proof of the first assertion of Theorem 1 consists of the following two parts: 1) The first part is equivalent to formula (7), i. e., the first part is the proof of Theorem 1 for systems with constant coefficients in Iw”. Here, one uses the explicit formula for the solution of the system and a technique from the branch theory of integrals depending on a parameter. 2) The second part is equivalenct to formula (8). With the help of a technique developed in the theory of elliptic problems on bounded domains, one builds a paramettia: for problem (2) ( an “approximate inverse operator”) which allows us to reduce problem (2) to the equation Bku = cp, where the following theorem is applicable to the operator Bk. Let HI and Hz be Hilbert spaces, and U be a connected domain in the complex plane. The family of operators Ax : HI 4 Hz, X E U is called a finitely meromorphic Fredholm family, if 1) the operator Ax depends meromorphically on X for X E U, where the corresponding Taylor and Laurent series converge in the uniform norm; 2) for any pole X = Xe of the family Ax the coefficients of negative powers of X  Xe in the expansion of Ax in a Laurent series are finitedimensional operators, that is, they map the space HI into a finitedimensional subspace of the space Hz; 3) for all points X E U at which the family Ax depends analytically on A, the operator Ax is Fredholm; at any pole X = Xe E U of the family Ax, the operator appearing as the coefficient of the zeroth power of X  Xe in the expansion of Ax is Fredholm. Recall that an operator is Fredholm if its domain of definition is closed and the dimensions of its kernel and cokernel are finite and equal. Theorem 2. (Blekher 1969) Let U be a connected domain in the complex plane, Ax : HI + Hz, X E U be a finitely meromorphic family of Fredholm operators, and let there exist a X = X0 E U such that the operator Ax, is invertible. Then the operators (AX)’ : Hz + HI, X E U, form a finitely meromorphic Fredholm family. 1.4. The Behavior of the Resolvent at High Frequencies. The asymptotic expansion of solutions u = Rkcp for real k, Ic + co (or for 1Re kl + IX) are called shortwave (highfrequency) asymptotic expansions. The origin of this term stems from the following fact. Let the oscillations of a medium be caused not by an initial disturbance but by a periodic forcing term, i. e., the initial data in (1) are equal to zero, but the righthand side of the equations is equal to ‘p(z)eikt on 0. Then if the solutions are asymptotically periodic, that is, ZJM u(z)eikt for t > 1, then the amplitude u of the steadystate oscillation is a solution of (2), with k equal to the frequency. It is fairly easy to estimate the resolve& Rk for Im k + CO, since we get further away from the continuous spectrum as Im k increases (we note that in fairly general circumstances problem (2) becomes an elliptic problem with a parameter when E < arg k < 7r  E, e > 0). However, we are interested in the behavior of solutions of (2) when k belongs to the continuous spectrum of the problem and the operator Rk is unbounded. It turns out that passing
II. Asymptotics from the operator & solutions of (2) through them as 1ReIcl + 00. than those which hold We denote by U,,p,
of Solutions to Exterior BVP
61
to & allows not only a meromorphic continuation of the continuous spectrum but also yields estimates of The (sharp) estimates obtained are somewhat worse as Im Ic + 00. a, /?I > 0 the domain in the complex Ic plane for which [Imkl
< culnIReIcl
p.
In the case of even n, when the operator & has branch points at Ic = 0 and k = 00, we need to add the condition 1r/2 < arg k < 3x12 to the inequality. We recall that P = P(t) : Ho 4 Brn‘,y denotes the operator which transforms functions ‘p E Ho into solutions of (1). In view of condition C, it exists and is bounded. Let the function 5 = c(t) be infinitely differentiable for t # 0, equal to unity for 0 2 t 2 T 1 and equal to zero for t > T and t < 0. Here T = T(b) + 1, where T(b) is determined in condtion ‘D. We will denote the Fourier transform (from t to k) by F +k. In the sequel, functions in H”(R) (or BS) will be considered as elements of the space Hrb,. In these cases, it is understood that the operator Ji is applied to the functions. However, in order not to make the formulas too cumbersome, we will not write this restriction operator explicitly. Theorem 3. Let the conditions AD hold and let ‘p E H,“. Then the operator Ftk(0 and lkl > E we have the estimates
lIFt+k(CP)vll,j,(b) 5 C(~)IkI1jeT~lmk~Il~ll~,a, For any c~ > 0 there
exist constants
O<j<m+l.
(9)
p and Ci such that for k E U,>p and any
j>O
ll&f  &+k(
5
Cl~ll~eT’lmk~II~llo,a,
Then for
O<jlm+l.
(10) any (Y > 0
(11)
Theorems 3 and 4 contain the main technical load of this chapter. We will discuss the idea behind them. If condition C is fulfilled, then Theorem 4 is clearly a consequence of Theorem 3. If, however, condition C’ is fulfilled without condition C then a weaker variant of Theorem 3 is satisfied, and theorem 4 follows similarly. The estimates (9) are elementary. The main content of Theorem 3 is contained in the estimate (10). Since the function & is an exact solution of (2) for z E 6$,, and the difference between
B. R. Vainberg
62
problem yields the shortwave (I Re ICI ) oo) asymptotics of the solution of the corresponding stationary problem. The nontriviality of estimate (10) (and also the inaccuracy of the formulated heuristic statement, if conditions of the type AD are not fulfilled) is connected with the fact that the Fourier transform of a smooth function need not decay at infinity (for example, w = et*). It is easy to check the assertion of Theorem 3 if we know the behavior of solutions of the nonstationary problem for 1x1 < b as t + co. Since nothing is known a priori about the behavior of solutions of (1) as t + 00, the proof of Theorem 3 is fairly difficult. Vice versa, it will be easy in the following sections to obtain the full asymptotic expansion of solutions of (1) for 1x1 < b as t + co from Theorems 3 and 4. Theorem 3 also yields the shortwave asymptotic expansion of solutions to the stationary problem in those cases when a parametrix of the nonstationary problem (that is, an operator differing from P by a smooth operator) is known. Concrete examples of this situation will be examined in the following parts. Remark. If for all s 2 0 we have, instead of estimate (3),
lll4ll~+~l,Y 5 Csll~lls> then we may substitute
$2. The Expansion
s for m and s for 0 in (lo),
s E Iw.
Longwave Approximation and the Asymptotic as t + 00 of Solutions to Mixed Boundary Value Problems
2.1. Odd n. It follows from condition C that for I Im ICI > y the operator & has no poles (since Rkcp = Ft+kPp for these values of k, and the function Ft,kPp depends analytically on k, Im k > y). From this and from Theorem 4 it follows that any halfplane Im k > const contains no more than a finite number of poles of the operator &. We enumerate them in such a way that Imkk 2 Im kj+l. Then Im Ic, + co for j + 03, moreover the successive poles get further from the real axis at least at a logarithmic rate. Theorem 5. Let n be odd, conditions A, D, C’ and D hold, cp E H,” and v E Brn‘,+t’ be a solution of (1). Then in the domain Q, the following expansion is valid for any N
V(t, X) = i 2 k~@&Xik”] j=l 2
+ wj,T(t, x) ,
where for any E > 0 and any s, j = 0, 1,2,. , ., there is a T such that fort the following estimate holds
aj 5 C(N, atjWN I/ I/s,(b)
s, j, e)e
kN+l+E)t IIpllo,a .
(12) > T
II. Asymptotics Remarks.
of Solutions to Exterior
BVP
63
1. Clearly formula (12) may be rewritten in the form
v = i:
fpuq,i(x))“~li3t
+ WN )
q=o
j=l
w.here pj + 1 is the order of the pole k = kj of the operator function & uqj
and
E c(flb).
2. The poles of the operator & which lie in the upper halfplane correspond to the terms in (12) which grow exponentially as t + oo. Poles lying on the real axis correspond to terms in the form of a product of an oscillating exponential function (or unity, if kj = 0) and a polynomial in t. Poles in the lower halfplane correspond to the terms which decay exponentially as t + co. There are no more than a finite number of poles in the halfplane Im k > 0 and their locations are determnined by Theorem 1 if k # 0. In Vainberg (1975 and 1982) sufficient conditions for the absence of poles at the point k = 0 can be found. For the proof of Theorem 5, we write the solution of (1) in the form 1 v =
G
i(y+l)+m J Rkcpeikt
dk,
(14)
i(y+l)lx
where y is defined by condition C. Then for x E fib and cpE H,” the operator & in (14) may b e replaced by & and the contour of integration moved to the lower halfplane, changing it to the line 1 : Im k = Im kN+l + E, where E > 0 is chosen so that no poles of & lie directly on 1. Then the estimates (13) will hold for the integral along I, and the difference between the integral (14) and the integral along 1 is equal to the sum of the residues appearing in the righthand side of (12). Both of these assertions follow easily from Theorem 4. These arguments lead to the following asymptotic expansion of the solution ‘u of the problem of forced oscillations of a medium: ( Lv(t, x) = cp(x)eCiwt,
x E 62, t > 0, (15)
Theorem 6. Let n be odd, conditions A, I?, C’, 2, hold, ‘p E H,” and u E Brn‘,y be a solution of (15), where w is an arbitrary complex number. Then in the domain f&, for any N < co we have the following expansion
21= 5 ,rrtj3p,:&e8”“]
+ x&‘pei”t
+ WN
,
j=l
where the estimates (13) hold for the function wN and x = 0 if w coincides with one of the poles kj or if Im kN+l 2 Im w. Otherwise, x = 1.
64
B. R. Vainberg
In this way, a periodic forcing term (w is real) can induce exponential growth in the solution (in the presence of eigenvalues of the stationary problem in the upper halfplane) or excite a finite number of modes, one of whose frequencies equals w, while the others’ do not depend on w. 2.2.
Longwave Asymptotic Expansion. The longwave (or lowfreasymptotic expansion is the asymptotics of solutions of (2) as k + 0. In the case of a space of odd dimension, the asymptotic expansion is given by Theorem 1, by which the solution Rkcp, cp E H,” of (2) is expressed in a series in powers of k for x E &. This series may start with a negative power of k if & has a pole at the point k = 0, or with a zero power if there is no such pole. Sufficient conditions for the absence of such a pole can be found in Vainberg (1975, 1982). In the case of a space of even dimension, Theorem 1 says nothing about the behavior of the operator & for k + 0. The fundamental lemma, with whose help the behavior of the operator & is studied as Ic + 0 (among other applications), has the following form. We denote the following region on the Riemann surface of Ink by GT,E: quency)
f2,,, = {k : ) argkl Lemma for complex form:
< y,
I/C/ < E} .
1. Let Sk : H + H be an operator on the Hilbert space H defined k # 0, 1kl < 1, Fredholm for these values of k, and having the N Sk =Tklnk+CAjkj+Gk. j=l
Here
Tk, Gk are analytic operators on the disk Ikl < 1, the s > 0, and Aj, 1 < j 2 N, are finitedimensional, the operator Go is Fredholm, and the operator Sk0 is invertible at least for one value k = Ice # 0, lkal < 1. Then for any y < 00 and for some E = E(Y) the operator (Sk)’ may be represented in the form operators
the operators &Tklkzoj
cc
(sk)l = Nk + f(k)
i
c x(5& i=o 3=0
ld k)ki
,
where Nk is an analytic operatorfunction in the disk lkl < E(Y), the operators Szj are finitedimensional, the series converges in the uniform operator norm for k E fl,,,, and the function f has the form
(16) Here a 2 0 is some integer, P and P, are some polynomials of order than a and s(a + 1)) respectively, Pa(t) E 1 and the series (16) absolutely and uniformly for k E Q?,,.
no higher converges
II. Asymptotics
of Solutions to Exterior BVP
65
The first term of the expansion for the operator (Sk)’ has the following form: there exist integers a, /?, b (p 2 0), a polynomial P and operators D, such that for k E Q,,,
/I&rl  &
P CD, s=o
In” k < Clk”+l Ins ICI. II
Lemma 1 may be applied to the operator BI, which was examined above in the discussion of the proof of Theorem 1. (For this, it is first necessary to study the behavior of the operator & as k + 0 corresponding to a system with constant coefficients in R?.) As a result, we arrive at the following assertion. Theorem 7. Let n be even and conftitions A, B, C’, V hold. Then for any y > 0 and some E = ~(7) the operator RI, may be written in the form l&
= Lk + kOf(k)
F
k
Ri,ki
lnj k,
(17)
where the operators Lk, Rij : H,” + H$) are bounded, the operator Lk depends analytically on k in the disk lkl < E(Y), the operators Rij are finitedimensional, the function f has the form (16), ,D = m  n for n < m, p = 1 for n = m, ,B = 0 for n > m, and the series (17) converges uniformly in the operator norm for k E GT>,. Remark. Expansion (17) may be rewritten in the following simple (but a little less explicit) form. fik = kKA
2 ki i,j=o
[ 1 L
P(ln k)
’ Pi,j(ln
k) ,
where A is an integer, P = P(X) is a polynomial in X, Pi,j are polynomials with operator coefficients. The analogous expansion is valid for the operator (Sk)l in Lemma 1. Clearly, the series (17) may be written in the following, outwardly simpler, form: CorxJ & = kA 71 x Bij (k lna+’ k)i In’ k, (18) ix0 j=o
where A is some integer, the operators Bij : H,” + Hgj are bounded, all of the operators Bij, except possibly those for which i 2 A, j = i(a + l), are finitedimensional, and the double sum converges uniformly in the operator norm. We may write the operator (Sk)’ analogously in Lemma 1. The merit of the expansion (17) lies in the fact that with the help of a finite number of terms, it allows one to obtain an approximation to the operator & to any degree of accuracy (in k). In (18), an infinite number of terms differ by a logarithmic factor from the first term of the asymptotic expansion. We
66
B. R. Vainberg
note, additionally, that the proof in Vainberg (1982) of Lemma 1 is such that it allows one to refine the result in concrete situations, when one has further information about the operators Tk and Aj. 2.3. Mixed Problems, Even n. We denote the “singular part” of the operator & by ri’,. It is obtained from the sum (18) by throwing out the terms for which i 2 A and j = i(u + 1). For ]k] + 0 and ] argk] < 27r, let pk = BkP lnq k + O( 1kp lnqel ICI) ,
where the operator B is nonzero (clearly, it is finitedimensional). Theorem v E Bm‘ly holds
8. Let n be even, conditions A, l3, C’, V hold, ‘p E H,” and be a solution of (1). Then in the region fib the following expansion
21 = i
c
;zr.[&pePikt]
Imk3>0
+ c,tp1(ln4+‘P
t)Bq~ + wP.
’
Here p = 1 for p < 0, p = 2 for p > 0, cl = (i)“qp[(p
 l)!]l,
c2 = ~(i)~~+~p!
and for any s, j = 0, 1, . . . , and some T, t > T
aj 5 C(s, j) dtjWp /I II,s@)
2
[tp1 lnqPfi
t11IIff4lo,a.
In an expansion of the type (18) for the operator kk,w E (k  w)l Rk, we leave only those terms which contain a power of Ink or a negative power of k. We denote the remaining sum by iilk,,. For lkl + 0, I arg kl < 27r, let pk,,
= Dk’
lnh k + 0( Ik’ Inhe ICI) ,
where the operator D is nonzero (and finitedimensional). Theorem an arbitrary the domain
9. Let conditions A, B, C’, D hold, with n even, cp E Hz, w be complex number. Let u E Brn‘,y be a solution of (15). Then in fit, the following expansion holds
Here p = 1 for r < 0, p = 2 for r 2 0; d, = const # 0; x = 0, if Imw < 0, or w = 0 or w coincides with one of the poles kj, for which Im kj > 0. Otherwise, x=1. Foranys,j=0,1,2,... andsomeT witht>T
aj 5 C(s, j) mwp /I II,s(b)
&
[t‘llnh‘” tII IIdlo,a
Theorems 8 and 9 are proved along the same lines as Theorems 6 and 7. Only now in order to choose a singlevalued branch of the function & it is necessary to make a cut in the kplane along the negative part of the imaginary axis and choose the contour 1 so that it passesabove the cut.
II. Asymptotics 2.4. Problems condition hold
with
of Solutions to Exterior BVP
Bounded
Condition E. For any solution lowing estimate holds
Energy. u E B”,r
ll4lL*(n, 5 c(cp),
67
Let the following
additional
of (1) with ‘p E C?(n) t >o.
the fol
(19)
It has been shown in this case that the operator & cannot have poles in the upper halfplane and that it can have poles on the real axis of no higher than first order, and further that the domain of definition of the residues of the operatorfunction & at these poles coincides with the eigenspace of (2). In the case of even n, in addition II&II
< Clkll
for
IlcJ+ 0, I arg ICI< 2X.
(20)
This leads to the following simplified variant of Theorems 6 and 8. Theorem 10. Let conditions A, i3, C’, ID), I hold. Then (2) has no more than a finite number of eigenvalues k = wj, 1 < j 5 r, in the upper halfplane, all of which are real, their corresponding eigenspaces ‘ltj are finitedimensional, for wj # 0 the eigenfunctions have compact support in a,, and there exist bounded operators Bj : H,” + 7$ such that in the domain Q, for solutions u E BrnP1ly of (1) with cp E H,” the following expansion holds
For the function w for any s,j = 0, 1,2,. . ., some T and following estimates hold: if n is odd, then for some S > 0
any t > T the
IIII SW do
and
if n
L: C(s, ~F”tllvllo,a ) s,(b)
is even, then
s,(b) Remarks. 1. If condition I holds, then similar variants of Theorems 7 and 9 also hold (see Vainberg 1982). 2. If the estimate (20) IS . k nown, then we can require, instead of (19), that for some j 2. 0, the following estimate hold
B. R. Vainberg
68
2.5. The KleinGordon Equation. The scheme developed above for obtaining the asymptotic expansion as t + co of solutions of hyperbolic equations is applicable also in more general situations than indicated earlier. As an example, we consider the problem
for the linear KleinGordon equation, describing the motion of a particle of mass ma and charge e in a constant magnetic field with vector potential A = (Al(z),... , A,(z)) E C”. Here m = moc2, c is the speed of light, z E Eta, v = (& )...) &). w e consider the case of the shortrange field, that is, we assume that the vectorfunction A(z) has compact support. The reason why we may not apply the methods obtained in $2 to (21) in a strictly formal manner lies in the fact that (21) with m # 0 does not satisfy A. However, a method for studying (21) remains the same. But the result obtained in this case is very different. If the asymptotics as t + co of the solution of (21) f or m = 0 and odd n has an exponential character and is determined by the poles of the operator &, then for the samen and m # 0, the asymptotics has a powerlaw character for real A(z) and is determined by the behavior of the resolvent near the points fm. We mention the corresponding result (for the case of even n, see Vainberg (1974)). Let RI, be the resolvent obtained for the operator of (21) for m = 0. Then
For any N
where J(u) = ctPiu++)sin q=o
[mt
+
(q

i)
51
~h,,,H2p+lu,
p=o
b,,, are some constants, and we have for the functions ‘ZUNfor t >> 1 and any
s, j = 0, 1,2, . . ., the estimates 5
s>(b)
C(N,
s, j)t
(
N+s) [IIPllo,a+ II~llo,al.
II. Asymptotics
of Solutions to Exterior BVP
69
Part II The Scattering Problem 5 1. Quasiclassical Asymptotic Expansion of the Solution to the Scattering Problem 1.1. Introduction, the Geometry of Rays. In this chapter, we study primarily the scattering of plane waves in an inhomogeneous medium. The process of scattering is described by a function + = +(L, X) satisfying the equation
[A + ~2hm = 0, and having the form $(lc,z) the radiation conditions
x E Et”, k>O,
= eikxn + u(lc,~),
21= f(O, IC)r(1“)/2eikr(1
+ O(rl)),
(1)
where the function r + cm,
f&:.
u satisfies
r
(2)
The function f is called the scattering amplitude. Here q E CW(IP); q(x) > 0; q(x) = 1 for r = 1x1 > a. The function eikxn corresponds to a plane wave propagating along the axis x, and the function u describes the scattered wave caused by the inhomgeneities of the medium (U = 0 if q = 1). Supposing that the nontrapping condition holds (see part 1) for the operator (l), we obtain the asymptotic expansion of the function $J as k + co, 1x1< b < co and the asymptotic expansion of f as k + co. The constants a and b are arbitrary and fixed, and we will not track the dependence of any quantities on a or b. In the problems of quantum mechanics, q(x) = E  U(x), k2 = 2mi’‘, where E is the energy, U is the potential, m is the massof the particle, fi. is Planck’s constant, and the asymptotic expansions as lc + co are called quasiclassical. The origin of this term stems from the fact that the laws of classical mechanics can be obtained from their quantum mechanical analogues by passing to the limit ti + 0. These asymptotics are also called WKBasymptotics in honor of Wentzel, Kramers and Brillouin, who first applied these asymptotic series in the study of onedimensional quantum mechanica. problems. In acoustics, k is proportional to frequency and the asymptotic expansion as k + oc is called highfrequency or shortwave. Equation (1) corresponds to the hamiltonian H = IpI2  q(x), p E R” and the hamiltonian system dx
 = 2p, ds
2
= Vq(x);
X(O) = (Y, a),
P(O) = (0,. . ., 071) >
(3)
where y E lR”‘. The initial conditions in (3) correspond to a plane wave propagating along the axis 2,. The projections of the phase curves of the system (3) into lF$ are called rays. It is easy to show that in regions where
70
B. R. Vainberg
q = 1, the rays propagate along straight lines. In particular, for IyI > a they coincide with the lines x’ = 1~, CQ < x, < 0~). Here x’ = (xi,. . . ,x,i). For IyI < a, the ray coincides with the line x’ = y while x, < Vq, that is while the ray remains outside the ball 1x1 < a. Inside this ball, it travels along some curvilinear path, and exiting from the ball, goes straight in the same direction as it exited with. A function J/JN satisfying equation (1) with error O(lCPM), where M + OCI as N + oo, is called a formal asymptotic solution (f. a. s.) of the equation. In some sufficiently small neighborhood of the initial hyperplane x, = a (which the rays emanate from), it is possible to construct a f. a. s. of equation (1) in the form ~+!JN= eilcscz) c,” aj(x)lcj, where 4~ = eikxn for x, < a. In a larger neighborhood, where it is possible for rays to intersect and the given f. a. s. does not exist, other solutions may be constructed with the help of the Maslov canonical operator. However, the relationship between these f. a. s. and the exact solution of the scattering problem is not clear a priori since (1) has infinitely many solutions if no boundary conditions are specified. The unique solution II, which interests us is determined by the farfield conditions (2). In this chapter, we briefly recall the construction of the canonical operator and use it to construct an f. a. s. $JN of (1) in the ball 1x1 < b. A method will then be described that is based on the use of the nonstationary problems and results of the previous chapter. It will allow us to estimate the difference +  IJN in the ball 1x1 < b, independently of the behavior of the constructed approximations 4~ for r + oo, and to prove that I,~,N is asymptotic to 4 in this ball as Ic + 00. 1.2. Maslov Canonical Operator (MCO). We explain only the formal construction of the MC0 and refer the reader to the previous article or to Maslov and Fedoryuk (1981), Mishchenko, Sternin, and Shatalov (1978), and Vainberg (1982) for details and proofs. The MC0 is contructed on an ndimensional lagrangian manifold, lying in 2ndimensional phase space. In the present case, the MC0 will be constructed on the manifold n c Iw$P), formed by the phase curves of (3). We note that the nontrapping condition allows us to define the MC0 on all functions in C(n), not only on functions with compact support. Let (Y = (oi,. . . ,as),/3 = (pr,. . . ,&s) be disjoint subsets of the set ofindices (1,2,... ,n)suchthatan/?=flandaU/?=(l,2 ,..., n),andlet I,0 = n  s. By x, , xcp, p,, pp we denote vectors consisting of the corresponding components of the vectors x and p. On the manifold A there exist global coordinates (y, s) called ray coordinates. Since the manifold A is lagrangian, there exists on it an atlas {.G’j}, each chart of which is simply connected and such that for some cx = aj J. _ dx,dm ~2dyd.s 3
c #O,
C,%>
a = et(j).
II. Asymptotics
of Solutions
to Exterior
BVP
71
Here (~~,pp) is determined by the system (3). If the vector (Y = o(j) can be chosen in more than one way, we fix it arbitrarily. In light of (4), on any chart fij, in addition to ray coordinates, there exist local coordinates (~,,po), (Y = o(j), and on the chart flj the manifold A may be written in the form xcp = “p(x:a,Pp),
Pa = PcY(xa,Pp);
where U, is some domain in KY. Let the function S E C(A) be defined
Q: = 4j),
(xCa,pa)
E uj,
(5)
by the formula
(6) where L(C) is an arbitrary smooth curve on n connecting the point with ray coordinates (y, s) = (0,O) with the point C (the function S does not depend on the choice of L). By S, = Sj (xcy , PO), (Y = o(j), we denote the value of the function S at a point of the chart f2j with local coordinates (xa,pp). Let @j = @j(GY, PP) = Sj(Xcx~ PP)  h3(xa~ The function @j is the generating that is, on that chart Vz_@p, = Pa(Gx,Pfi),
function
V,,@j
PO>, PO) .
of the manifold
(7)
A on the chart
flj,
= “p(GY,PpL
where the functions p, and x0 are defined in (5). We call a point C E A nonsingular if some neighborhood of it projects diffeomorphically onto some domain in IWE. We denote the set of singular points by E(A), and its projection into JR: is called the caustic set. We include a sufficiently small neighborhood G’j, of the point < E A in the atlas {Qj} with ray coordinates (y, s) = (0,O) such that fijO n E(A) = 8. We take x as the local coordinates on f2?, We denote by yj the KellerMaslov index of the chain of charts connecting fij and Qjnj,. The value of rj does not depend on the choice of the chain. This constant is an integer which, in the case where fij n E(A) = 8, is equal to the intersection index of any curve connecting f2jnj, and f2.j with E(A). Let {ej} be some partition of unity on A subordinate to the covering {Qj} and gj = gj(x) be arbitrary functions in Cr(IF) that are equal to one on the orthogonal projection of the chart Qj in RF and equal to zero outside a unit neighborhood of this projection. The canonical operator KAJ : C(A) + Cm(IRE) is defined by the formula
where
‘P E C(A),
a = a(j),
(wJy’)
= (e3pJJ1)
(xa,pa)
E C~(fY$).
B. R. Vainberg
72
This formula is distinct from the standard construction due to the presence of the multipliers gj. If cp E C,(A), their presence is not essential since it alters the MC0 by a quantity of order 0(X“), X + oo. For cp E C(A), thanks to the nontrapping condition and the presence in (8) of the functions gj, only finitely many of the terms in (8) differ from zero for 1x1 < c < 00, where c is arbitrary (that is, (8) is well defined). Finally, in this chapter it is convenient to take the atlas {Q} in a special form. Let Ac = {< E A : s = O}. Clearly, it possible to require A, c Uk>0&, p(2k) = 0, A,n&, = 0 for any,1 = 1,2,. . . . This will simplify finding initial conditions below. 1.3. Asymptotic Expansion of the Solution to the Scattering Problem. We will seek a formal asymptotic solution of (1) in the form
*N
=
f(ik)Jyj , [ 1 ‘pj E C’==(A).
K/l,li
j=o
If the functions ‘pj satisfy the transport equations (some ordinary differential equations along the phase curves of problem (3), which make up A, see the previous article or Maslov and Fedoryuk (1981), Mishchenko, Sternin, and Shatalov (1978), Vainberg (1982)), then in any ball 1x1< b [A + k’q(z)]+N
= O@C~+“‘~),
k+co,
(9)
that is, +,N is a f. a. s.of equation (1). Theorem 1. Let the nontrapping condition the transport equations and (POI,=~ = 1, ‘pjlSEo v= (VI,... , v,) and 1x1 < b
$b(k,
hold, the functions ‘pj satisfy = 0 for j > 0. Then for any
x)  ?,bN(k, x)]i 5 c(N, v)kN1+v+n’2,
k> 1.
In particular, if the domain V c IRZ does not contain any caustic points and if m rays of system (3) pass through each of its points, then, uniformly on compacta belonging to V,
j=l
where the functions Sj are determined by formula (6) with < = <j(x), Jj by (4) with IpI = 0 and C = <j(x), and the <j(.x) are points on A whose projections on II%: coincide with x E V. The constants 35 are equal to the intersection index of L(&(z)) and E(A).
II. Asymptotics
of Solutions
to Exterior
BVP
73
As we have already mentioned, (1) has an infinite set of f. a. s. in the ball 11~1< b. It is necessary to show that it is $N that is asymptotic to the solution $J of the scattering problem. The key to the proof of the theorem is given by formula (10) appearing below, which connects the function
Let
l ( Lv
( u(t,x)
q(x)$A
‘u=o,
t > 0,
XEEP,
t 5 0,
2 E R” )
>
= P(t)X(xn
 t),
where x E C(W), x(r) = 0 for r > a, x(r) = 1 for r < a  1. From the results of the previous chapter it follows that for a solution w of the problem Lw=O,
t > 0;
= 0,
Wlt,O
for any v, j, some T and t > T the following
w:itzo estimate
= f E ff,” holds:
where t3++’ may be replaced by eest for some b > 0 if n is odd. This allows us to prove the following assertion in a relatively simple manner. Let of ‘p be equal to P E Cl?@), (P(T) = 0 for T < 0 and r > 1 and the integral one. Theorem 2. Let the nontrapping the nonstationary scattering problem u = $~(k,
where
for some T’,
u of
x)ePt
+ u1 ,
any N and u, 1x1 < b and T > T’ vleiktcp(t
Thus,
condition hold. Then the solution be represented in the form
can
 T) dt
< C(N,
.~)k~,
k>
1.
for 1x1 < b and T > T’ +(k, x) = I”
w(t, x)etktcp(t
 T) dt + O(k“)
.
(10)
For the proof of theorem 1 it is now sufficient to substitute the wellknown (see the previous article or Maslov and Fedoryuk (1981), Mishchenko, Sternin, and Shatalov (1978), Vainberg (1982)) asymptotic expansion of the function w as k + 00 on the compact set 1x1 2 b, T 5 t 5 T + 1 into the previous integral and calculate the asymptotic behavior of the resulting integral. We note that this method allows us to avoid a twoparameter expansion of the function Q,, where k f cm and 1x1 4 co simultaneously, connected with the fulfillment of the radiation condition for the functions $J, ‘$N for all k >> 1, and also with the determination of the behavior of the righthand side of (9) for 1x1 + 03.
74
B. R. Vainberg
$2. Asymptotics
of the Scattering
Amplitude
We recall that for 121 > a the rays in problem (3) are straight lines. Let P be a map of JlJF’ to the unit sphere 57r which transforms each point y E IPl into a vector 0 E 57l associated with the direction at infinity of a ray of the system (3) with ray coordinates y. The modulus of the jacobian 1(g) of the map is equal to the angular density of the rays at infinity. Let m rays of the system (3) go to infinity along the direction 0 = Be E ,‘FP1 with y = yj, 1 5 j < m, and I(yj) # 0. Th en an analogous situation occurs for 10  00 1 < 1 also. Let yj (0) be the value of the ray coordinates of y for which the rays lj of (3) go to infinity along the direction 8. We will denote a segment of the ray Zj from the point with ray coordinates (y, s) = (yj(0), 0) to the point J: by Ii, and the corresponding segment of the phase curve of (3) by Lj,. Let 1j(0) = r(yj(0)) and
Jz
S,(x) = a + L3 (p, dx) . The
function
x E P,
Fj(@) = S,(x)  (Q,x), depends
only on 6’.
Theorem 3. Let the nontrapping (3) go to infinity along the direction l0&/<1 andkAoo
j=l
where
1x1 z+ a,
condition hold, m rays of the system 00, and I,(&) # 0. Then for any N,
s=o
ajo E 1 and ~j is the intersection
index
of Lj, with
E(A)
The asymptotic expansion of the function f for any 8 is expressed through the canonical operators constructed on the (n  l)dimensional lagrangian manifolds L+, LO c T*(S”l). We pass to the coordinates (0,r,q8,qr) in (3), where (qo, qT) are impulses corresponding to the spherical coordinates (0, r). We then have L+ = lim,,,{(8,q,) : y E EY‘}, and LO is the analogous manifold for q(x) E 1. In order to obtain the asymptotic expansion of the scattering amplitude f we can use the following integral representation of f in terms of the scattered wave u: f = pn I=,
[g w&k)?
+ il~(~, ~)u]
eik(elx)
dS,
II. Asymptotics
of Solutions
to Exterior
BVP
75
where R is arbitrary. This representation is a consequence of Green’s formula for u and the Green’s function of the Helmholtz operator. Then its quasiclassical asymptotic expansion from Theorem 1 is substituted for u = 1c, eikz, and it remains to solve the nontrivial problem of the asymptotic evaluation of the resulting integral.
76
B. R. Vainberg
Part III The Parametrix and the Full Asymptotic Expansion of the Spectral Function of Differential Operators in JR” $1. The
Parametrix
for Hyperbolic
Equations
and Systems
1.1. Introduction. A parametrix (of a Cauchy problem) for hyperbolic equations or systems is a function EN, differing from the Green’s function E by a function in the class CM, where M + co as N + 00. Having an explicit formula for EN, it is possible to study the discontinuities of the function EN and thereby to study the discontinuities of the Green’s function. In particular, it gives the wave fronts and the singularities on the fronts for the problem of oscillation of a medium excited by point sources. The parametrix enables the solution of many other problems also, for example, it reduces the solution of the Cauchy problem to an integral equation with a smooth kernel. The parametrix may be constructed in the usual way (cf. the previous article or Maslov and Fedoryuk (1981), Mishchenko, Sternin, and Shatalov (1978), V. (1982)) with the aid of the Maslov canonical operator or a Fourier integral operator connected with an (n + 1)dimensional lagrangian manifold lying in extended phase space. In this section a new, simpler formula for the parametrix is given for the case of equations in which the notrapping condition holds (cf. chapter 1) and systems whose coefficients do not depend on t. This formula has the form: EN(t,x)
=
y)eixt d, Jrn 03(X7 gX,N
(1)
where gX,%J is constructed on the stationary part of the operator and is related to an ndimensional lagrangian manifold lying in the phase space lKtz,P). Formula (1) is the analogue of an expression of the Green’s function in terms of the spectral function corresponding to the stationary operator. In contrast to the latter expression, the formula for the parametrix does not suppose selfadjointness of the operators or the fulfillment of any conditions on their spectra. In some cases of nonselfadjoint operators, it may substitute for the theorem of spectral decomposition. One of the applications of the new formula for the parametrix is the possibility of obtaining, with its help, the full asymptotic expansion of the spectral function corresponding to an elliptic operator on Iw” in the selfadjoint case. We leave this discussion for section 2. 1.2. Second of the form
Order
Equations.
Let z E Iw”, L be a hyperbolic
operator
II. Asymptotics
L==A
d2
( x,~ “>,
of Solutions
A=2
to Exterior
BVP
77
aij (xl i,j=l
where the operator A has infinitely notrapping condition. Let a(a,p) n,dimensional lagrangian manifold the problem
differentiable coefficients and satisfies the = C aij(x)pipj, p E JR?, and A = Ay be an in IR$) consisting of the phase curves of
40) = Y, (3) P(O) = &f&,
w E 9l
Here, p(0) is normalized so that a(y,p(O)) = 1. Let KA,X : Cm(A) + Cm(R~) be a Maslov canonical operator (MCO) which depends on the parameter y and is constructed on the manifold n = AY with the invariant measure dw ds on it (dw is the element of surface area on the sphere 9l). The construction of the MC0 is described in the previous chapter. In order for it to be applicable to the lagrangian manifold described above, it is necessary to substitute the measure dw ds for 2 dy ds in the formula for J3, fix a generating function @j on the chart fij (they are determined only up to an additive constant), and indicate the value of the Maslov index on one of the charts. As a generating function @j = @j (x,, pp) on the chart Qj with local coordinates (x,, pp), a = a(j), we take @j =
s
LW4
= 2s(Gx>PP)
 (Q(G,P~),P~)  (xp(xci,
PP), PP)
Here C is a segment with ray coordinates
of the phase trajectory in problem (3) going from the point (w, s) = ( w”, 0) to a point < E flj with local coordinates s corresponding to the point (G, PP); S(G, PP) is th e value of the parameter <; the function xcp = x~(cca,pp) is determined by formula (5) of chapter 2. For 0 < s < t, where E > 0 is sufficient.ly small, the manifold Au projects diffeomorphically to GIBE.We include in the atlas {Gj} a chart Qj,, the points of which satisfy 0 < s < E, and where we take x as local coordinates and set “/jO = 0. Finally, we recall that t,he manifold A = Au depends on y E R”. We take all objects in the construction of K,I,x (cf. formula (8) of chapter 2) such that they depend smoothly on y (this is possible). We denote by ~A,N = j’~,~(x, y) the function
where the functions +j equations and the initial
E Cm(A) conditions
satisfy
the recurrent
system
of transport
B. R. Vainberg
78
&I,=, = $j(W,Y/)> j>o, which are taken such that for any N
W)ffi,Jr(G YY) dX S(n:  ECNn(R2n). JO" Y)
(5)
0
Here C is an arbitrary function in Cm(Iw) such that C(X) = 1 for X > 2, C(X) = 0 for X < 1, and the transport equations are written in the standard form so that on any compact set in Iw” /(A + X2)fx,NI
2 CXN+n3’2,
x+ccl.
Theorem 1. Let the notrapping condition be satisfied. Then 1. It is possible to take $J~(w, y), j 2 0 such that (5) holds. Relation (5) and the transport equations determine the r+!~j (w, y), j > 0 uniquely with +J(,,~
=
$$27ry
[a(y, w)]“e
77 i’+nT
4
.
2. For the function
Y)x dA Jm<(X)f,&&, sin JX
EN(t,x,Y)
=
0
the following
relations
hold
(7) ENI~=~
= 0,
dEN at
6(2  y) E cNyIw2y. t=o 
1. The integrals (5) and (6), just as several other improper integrals encountered below, are to be interpreted in the standard manner that integrals of rapidly oscillating functions are usually interpreted. 2. The second assertion of the theorem is a consequence of the first. Also, from (5) and the tranport equations, it follows that for integers j E 10,(N  n)Pl Notes.
J
O” &A)Xjfd,N(~,y)
dX  (A)jS(x
 y) E CNn2j(IW2n).
(9)
0
3. Let (A) be a selfadjoint operator with an absolutely continuous spectrum, (Ex} its spectral family, and ei the kernel of the operator dEx/dX. Then if we substitute ei for the function C(X)fA,, in the integrals (5), (6) and (9), then the left hand sidesof (5), and (7))(g) will be identically zero and the indicated relations will turn into the usual spectral decomposition of the operators (A)j, j > 0 and (sin )/a. In this way, Theorem 1 may be viewed as an analogue of the spectral decomposition theorem for nonself adjoint operators.
4. Sometimes
II. Asymptotics
of Solutions
it is convenient
to replace
to Exterior (6) with
BVP
79
the equivalent
formula
5. To apply this method further, we need the parametrix for all t > 0. If we construct it only for t E [0, T], T < oo, then we may easily modify (6) and (6’) with the help of cutoff functions and still drop the notrapping condition.
1.3. Higher
Order
Equations
and Systems.
Let
.;.,(x.&,2&)
(10)
be a hyperbolic operator of order m with a unit coefficient for (i3/iat)m, Lo = Lo(z, X,p) be the leadingorder homogeneous part of its characteristic polynomial, X = &.(x,p), 1 5 T < m, be the (smooth) roots of the polynomial Lo with respect to the variable X. We assume that Lo(z,O,p) # 0 for p # 0, that is, X,(x,p) # 0 f orallr,xandp#O.Let+=sgnX,(x,p), p# 0; G(X,P> = lL(&P)l. We denote by A, = A,,, the ndimensional lagrangian manifold in lRFzp) consisting of the phase curves of the hamiltonian system with hamiltonian H, = a,(x,p) and initial conditions
The notrapping condition for the operator (10) is formulated the same way as for (2). In this case, the projections onto R” of the phase curves of each of the above hamiltonian systems, 1 5 T 5 m, are called rays corresponding to (10). The operator KA,,x is constructed exactly as the operator KA,~ was constructed in the previous section. Let
where initial
the functions conditions
which
are taken
$r,j
E C” (A,)
satisfy
?%,j Is=0 = h,3 (w, Y),
the transport
equations
and the
j >o,
so that
lrn ((A) g [&?$(Z, g)(itJ,“‘1
dX  6(x  z/Y)E cNn(~2n).
(12)
80
B. R. Vainberg
Theorem 2. Let the operator 1. There exist unique functions (ll), (12) are satisfied. Moreover,
(10) satisfy the notrapping condition. Then $r,j(w, y), j > 0, for which the inclusions
Here A, = X,(y, w). 2. For the functions
the following
relations
hold LEN
d”EN
EC t=” d”lE awN
atk
E CN“(R2n+1),
Nfmnkl(R2n),
t=”
O
 S(x  y) E CN7yW2y.
For simplicity in the case of a system, order system. Let
we show the result
only
for a first
(13) be an elliptic
matrix
operator
of dimension
m x m such that
the operator
L=&B
(14)
is hyperbolic. Here I is the identity matrix. This means that for all x,p the eigenvalues X,(x,p), 1 2 s 5 m, of the matrix Be(x,p) = C Mj(x)pj real and distinct. In view of the ellipticity of the operator (13), X,(x,p) for P # 0. Let E,~ = wL(x,p), P # 0, ~(55~) = I&(x,P)~. The lagrangian manifolds A, = Ar,Y, the notrapping condition and operators KA,,~ are defined for the operator (13) exactly as they were equation (lo), whose characteristic polynomial has the same roots A,, s < m. By gl,N = gi,N(x, y) we denote the vector function
# 0 are # 0 the for 1 <
(15) where
the vector
functions
$)T,j E Cm(&)
~T&=O =
~T,j(W>Y),
are fixed jL0,
by the initial
conditions
(16)
II. Asymptotics
of Solutions
to Exterior
which will be specified later, and by the requirement mate holds uniformly on compacta in IR”:
BVP that
81 the following
esti
This estimate means that the vector function (15) is an f. a. s. of the system (B  E,X)U = 0. From this estimate, it follows by standard methods that $+,j = ~~,j(~r+dr,j, where: 1) the vector functions d,,j are determined uniquely by recurrence from an algebraic system of equations containing $r,n, 0 < Ic < are the normalized eigenvectors of the matrix Ba with j (4,o = 0); 2) (PT eigenvalues X,(z,p); 3) the coefficients ,cL~,~ satisfy the transport equations. The arbitrariness in the choice of the initial conditions (16) lies in the fact that the initial conditions for the coefficients pr,j lsZo = pL,,? (w, 1~) may be prescribed in various ways. We take them so that
dX 6(x  Y)hk E CNn(R2n))
lW C(X)@,N
(17)
where hk is a vector whose kth component is one and all the others are zero. Let fk = C rCrCrn gi,,,, and let f$ be the columns which are obtained if only terms for which fe, > 0 are left in the last sum, and gzN be the matrices consisting
of the columns
fkf, 1 5 Ic 5 m.
Theorem 3. Let the notrapping condition hold Then 1. The initial conditions pu,,j lSzo may be specified, that the relation (17) holds. In this case pu,,o(w, y) = (27r)*ei+n~A,(w, 2. For the matrix
LEN
relations
s0
O” <(X)[gT,NeiXt
Second
in fact,
operator
(13).
uniquely,
such
(hk, (~~l~,~).
E CNn(iR2n+1
Order
+ gi,NeiXt]
dX,
hold ),
$2. Asymptotics 2.1.
the
function
EN = the following
y)I*
for
ENI~,~
 6(x  y)I E CNn(IR2n).
of the Spectral Function
Equations.
Theorem 4. Let the operator A, defined in formula (2), be formally selfadjoint, satisfy the notrapping condition, and coincide with the laplacian for 1x1 >> 1. Let ei be the kernel of the operator dEx/dX, where {Ex} is the spectral
82
B. R. Vainberg
family of the operator (A), and the function ~A,N be defined in (4). Then for any j 2 0, CI = (CQ, . . . , CY,), /3 = (PI,. . . , /3,), b < co and N, there exists a C = C(d, b, N) (h ere d = Ial + IPI +j), such that for 1x1, IyI 5 b and X f CC
This assertion is a direct consequence of the results in chapter 1 and the formula for the parametrix, introduced in $1 of that chapter. In fact, corresponding to formula (6’) we obtain ENeiXt
dt .
On the other hand, for X > 0 in D’(IwF) we have the equality
sy
E(t, x, y)eixt
dt.
lxJ
It follows then for X > 1 that ei2 
f&N
= g
1
m (E  EN)eiXt
dt.
s co
The difference EEN is a sufficiently smooth function if N is large. Therefore for the proof of the estimate (18)) we need only obtain some information about the behavior of the function E  EN and its derivatives as t + 00. From (4) and (6’) it follows that EN and its tderivatives decay faster than any power of t as t + 03. The information about the behavior of the function E as t + co necessary to obtain (18) is contained in Chapter 1. Theorem 4 gives the asymptotics of ei when (xc,y) belongs to any compact set in ll%2n.If the point (x0, y”) is such that the point x0 is not caustic for the ray family (3) for y = y”, then in some neighborhood of the point (x0, y”) (meaning also in any connected domain in Tw2” consisting of points of the indicated form) it is possible to make Theorem 4 more precise. This refinement allows one to write in a simple way the function fX,N in the formulation of Theorem 4 with the help of (4) and (8) from Chapter 2 and under a certain choice of the atlas (52) in the construction of the operator KA,x. Assume exactly one ray of (3) goes from point y to point II: for s > 0. Let C be the segment of the phase curve corresponding to it and let (w’, so) be the value of the ray coordinates (w, s) for which the ray passesthrough the point x. If x # y and (19) where x = x(w,z) is determined from (3), then the point x is called noncaustic. Let the function S = S(x, y) be equal to J,(~,dx) and y be the intersection index of C with the set of singular points E(n) = {< E A : &I6 = 0). If a single ray passesthrough the point x m times, we say that m rays pass through the point.
II. Asymptotics
of Solutions
to Exterior
BVP
83
Theorem 5. Let all the conditions of Theorem 4 befulfilled. Then 1) If exactly one ray from y” passesthrough the point x0 # y” for s > 0 and
the point x0 is not caustic, then this condition holds for all the points (x, y) in some neighborhood U c Iw2” of the point (x0, y’), and there exist functions aj(x, y) such that ei(x, y) = 5
aj(z, y)X”t”
x
j=o
(20)
cos
,++j)]
dis(x,y); [
Here for any compact set K
dd dXjdxc,dyP Further,
+ ew(x,
( c
Y) .
U, any j 2 0, a, ,B, (x, y) E K and X + M
eA,N(x, y) 2 CA*+*,
C=C(d,K,N).
if A = A, then aj = aj for all j and ao(x, y) = 1/5(271.)9
[a(y,w)]2
,
where w is the value of the ray coordinate corresponding to the ray going from the point y through the point x. 2) If m rays passfrom y” through x0 for s > 0 and condition (19) holds for them all, then it also holds for all points in some neighborhood U of the point (x0, y’), and ei is represented for (x, y) E U in the form of a sum of terms, each of which corresponds to its ray and has the form (20). 3) If the rays emanating from the point y = x0 do not return to that point, then this condition holds for all points x in some neighborhood V c IP of the point x0, and there exist functions a3(x) such that ei(x,x)
= ~ak(x)X~.‘”
+ eA,N(x) .
(21)
k=O
Here a’(x) = :(27r” and for all compacta Q @+lal eA,N(x) dXjdX”
c
/
dp
4z,P)0, CX,x E Q and X + 00 5 CXN2+4j,
C=W,a,Q,W.
If exactly m rays emanating from the point x = y” return back to that point for s > 0 and satisfy condition (19) at their return, then this condition holds for all points x in some neighborhood V c R2” of the point x0 and ei(x,x) is representable in the form of a sum of the expressions obtained for ei in
sections 3 and 2 (in the latter it is necessary to set y = x).
B. R. Vainberg
84
We note that the principal term in the asymptotics of the function ei(2, X) depends on the value of the coefficients of the operator A only at the point 2. The lower order terms do not have this local character and depend on the geodesic loops. Moreover, the main contribution of derivatives in x or X to the asymptotics is brought by geodesic loops. For n = 2, this is true for first order derivatives already, for n = 3, second order. For n = 3, the degree of the first derivative of expressions (20) for y = 5 and (21) is equal.
2.2. Systems proved
exactly
of First
as Theorem
Order
Equations.
The
following
assertion
is
4 is.
Theorem 6. Let the operator (13) satisfy the notrapping condition and be formally selfadjoin& and let the matrices Mj(x) be independent of x and the matrix N(x) be zero for 1x1 >> 1. Let ei be the kernel of the (matrix) operator dEx/dX, where {Ex} is the spectral family of the operator (13). Then for any b < DC), j > 0, Q, ,B and N, 1x1, lyl < b and X + fco
there
exists
a C =
C(d, b, N)
such
that for
A consequence of this theorem is an analogue of Theorem 5. Commentary
on the Literature Chapter
1
There are five different approaches to the study of the behavior for t + 00 of the solutions of exterior mixed boundary value problems for hyperbolic equations and systems. One of them (cf. Eidus 1964, 1969, Mikhailov 1967, Mizohahta and Mochizuku 1966, Mochizuku 1969, Matsumura 1970) is based on representing the solution, via the theorem on spectral decomposition, in the form of an oscillating integral over the spectrum of the corresponding stationary problem. This integral can be successfully investigated if the properties (smoothness, estimates) of the spectral function are known. This method usually does not allow obtaining the asymptotic expansion of the solution, but only estimates of it, which are additionally not uniform in the set of initial conditions. The latter seriously reduces the possibility of applying results obtained by this method to the asymptotic investigation of the corresponding stationary problems. A further inadequacy of the method lies in the requirement that the problem be symmetric. In the articles by Morawetz (1962) and Lax and Phillips (1963) (see also Zachmanoglou (1963) and Morawetz, Ludwig, and Strauss (1977)), new energy estimates are found which allow determination of the decay rate for t + 00 of the local energy of solutions of the exterior problem for second order equations. More refined results for the wave equation on the exterior of a strictly convex body in two and three dimensions can be obtained with the help of a special reduction of the problem to an integral equation on the boundary, see for example Babich (1972), Muravei (1970, 1973), Babich and Grigor’ev (1974).
II. Asymptotics
of Solutions
to Exterior
BVP
The fourth approach to studying the behavior as t + co of solutions of nonstationary problems, presented in Lax and Phillips (1967) (see also Lax and Phillips (1971), Iwasaki (1969)), is b ase d on using the group properties of the solutions and the theory of scattering. Finally, the fifth approach, developed in this section, is due to this author (cf. 1974, 1975, 1982, and the references given there), and the article by Ladyzhenskaya (1957) served as a starting point. We note that this last approach allows one to obtain all of the results which follow from the theory of scattering, but without the restrictions imposed on the problem when studying it using that theory. Problems in which the notrapping condition does not hold are examined in Ralston (1969), Majda and Ralston (1978a, 197813, 1979), Lazutkin (1981), Bardos, Guillot, and Ralston (1982), Petkov and Popov (1982), and Ikawa (1983). Several other results on the asymptotics for t + 00 of solutions to wave problems are contained in Wilcox (1978)) Mochizuku (1982)) and Gushchin and Mikhailov (1986). Further development of the results covered in this section is contained in Rauch (1978), Ralston (1976), Murata (1983), Melrose (1983), and Menzala and Schonbek (1984). A large list of works on the properties of the resolvent of a stationary operator is included in V. (1982, , see the supplement to chapter 9). Chapter
2
Theorems l3 are obtained in V. (1977a, 1977b) (see V. (1982) also). A theorem analogous to Theorem 3 was announced in Guillemin (1977). The asymptotics of a function f for arbitrary 0 is contained in Protas (1982). The asymptotics for Ic + 00 of the Green’s function for equation (1) was obtained earlier in Babich (1965) and in Kucherenko (1969). The quasiclassical aymptotics of the scattering amplitude of waves in a homogeneous medium in the exterior of a bounded object are investigated in Majda (1976), Majda and Taylor (1977), Melrose (1980), and Petkov (1980). Additional references can be found in Babich and Buldyrev (1972), V. (1982) (see Kucherenko and Osipov (1983) also). Chapter
3
A good familiarity with the asymptotics of the spectral function of elliptic operators on unbounded domains can be obtained from the works Arsen’ev (1967), Babich (1965, 1980), Kucherenko (1969), Buslaev (1971, 1975), Majda and Ralston (1978a, 197813, 1979), Morawetz, Ralston, and Strauss (1977), Ivrij and Shubin (1982), Popov and Shubin (1983), V. (1983, 1984, 1985), and Popov (1985). In particular, in Buslaev (1971, 1975) the asymptotic behavior of the spectral characteristics (including the kernel of the resolvent) of exterior problems for operators with constant coefficients in their principal terms. In Babich (1965) and Kucherenko (1969) the shortwave asymptotics for second order operators with variable coefficients in Iw” are obtained. In Popov and Shubin (1983) the full asymptotic development of the function ei for X + 00 is obtained for a second order operator, assuming the conditions of theorem (4) are fulfilled and with the additional assumptions that IZ  y[ < 1 and that the operator A has no geodesic loops, that is, no ray corresponding to problem (2) crosses any point twice. The answer in this case is given by an integral over the sphere Ye1 of some rapidly oscillating (as X + oo) function depending on the parameter w E s”1. Some stronger results are contained in Popov (1985). The asymptotics of ei was obtained without the above additional assumptions by this
85
86
B. R. Vainberg author in (1983, 1984) in the form of an integral over S”l of some family of canonical operators. The results of this section were published by the author in (1985). Simpler formulas for e; were obtained in Babich (1980), where the remainder term was small only in an average sense. In this last work the problem was investigated on a bounded domain, the asymptotics of ei were found for 2, y lying in some compact set in the domain, and the assumption that the operator was of second order was used in an essential way.
References* Agranovich, M. S., and Vishik, M. I. (1964): Elliptic problems with a parameter and parabolic problems of general type. Usp. Mat. Nauk 19, vol. 3, 53161 (Russian), Zbl. 137,296 Arsen’ev, A. A. (1967): Asymptotics of the spectral function for the Schrodinger equation. J. Vychisl. Mat. Mat. Fiz. 7, 12981319 (Russian), Zbl. 162,163 Babich, V. M. (1965): The shortwave asymptotic form for the problem of a pointsource in an inhomogeneous medium. Zh. Vychisl. Mat. Mat. Fiz. 5, no. 5, 949951, Zbl. 167,243. English transl.: USSR Comput. Math., Math. Phys. 5, no. 5, 247251 (1968) Babich, V. M. (1972): On the asymptotics of the Green’s function for some wave problems. Mat. Sb. 87, Ser. 87(129), 4457 (Russian), Zbl. 251.35015 Babich, V. M. (1980): Hadamard’s method and the asymptotics of the spectral function for second order differential operators. Mat. Zametki 28, vol. 5, 689694. English transl.: Math. Notes 28, 800803 (1981) Babich, V. M., and Buldyrev, V. C. (1972): Asymptotic Methods for Short Wave Diffraction Problems. Moscow: Nauka, 456 pp. (Russian), Zbl. 255.35002 Babich, V. M., and Grigor’ev, N. S. (1974): The analytic extension of the resolvent of outer threedimensional problems for Laplace operators in the second sheet. Functs. Anal. Prilozh. 8, no. 1, 7172. English transl.: Funct. Anal. Appl. 8, 62263 (1974) Zbl. 292.35064 Bardos, C., Guillot, J. C., and Ralston, J. (1982): La relation de Poisson pour l’equation des ondes dans un ouvert non borne. Application a la thkorie de la diffusion. Commun. Partial Differ. Equations, no. 7, 905958, Zbl. 496.35067 Blekher, P. M. (1969): On operators depending meromorphically on a parameter. Vestn. Mosk. Univ., Ser. I, no. 5, 3036, Zbl. 187,393. English transl.: Moscow Univ. Math. Bull. 24 (1972), Zbl. 243.47010 Buslaev, V. S. (1971): Scattering of plane waves, spectral ssymptotics and trace formulas in exterior problems. Dokl. Akad. Nauk SSSR 197, no. 5., 9991002. English transl.: Sov. Math. Dokl. 12, 591595 (1971), Zbl. 224.47023 Buslaev, V. S. (1975): On the asymptotic behavior of spectral characteristics of exterior problems for the Schriidinger operator. Izv. Akad. Nauk SSSR., Ser. Mat. Mekh. 39, no. 1, 149235 (Russian), Zbl. 311.35010 Eidus, D. M. (1964): The principle of limit amplitude. Dokl. Akad. Nauk SSSR 158, no. 4, 794797. English transl.: Sov. Math. Dokl. 5, 13271330 (1965), Zbl. 141,299 * For the convenience of the reader, references to reviews in Zentralblatt fur Mathematik (Zbl.), compiled using the MATH database, and Jahrbuch iiber die Fortschritte der Mathematik (Jbuch) have, as far as possible, been included in this bibliography
II. Asymptotics
of Solutions
to Exterior
BVP
87
Eidus, D. M. (1969): The principle of limiting amplitude. Usp. Mat. Nauk 24, vol. 3, 91156, Zbl. 177.142. English transl.: Russ. Math. Surv. 24, no. 3, 97167 (1969), Zbl. 197.081 Guillemin, V. (1977): Sojourn times and asymptotic properties of the scattering matrix. Publ. Res. Inst. Math. Sci., Kyoto Univ. 12, Suppl. 6988 Gushchin, A. K., and Mikhailov, V. P. (1986): On uniform quasiasymptotics of the solution of the Cauchy problem for a hyperbolic equation. Dokl. Akad. Nauk SSSR 287, no. 1, 3740. English. transl.: Sov. Math., Dokl. 33, 326329 (1986), Zbl. 629.35072 Hormander, L. (1968): The spectral function of an elliptic operator. Acta Math. 121, 193218, Zbl. 164,132 Hormander, L. (1971): Fourier integral operators. Acta Math. 127, no. 2, 79183, Zbl. 212,466 Ikawa, M. (1983): On poles of the scattering matrix for two strictly convex obstacles. J. Math. Kyoto Univ. 23, no. 1, 127194, Zbl. 561.35060 Ivrij, V. M. (1978): The propagation of singularities of solutions of the wave equation near the boundary. Dokl. Akad. Nauk SSSR 239, 7722774. English transl.: Sov. Math. Dokl. 19, 400402 (1978), Zbl. 398.35057 Ivrij, V. M., and Shubin, M. A. (1982): On the asymptotics of the spectral shift function. Dokl. Akad. Nauk SSSR 263, no. 2, 283284. English transl.: Sov. Math. Dokl. 25, 332334 (1982), Zbl. 541.58047 Iwssaki, N. (1969): Local decay of solutions for symmetric hyperbolic systems with dissipative and coercive boundary conditions in exterior domains. Publ. Res. Inst. Math. Sci. 5, no. 2, 1933218, Zbl. 206,400 Kreiss, H.O. (1970): Initial boundary value problems for hyperbolic systems. Commun. Pure Appl. Math. 23, no. 3, 277298, Zbl. 193,069 Kucherenko, V. V. (1969): Quasiclassical asymptotics of a pointsource for the stationary Schrodinger equation. Teor. Mat. Fiz. 1, no. 3, 384406 (Russian) Kucherenko, V. V., and Osipov, Yu. V. (1983): The Cauchy problem for nonstrictly hyperbolic equations. Mat. Sb., Nov. Ser. 120, no. 1, 84111 (Russian), Zbl. 519.35048 Ladyzhenskaya, 0. A. (1957): On the principle of limiting amplitude. Usp. Mat. Nauk 12, vol. 3, 161164 (Russian) Lax, P., and Phillips, R. (1967): Scattering Theory. Academic Press, 276 pp., Zbl. 186,163 Lax, P., and Phillips, R. (1971): Scattering theory. Rocky Mt. J. Math. 1, no. 1, 173223, Zbl. 225.35081 Lax, P., Morawetz, C., and Phillips, R. (1963): Exponential decay of solutions of the wave equation of a starshaped obstacle. Commun. Pure Appl. Math. 16, no. 4, 4777486, Zbl. 161,080 Lazutkin, V. F. (1981): Diffractive losses in open resonators: A Geometric Approach. Dokl. Akad. Nauk SSSR 258, no. 5, 10891092 (Russian) Lions, J., and Magenes, E. (1968): Problems aux Limites Non Homogenes et Applications. Paris 1, 372 pp., Zbl. 165,108 Majda, A. (1976): High frequency asymptotics for the scattering matrix and the inverse problem of acoustical scattering. Commun. Pure Appl. Math. 29, no. 3, 261291, Zbl. 463.35048 Majda, A., and Osher, S. (1975): Reflection of singularities at the boundary. Commun. Pure Appl. Math. 28, 479499, Zbl. 307.35077 Majda, A., and Ralston, J. (1978a): An analogue of Weyl’s formula for unbounded domains I. Duke Math. J. 45, 183196, Zbl. 408.35069 Majda, A., and Ralston, J. (197813): A n analogue of Weyl’s formula for unbounded domains II. Duke Math. J. 45, 513536, Zbl. 416.35058
88
B. R. Vainberg
Majda, A., and Ralston, J. (1979): A n analogue of Weyl’s formula for unbounded domains III. Duke Math. 3. 46, 725731, Zbl. 433.35055 Majda, A., and Taylor, M. (1977): The asymptotic behavior of the diffraction peak in classical scattering. Commun. Pure Appl. Math. 30, no. 5,639%669, Zbl. 357.35007 Maslov, V. P. (1973): Operator Methods. Moscow: Nauka, 543 pp. (Russian) Maslov, V. P., and Fedoryuk, M. V. (1981): S emiClassical Approximation in Quantum Mechanics. Dordrecht, Holland, 301 pp. Matsumura, M. (1970): Comportement asymptotique de solutions de certains problemes mixtes pour des syst&mes hyperboliques sym&riques a coefficients constants. Publ. Res. Inst. Mat,h. Sci. Kyoto Univ. 5, no. 3, 301360, Zbl. 242.35056 Melrose, R. (1980): Forward scattering by a convex obstacle. Commun. Pure Appl. Math. 30, no. 5, 461499, Zbl. 435.35066 Melrose, R. (1983): Polynomial bound on the number of scattering poles. J. Funct. Anal. 53, no. 1, 287303, Zbl. 535.35067 Melrose, R., and Sjijstrand, J. (1978): Singularities of boundary value problems I. Commun. Pure Appl. Math. 31, 593617, Zbl. 378.35014 Menzala, G. P., and Schonbek, T. (1984): Scattering frequencies for the wave equation with a potential term. J. Funct. Anal. 55, no. 3, 297322, Zbl. 536.35060 Mikhailov, V. P. (1967): On the stabilization of the solution of a nonstationary bounded problem. Tr. Mat. Inst. Steklova 91, loo112 (Russian), Zbl. 162,152 Mishchenko, A. S., Sternin, B. Yu., and Shatalov, V. E. (1978): Lagrangian manifolds and the canonical operator method. Moscow: Nauka, 352 pp. (Russian) Mizohata, S., and Mochizuku, K. (1966): On the principle of limiting amplitude for dissipative wave equations. J. Math. Kyoto Univ. 6, no. 1, 109$127, Zbl. 173,371 Mochizuku, K. (1969): The principle of limiting amplitude for symmetric hyperbolic systems in an exterior domain. Publ. Res. Inst. Math. Sci. Kyoto Univ. 5, no. 2, 259265, Zbl. 206,110 Mochizuku, K. (1982): Asymptotic wave functions and energy distributions for long range perturbations of the d’Alembert equations. J. Math. Sot. Jap. 34, no. 1, 143171, Zbl. 475.76077 Morawetz, C. (1962): The limiting amplitude principle. Commun. Pure Appl. Math. 15, no. 3, 349361, Zbl. 196,412 Morawetz, C., and Ludwig, D. (1969): The generalized Huygens’ principle for reflecting bodies. Commun. Pure Appl. Math. 22, no. 2, 189206, Zbl. 172,383 Morawetz, C., Ralston, J., and Strauss, W. (1977): Decay of solutions of the wave equations outside nontrapping obstacles. Commun. Pure Appl. Math. 30, no. 4, 447508, Zbl. 372.35008 Murata, M. (1983): High energy resolvent estimates I, first order operators. J. Math. Sot. Jap. 35, no. 4, 711733, Zbl. 522.35013 Muravei, L. A. (1970): Asymptotic behavior of solutions to the second exterior boundary value problem for the twodimensional wave equation. Differ. Uravn. 6, no. 12, 22482262, Zbl. 218,232. English transl.: Diff. Eq. 6, 17091720 (1973), Zbl. 263.35054 Muravei, L. A. (1973): Largetime symptotic behavior of solutions to the second and third boundary value problems for the wave equation with two space variables. Tr. Mat. Inst. Steklova 126, 73144 (Russian), Zbl. 284.35005 Petkov, V. (1980): High frequency asymptotics of the scattering amplitude for nonconvex bodies. Commun. Partial Differ. Equations 5, 293329, Zbl. 435.35065 Petkov, V., and Popov, G. S. (1982): Asymptotic behaviour of the scattering phase for nontrapping obstacles. Ann. Inst. Fourier 32, no. 3, 111149, Zbl. 497.35009 Popov, G. S., and Shubin, M. A. (1983): Asymptotic expansion of the spectral function for secondorder elliptic operators in Iw”. Funkts. Anal. Prilozh. 17, no. 3, 3745. English transl.: Funct. Anal. Appl. 17, 193200 (1983), Zbl. 533.35072
II.
Asymptotics
of Solutions
to Exterior
BVP
89
Popov, G. S. (1985): Spectral asymptotics for elliptic secondorder differential operators. J. Math. Kyoto Univ. 25, 6599681 (1985), Zbl. 598.35081 Protas, Yu. N. (1982): Quasiclassical asymptotics of the scattering amplitude of a plane wave in an inhomogeneous medium. Mat. Sb. Nov. Ser. 117 (159), no. 4, 494515 (Russian), Zbl. 508.35065 Ralston, J. (1969): Solutions of the wave equation with localized energy. Commun. Pure Appl. Math. 22, no. 6, 807823, Zbl. 209,404 Ralston, J. (1976): Note on the decay of acoustic waves. Duke Math. J. 46, no. 4, 799804, Zbl. 427.35043 Rauch, J. (1978): Asymptotic behaviour of solutions to hyperbolic partial differential equations with zero speed. Commun. Pure Appl. Math. 31, no. 4, 431480, Zbl. 378.35044 Sakamoto, R. (1970): Mixed problems for hyperbolic equations, I. Energy inequalities. J. Math. Kyoto Univ. 10, no. 2, 3499373, Zbl. 203,180 SanchezPalencia, E. (1980): Nonhomogeneous media and vibration theory. Lect. Notes Phys., 127 pp. Taylor, M. (1976): Grazing rays and reflection of singularities of solutions to wave equations. Commun. Pure Appl. Math. 29, no. 1, l38, Zbl. 318.35009 Tikhonov, A. N., and Samarskij, A. A. (1953): The equations of mathematical physics. Moscow: Gostekhizdat, 679 pp. English transl.: Oxford, 1963, Zbl. 111,290 Vainberg, B. R. (1974): Behavior for largetime of solutions to the KleinGordon equation. Tr. Mosk. Mat. O.va 30, 139158 (Russian), Zbl. 318.35051. English transl.: Trans. Moscow Math. Sot. 30, 139159 (1974) Vainberg, B. R. (1975): On shortwave asymptotic behaviour of solutions to stationary problems and the asymptotic behaviour as t + cc of solutions of nonstationary problems. Usp. Mat. Nauk 30, vol. 2, 355, Zbl. 308.35011. English transl.: Russ. Math. Surv. 30, no. 2, l58) (1975), Zbl. 318.35006 Vainberg, B. R. (1977a): Statinonary problems in scattering theory and the largetime behavior of solutions of nonstationary problems for large values of time. Moscow: Mosk. Univ. Publ., 5862 (Russian), Zbl. 498.35068 Vainberg, B. R. (197713): Quasiclassical approximation in stationary scattering problems. F’unkts. Anal. Prilozh. 11, no. 4, 618, Zbl. 381.35022. English transl.: Funct. Anal. Appl. 11, no. 4, 427457 (1978), Zbl. 413.35025 Vainberg, B. R. (1982): Asymptotic Methods in Equations of Mathematical Physics. Moscow: Mosk. Univ. Publ., 294 pp., Zbl. 518.35002. English transl.: New York: Gordon and Breach Science Publ., 498 pp., 1989, Zbl. 743.35001 Vainberg, B. R. (1983): A complete asymptotic expansion of a spectral function of elliptic operators in Iw”. Vestn. Mosk. Univ., Ser. I, no. 4, 2936, Zbl. 547.35050. English transl.: Moscow Univ. Math. Bull. 38, no. 4, 32239 (1983) Vainberg, B. R. (1984): The complete asymptotic expansion of the spectral function of secondorder elliptic operators in Iw”. Mat. Sb., Nov. Ser. 123, no. 2, 195211, Zbl. 573.35070. English transl.: Math. USSR, Sbornik 51, no. 1, 191206 (1985) Vainberg, B. R. (1985): On the parametrix and the asymptotics of the spectral function of differential operators in Iw”. Dokl. Akad. Nauk SSSR 282, no. 2, 265269, Zbl. 617.35098. English transl.: Sov. Math. Dokl. 31, no. 3, 456460 (1985) Vainberg, B. R. (1990): Asymptotic behaviour as t + 00 of solutions of exterior mixed problems periodic with respect to t. Mat. Zametki 47, no. 4, 616. English transl.: Math. Notes 47, no. 4, 315322 (1990), Zbl. 708.35015 Wilcox, C. (1978): Asymptotic wave functions and energy distributions in strongly propagative anisotropic media. J. Math. Pures Appl., IX. Ser. 57, 2755321, Zbl. 409.35064 Zachmanoglou, E. (1963): The decay of the initialboundary value problem for the wave equation in unbounded regions. Arch. Ration. Mech. Anal. 14, no. 4, 312325, Zbl. 168,80
III. The HigherDimensional WKB Method or Ray Method. Its Analogues and Generalizations V. M. Babich Translated from the Russian by J. S. Joel
Contents Introduction..................................................... Chapter
1. Fundamental
NonLocal
93 ShortWave
Expansions.
..........
94
31. The Classical Ray Method. .................................... 94 94 1.1. The Starting Formulas ................................... 96 1.2. The Eikonal Equation ................................... 97 1.3. Rays and Ray Coordinates ............................... 98 1.4. Integration of the Transport Equations. .................... 99 1.5. Discussion of the Our Results ............................. 100 1.6. Reflection and Refraction of Ray Solutions ................. 102 $2. Point Source of Vibrations in an Inhomogeneous Medium. ......... 2.1. Ansatz.................................................lO 2 .103 2.2. Transport Equations. ................................... 104 2.3. Discussion of Our Results ................................ §3. ShortWave Expansion in a Neighborhood 105 of a Nonsingular Piece of a Caustic ............................. 3.1. Basic Assumptions .105 ..................................... 3.2. The Analytic Character of an Eikonal Close to a Caustic ..... 106 108 3.3. Asymptotics of a Solution ................................
V. M. Babich
92
Chapter
2. Some Modifications
of the Ray and Caustic
Expansions.
$1. Asymptotics of Vibrations of Whispering Gallery Type 1.1. The Ray Method in the Small . . . . . . . . . . . . . . . . 1.2. Boundary Conditions and Transport Equations . 1.3. Analogues and Generalizations . . . .. . .. . . . .. . . . $2. Surface Wave Propagated Along an Impedance Surface 2.1. Initial Statements. Ansatz . . . . . . . . . . . . . . . . . 2.2. Construction of the Asymptotics of a Solution . . Chapter
3. Gaussian
Beams and Their Applications
.. .. .. .. .. .. ..
.. ..
4. On Other ShortWave
Diffraction
.. .. .. ..
.. .. ..
110 110 112 113 114 114 115
. . . . . . . . . . . . . . . . .116
$1. Solutions Concentrated in a Neighborhood of a Fixed Ray . . . 1.1. Statement of the Problem. Ansatz . . . . . . . . . . . . . . . . . 1.2. Determination of r(O), r(l), and T(~) . . . . . . . . . . . . . . 1.3. Determination of ~i~,,.i~, ujil,,,ih . . . . .. . .. . . . . . .. $2. The Case of a Closed Ray .. . . .. . . .. . .. . . .. . . .. . .. . .. 2.1. Properties of the System of Jacobi Equations . . . . . . . . . 2.2. Construction of Quasimodes in the Case of a Closed Ray in the First Approximation .. . . . .. . .. 2.3. Higher Approximations . . .. . .. . . . .. . . . .. . . .. . . . 93. Summation of Gaussian Beams. . .. . .. . . . .. . . .. . .. . . . .. . . 3.1. Basic Ideas of the Method (see Babich and P an k ra t ova (1973) and Popov (1981)) . 3.2. Application of the Method of Stationary Phase. . . . . . Chapter
. . . 110
Problems
.. .. .. .. .. ..
116 116 118 119 120 120
. . . . 121 . . . . 121 . . 123 . . . . 123 . . 125
. . . . . . . . . . . . . . 125
$1. The Case of Smooth Reflecting Boundaries .. . . . .. . .. . . .. . . . 1.1. Fock’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Point Sources of Vibrations on the Boundary of a Domain (see Babich (1979), Babich and Buldyrev (1972), and Buslaev (1975)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $2. Various Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The Problem of the Change of Sign of the Curvature (Babich and Smyshlyaev (1984), Popov (1979)) . . . . . . . 2.2. Problems with Sharp Edges and Screens . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 125 . . . .125
. . . ..
127 128
. . . .128 . . . . 129 . . . .130
III. The HigherDimensional
WKB
Method
or Ray Method
93
Introduction In this paper we shall discuss the construction of formal shortwave asymptotic solutions of problems of mathematical physics. The topic is very broad. It can somewhat conveniently be divided into three parts: 1. Finding the shortwave asymptotics of a rather narrow classof problems, which admit a solution in an explicit form, via formulas that represent this solution. 2. Finding formal asymptotic solutions of equations that describe wave processesby basing them on some ansatz or other. We explain what 2 means. Giving an ansatz is knowing how to give a formula for the desired asymptotic solution in the form of a series or some expression containing a series, where the analytic nature of the terms of these seriesis indicated up to functions and coefficients that are undetermined at the first stage of consideration. The second stage is to determine these functions and coefficients using a direct substitution of the ansatz in the equation, the boundary conditions and the initial conditions. Sometimes it is necessary to use different ansatze in different domains, and in the overlapping parts of these domains the formal asymptotic solutions must be asymptotically equivalent (the method of matched asymptotic expansions). The basis for success in the search for formal asymptotic solutions is a suitable choice of ansatze. The study of the asymptotics of explicit solutions of special model problems allows us to “surmise” what the correct ansatze are for the general solution. 3. The proof of the solvability of the problem of mathematical physics under consideration and the fact that the formal asymptotic solution constructed via suitably chosen ansatze actually represents the asymptotic behavior of a solution of this problem. In the present paper we shall mainly discussthe problem of finding asymptotic solutions via ansatze, that is, constructions making up the second part of this topic. We give some (well known) definitions. Let ‘pj(z, e), j = 1,2,3, be a function of a point x E 6) c R” and a parameter c, which varies in some fixed interval 0 < c < ~0. In addition, let f(z, E) be a fixed positive uniformly bounded function for x E R, 0 < E < ~0, defined where ‘pj(x, E) is. Definition
1. The series
is called an asymptotic Ipj(x,e)I
5 AjeB3f(x,e);
expansion
if the following
Aj, Bj = const,
estimates Bj + +oo
hold: as j ) +CO.
(2)
The following Definition 2 and the concept of an asymptotic expansion of a solution u (see formula (4)) goes back to the classical works of Henri
94
V. M. Babich
Poincare on the method of small parameters in celestial mechanics. Let C be a differential operator. Definition 2. An asymptotic expansion of the equation Cu = 0 if
(1) is called
a formal
asymptotic
solution
IC(cpl + (p2+ . . . + cpj)l 5 CjED” f(x, t);
03 + +CC as j + +CO.
(3)
For many problems of mathematical physics, in particular, for practically all problems that are of interest for applications, the following marvelous thing happens: if we have been able to construct some asymptotic solution of the problem under consideration (where all the ‘pj are smooth and constructed uniquely), then for sufficiently small Ethere exists a unique classical or generalized solution of the problem of mathematical physics, and the formal asymptotic solution gives the asymptotics of this ordinary solution, that is, in its domain of definition the following inequality holds:
Here u is the ordinary solution and Fj + +co as j + DC).The proof of inequalities of the type (4) .1soft en a very difficult problem, which requires, as a rule, the application of a mathematical apparatus that is different from the one required for the construction of the ‘pj. This mathematical apparatus is based principally on the technique of a priori estimates of solutions of problems of mathematical physics.
Fundamental
Chapter 1 NonLocal ShortWave
Expansions
fjl. The Classical Ray Method 1.1. The Starting Formulas. The shortwave asymptotics of problems of mathematical physics is important by virtue of its applications,both inside of mathematics (for example, the problem of finding the asymptotics of the spectral function of elliptic differential operators) and in physics and engineering physics (for example: the construction of theoretical seismograms). Here we shall not concern ourselves with the numerous and important applications of shortwave expansions, but endeavor to give a brief survey of the main ideas of this area of mathematical physics. These ideas can be conveniently illustrated using the comparatively simple model example of the wave equation with variable velocity: 1
wtt(x)C2(X)
Aw = 0;
x= (xl,...
(1)
III. The HigherDimensional
WKB Method or Ray Method
95
Here the function C(X) is positive and infinitely differentiable in the domain in which it will be considered. The surfaces and functions that occur in what follows will be assumed to be infinitely differentiable. Instead of “infinitely differentiable function” (or surface) we shall generally write “smooth function” (or surface). ,In the present paper only solutions of equation (1) are considered that depend harmonically on time. More general variants of the dependence of w on time, which are important from both the practical and the theoretical point of view of spacetime ray expansions, their analogues and modifications (see Babich, Buldyrev and Molotkov (1985) and Whitham (1974)) will have to be omitted in order to keep this paper to a manageable size. We set w = u(x) exp(iwt). (2) The function
u satisfies
the Helmholtz
equation
with variable
coefficients:
(A+&)u=O,
(3)
The number w is a large parameter for the problem. The term “nonlocal” in the title of this chapter means that the range of applicability of the expansions obtained in this chapter does not depend on w. Of fundamental significance in what follows is the following ansatz, sometimes called a ray expunsion: u
=
y = const,
ewl) g
(Y$i+,)
(4) C4 j+‘=exp[q(j+y)], (where T and uj are real functions subject to determination). As a rule ray expansions diverge. Formula (4) simply means that the letter u denotes the formal series exp iwr C CiIj3+7. The ray expansion (4) is obviously a natural multidimensional analogue of the WKBexpansion [Fedoryuk (1983)]. Substituting expansion (4) in equation (3), assuming that 2~0# 0 and setting the coefficients of successive powers of w equal to zero, we arrive at the equations:
(W”=& 2VrVuj
+ujAr
= AIL~,
v= (& >...>& ) , u~ ~0; j = O,l,....
(5)
(6)
The desired function r is called an eikonal and equation (5) is called the equation. The recursive equations (6) are socalled transport equations.
eikonal
V. M. Babich
96
1.2. The Eikonal Equation. The eikonal equation (5) is integrated by a traditional method using the method of characteristics (see the classical treatments by Smirnov (1981) and Courant (1962)). Before we give a system of characteristic equations, we write equation (5) in the form: H 3 ;c’(x)
&I;
= a;
dr pj=dlcj.
j=l
The
characteristic
system dxj ds=$
of equations dH
dp, ds
for (7) has the canonical aH, dxi ’
dr ds
form:
l
In order to define a Cauchy problem with initial data on some m  ldimensional hypersurface C c R” it is necessary that the eikonal r on C coincide with a given function which is denoted by I/X rlc = $, and necessary that the sign of 2 be given (n is a continuously varying normal to C). In what follows C will simply be called a surface. For the (local) Cauchy problem to be solvable it suffices to require that any derivative of the initial function $ along the surface C be less than l/c in modulus. The method of characteristics in this problem can be presented using variational ideas. Indeed: it is easy to show that the characteristic system (8) is equivalent to the system of Euler equations for the functional (which is sometimes called the Fermat functional)
(9) if as a parameter on the extremals we take the value of the integral s” ds/c, counted from a fixed point on the corresponding extremal. These extremals, following a longestablished tradition, are called rays. This terminology is completely compatible with the physical concept of a ray. The first m equations of the system (8) can be written in the form
We have used the eikonal equation and the equalities dr/dxj = pj (see (7)). We return to the Cauchy problem. We outline the process for solving it. Knowing ~1~ = 11, and the sign of dr/dn and using the eikonal equation we can find all the derivatives dr/dxi, j = 1,2,. . . , m, and, hence, we can find the direction of the vector VT. From each point x0 E C a ray is emitted in the direction of VrlzO. Such a ray exists and there is only one of them. We now set
III. The HigherDimensional
WKB
Method
or Ray Method
97
z$ r(x)=$(x0) f J20 The integral is taken along a ray joining 20 and IC, with a “+” (respectively, a “” ) sign if LI: is located on the side of C to which the normal n is directed (respectively, if 2 is in the domain for which n is an exterior normal). Formula (1.1) also defines a solution of the Cauchy problem for the eikonal equation. It is not hard to see that this recipe for constructing a solution of the Cauchy problem is the same as the usual method of constructing this solution by the method of characteristics. The distinction is only terminological. 1.3. Rays and Ray Coordinates. A solution to the eikonal equation is easy to construct using rays, that is, extremals of the Fermat functional (9). These extremals are obviously geodesics of the Riemannian metric with square of the length element (ds2) equal to
2 (dx92 j=l c”(x). Thus, the theory of rays and along with it the theory of the eikonal equation can be considered as a special case of Riemannian geometry. Suppose the function q = C = const (see 31.2 above). The gradient of the eikonal T must be orthogonal to this level surface, and thus the rays from which T(X) is constructed must be orthogonal to C, that is, because of (10) VT and dzldH are only different by a positive multiplicative factor. Suppose the points on C are characterized by the parameters ol, cr2,. . . , am‘. We shall characterize the points on the ray passing through ZO((Y’, . , cP‘) E C in terms of the value of the integral f srz1 ds/c = 7 ( 5, ~0). The “+” sign is taken in case z is located on the side of C to which the normal n is directed and the “” sign otherwise. The coordinate system (cx’, CX~, . . . , CL?‘, r) is semigeodesic in the terminology of Riemannian geometry. It is regular close to C. In works on diffraction theory o?, 02, . . , o?’ , r are called ray coordinates. Suppose that theformulaszj =~j(c~~,c~~,...,cP~, T), j = 1,2,. . , m, or, in vector form, x = x(d,a2,. . . ) al, r) realize the transition from the coordinate system CY~,r to Cartesian coordinates. Fixing cy’, . . , Gl, we obtain the parametric equation of a ray, and, having fixed r, we obtain a level surface of the eikonal Another variant of ray coordinates is obtained by considering the central field of rays, that is, the set of rays issuing from a single point ~0. In order to define a ray it suffices to define a unit vector se directed along the ray at the point ~0. The end of the vector SO traces out the unit sphere [se/ = 1. Let al,. . . , aml be coordinates on the sphere. By defining ol,. . . , am’ we define se and thus a ray. The points on the ray will be characterized by the magnitude of the integral sZ: ds/c (th e integral is taken over a ray joining 20 and x). Thus a coordinate system ol, . , am‘, r is defined in a neighborhood
V. M. Babich
98
of the point 20, and this coordinate system is also called “ray coordinates”. Let si be the components of the unit vector se. We can also take r& = Xj as coordinates. The coordinates {Xj} are the classical Riemannian normal coordinates. The Riemannian normal coordinates will be essential in Section 2. The ray coordinates o?, . . . , cZ’’ , r give us the possibility of integrating the transport equations (6). 1.4. Integration of the Transport Equations. We shall assumethat a ray coordinate system ol,. . . , am‘, r has been introduced. We first turn to the first transport equation
2VueVr Multiplying
+ t&J& = 0.
(12)
this equation by ~0, we give it the divergence form div(z@r)
= 0.
(13)
We shall derive a formula that allows us to integrate equation (13). Suppose the vector A in the coordinate system o?, . . , cP‘, r has contravariant components 0, 0, . . . , z$/c” . In Cartesian coordinates its components will be 87 z&c” = 21;:
dXi
. dr
8x2
(we have used formula (10) here). Writing the expression for div(U$r) coordinate system (aj, r), we obtain the equality divA
= div(uiVr)
where
Here D(xj)/D(
(14)
= +c&
1 D(ri,...,xm) c D(d, . . . ) a1,
J=
in the
T) ’
(15)
oj, r) is the Jacobian. The quantity J is called the geometric
spreading.
In the caseof the plane m = 2 and three dimensions m = 3, J is respectively equal to J=
and
dz I da’
J=i&x&i,
I
where x denotes the vector product. Formula (14) also gives the possibility of solving equation (12) or the equivalent equation (13). It follows from (13) and (14) that ug 5 = const on each ray, from which we find uo = ?fQo(&
. . . ) a”‘)~.
(16)
The modulus symbol has been omitted on the lefthand side by assuming that ue does not change sign. The function $0 depends only on the ray. The
III. The HigherDimensional
WKB
Method
or Ray Method
99
equation for the uj (see (6)) is easy to integrate by the classical method of variation of constants. Setting uj = ZL~U~, where II~ is a new unknown function, which is easy to find, using formula (lo),
from which Uj
we find =I@
$j(C2,...,CEm1 ) +
~~(al,,,,,aml,
,?AuildT] 0
[
.
(17)
The
integral is taken along the ray. The functions $i, usually called the diffraction coefficients, and the function T&G,. . ) aml) remain undetermined for now. In order to find them we need additional information about the desired expansion. 1.5. Discussion of the Our Results. We return to the ray expansion (4). The expression exp(iwt + iw7) is called the phase factor. It remains constant on the moving surfaces S: t  T(X) = C = const, called wave fronts. We define the velocity u of the moving surface 5’. Suppose that a point x lies on S at time t. We construct a normal to 5’ at x and fix it. The velocity ~(5, t) of the moving surface S is the velocity of the point of intersection of S with this normal at time t. It is not hard to see that in our case v(x, t) = C(X)UT/lUTl
= c(x)so,
(18)
where so is a unit vector directed along the ray at the point 20. We return to the first approximation of the ray method uo exp(iwt + iw~(z)). Using the approximation uo (x) X5 uo(xo), close to 5, we find up to principal uo exp(iwt
+ LX(X))
4x1
= 7(x0)
terms
+ P~)z,(X
 x0)
that
z exp(iwt+iW(VT),O(x
 x0))
~uoexpiw~(xo).
Thus, in a neighborhood of a given point the first approximation of the ray method represents a plane wave, which is an exact solution of the wave equation Au  utt/c2(z0) = 0. The ray approximation can therefore be considered as a perturbation of a plane wave. In physics literature the vector wVr is denoted by k and is called the wave vector. The eikonal equation can be written in the form w = w(k), where w(k) = clkl. Physicists usually call the equation that expresses w as a function of k a “dispersion equation”. The equation w = cl ICI,equivalent to the eikonal equation, is the simplest example of a dispersion equation. Formula (13)) which gives the possibility of integrating the transport equation, admits an interesting energetic interpretation (see the book by Babich and Buldyrev (1972)), but we shall not go into this.
V. M. Babich
100
1.6. Reflection and expansion of the form U
inc
Refraction
of Ray
=
) .z
exp(iw?
Solutions.
Let uinc be a ray
?A~“,(iwy
(19)
defined on one side of the surface C (see Fig. 1 below) and suppose that the normal n to C is directed to the side “illuminated” by the incident wave. We shall assume that drinc/dnlc < 0. The problem is to find a “reflected” ray expansion uref : u ref = exp(iwrref)
2 uTf/(iw)j, j=O
such that drref/dnl c > 0 and such that formally relation is equivalent to the series of equalities
(20)
uinc + uref 1c = 0. The
last
The condition drref/dnlc > 0, equality (21) and equation (5) determine rref. We can regard relation (22) as Cauchy data for the transport equations (6), in which we must substitute rref (respectively, uTf) for r (respectively, for uj). Thus, uref is uniquely determined from the incident ray solution uinc. The case of the boundary condition
&(uinc + Pf)lC = 0 is considered
analogously.
Fig. 1 We consider an example. Let R c R” be a bounded domain, bounded by a surface C. Suppose C has positive curvature everywhere. This means that the Gauss second fundamental form of the surface C is positive definite at every point. We consider the case C(X) EE 1. Assume that the following plane wave is incident on C: m U
inc
= exp(i&“‘),
7inc = xjpj;
C$ j=l
= 1.
III. The HigherDimensional
WKB Method or Ray Method
101
(A plane wave is obviously a special case of a ray expansion.) The problem of finding the reflected wave, satisfying the Helmholtz equation (3), for c = 1, the radiation conditions, and whose sum with the incident wave vanishes on E is a classical wellposed problem. The question arises of how to find the shortwave asymptotics of the reflected wave as w + 0~). The ray method gives the possibility of finding this asymptotics in the domain illuminated by the incident wave (see Fig. 2; the rays in the case C(X) = 1 are straight lines), following the scheme outlined above. One can show that the ray expansion of the reflected wave ceases to be an asymptotic expansion in approaching the shadow zone from the illuminated domain (in the socalled penumbral zone). The shortwave asymptotics in the shadow and penumbral zones have a considerably more complicated form than the wave expansion.
Rays of the incident wave
Rays of the reflected wave
Fig. 2 The natural question arises of what the formal constructions outlined here are related to. Indeed, in spite of the naturality of our constructions from the physical point of view and their relative simplicity, it does not at all follow from them that our ray series exp(iwVef) . Cc, uTf/(iw)j actually gives the asymptotics of the reflected wave. In the case when the reflected wave can be constructed explicitly (a ball, an ellipsoid), the proof of the fact that the ray expansion gives the asymptotics of the reflected wave can be obtained by analyzing the explicit solution. However this approach is not simple even in the case of a ball, and in the case of an ellipsoid the corresponding computations are extremely cumbersome and the resulting technical difficulties have not been overcome by anyone. In the general case the justification of the ray expansion requires the consideration of formal shortwave asymptotics of the reflected wave in the whole domain Rm\Q and successive estimates of the remainder term. Problems of this kind were posed rather recently and were studied by Ursell, Babich, Ludwig, and Morawetz. The most complete results, from which, in particular, the justification of the expansion for the
V. M. Babich
102
reflected wave constructed here follows, were obtained by Buslaev [Buslaev (1975)]. References to earlier works are contained in the book by Babich and Buldyrev (1972) and the article by Buslaev (1975). The problem of the rigorous justification of asymptotics in the deep shadow zone (see Section 1 of Chapter 4 in this connection) was solved rather recently [Zayaev and Filippov (1984)] (in the plane case) using very delicate analytic considerations. We briefly discuss the question of refraction. Suppose now that the surface C separates two domains L’i and 02 in which the velocities cl in L’r and c2 in K&J are generally different. The incident wave will be defined by its ray expansion in the domain fir, close to C, and it is assumed that drinc/dnlc < 0. (It is assumed that the normal n is directed to the interior of L’i.) Suppose that the classical boundary conditions
7J1qc = u(2)Ic,
&(l)/dn[,
= dt~(~)/thl~
are defined on C. (Here u(~)]~ and &A(~)/&]~ are the limit values of the corresponding functions, starting from the values in &.) The reflected and refracted waves are sought in the form of ray expansions, where the corresponding eikonals satisfy the equalities
and the inequalities d7‘ef/lh(c
> 0,
d7refr/th(c
< 0.
If the tangential derivatives drinc/dZ of ?nclC satisfy the inequality of the reflected and refracted waves pT’““/dzl < l/ ~2, then the construction is carried out just like the construction of the reflected wave outlined above. If ]dGC/dl] > 1/ c2 f or certain tangential directions, then it is necessary to introduce a complex eikonal (see Chapter 2).
$2. Point
Source
of Vibrations
in an Inhomogeneous
Medium
2.1. Ansatz. The ray expansion loses its asymptotic nature in a neighborhood of the point set where the geometric spreading vanishes: the principal term of the ray expansion is already infinite for J = 0. In neighborhoods of the points where J = 0, another form of asymptotic expansion, different from (4), is necessary, i.e., another ansatz. In this section we consider the case of a point source of vibrations, when the equation describing the wave process has the form:
(A+&)
?A= S(z  2lj),
(23)
III. The HigherDimensional
WKB
Method
or Ray Method
103
with the Dirac hfunction on the righthand side. Equation (23) does not yet determine u completely. We construct a formal asymptotic solution of equation (23) in some neighborhood of the origin. The partial sums of this solution tend exponentially to zero as w + +co if z # 20 and argw is a fixed number E (0,~). In this sense our solution satisfies the limit absorption principle. It is natural to expect that outside of a small neighborhood of ~0 the asymptotic solution of equation (23) in which we are interested will become a ray solution, corresponding to a central field of rays centered at ~0. It is not hard to prove that the geometric spreading J of a central field is different from zero at points x that are close to 20 (Z # ~0). The shape of the desired asymptotic solution (ansatz) is not simple. The question of how to conjecture what form it should take will be discussed briefly afterwards. This ansatz is as follows:
where
the functions
f, are expressed
by the formula:
fp(w, 7) = &+J
27 p Hp(w7),
2
where
7 = ~(20, X) is the eikonal,
(
equal
ray joining the points 20 and x, Hil) smooth functions of their arguments.
W
>
to the integral .IS the Hankel
szO ds/c, function,
taken
over a
and the ~1 are
2.2. Transport Equations. An expansion of the form (24) for T 5 const is asymptotic not only with respect to the small parameter l/w but also with respect to the degree of smoothness in the following sense: the functions f, lose smoothness for T(X,XO) = 0 but the number of continuous derivatives of fp increases without bound as p + co. The most singular term in the expansion (24) is the zeroth term vof~. It is natural to require that the bfunction in the righthand side of equation (23) should actually occur because of this term. For the application of the operator A + $ to vofq to lead to a singularity whose principal term is equal to S(x  x0) it is necessary that $lnr ZlofE+
for m = 2

r; ( 2(m

>
2)7rm/2rm2
as 5 4 ~0
(26)
for m > 2
since applying the mdimensional Laplace operator to the righthand side of the relations (26)) we obtain 6(x  x0), and the term W~TJO~,Z/C~ is less singular. Taking the behavior of the Hankel functions for small values of the
104 argument obtain
V. M. Babich and the wellknown
relation
‘uo(T 20)/,,,” Now we substitute
the expansion
af,(wlr) rdr
w2.fp(w,
T)
=
~(2  20)  T/C(Q)
into
account,
we
11
2(C(Q))m2m
(24) into equation
(27)
ml (23).
The
=
?,fpl(U,
=
4(1  p)fpl(w, 7)  4t2fp2(w, ~1
relations
71,
once again give the possibility of representing this expansion as a series in the functions fP(w, r) with coefficients that are independent of w. Setting the coefficients of the function fP equal to zero, we obtain a recursive system of relations
The differentiation For vc we will find
IJO= For the ‘~1 we will
a/&
is taken
2
along
ml fw (c(xo))m227rz [I obtain
a ray leaving
T Ar2  2m/c2 47 0
from
the point
20.
d7
I.
the formulas
Here $~l+r = const on each ray. These are the diffraction coefficients. The “recipe” for choosing them is very simple. We set $l+r = 0. This ensures the smoothness of ‘u~+i(~, Q), I = 0, 1, . . .. The proof of the smoothness of ZQ and even its analyticity (this in the case of an analytic C(X)) is carried out by introducing Riemannian normal coordinates in a neighborhood of ~0 analogously to how the analyticity of the coefficients in the expansion of an elementary solution (see Hadamard (1932)) is proved.
2.3. Discussion of Our Results. In the considerations of the previous subsection there is a striking analogy between the recurrence relations for the ~11and the recurrence relations for the coefficients in the Hadamard expansion of an elementary solution (see Hadamard (1932)). This analogy is not random. It is explained by the fact that expansion (24) is obtained if we take the Fourier transform of the series giving the Hadamard elementary solution’
’ Hadamard considered expansions analogous to (28), only in the case of an even number of space variables (in this connection see Courant (1962)).
III. The HigherDimensional
WKB
Method
or Ray Method
105
(28) The
elementary
solution
satisfies
the conditions
1w,, c”(x)
 nw = S(x xo)b(t), Wlt<0= 0.
A formal application of the Fourier transform to the series (28) also leads to the functions f, (see Babich (1965)). Both the Hadamard expansion (28) and the expansion (24) are unique: similar expansions have not been written down for several general equations with variable coefficients (for example, for the dynamic equations of elasticity theory in the case of an inhomogeneous medium). It. is necessary to use an indirect procedure, connected with the Radon transform or socalled Fourier int,egral operators. The expansion (24) possesses a nonlocality property: it holds2 in a domain that does not depend on w. Nonlocal expansions of this kind are more the exception to the rule than the norm. The ray expansion (see Section 1) and expansions in a neighborhood of a caustic, treated in the following section, are also nonlocal.
$3. ShortWave Expansion in a Neighborhood of a Nonsingular Piece of a Caustic 3.1. Basic Assumptions. The ray expansion (4) ceases to be valid in neighborhoods of the points where the geometric spreading J (see (15)) vanishes. The set of points where J = 0 is called a caustic. We consider here the most regular case when the caustic is a smooth surface. We shall denote it by the letter S. We shall also make other, additional assumptions that indicate the nondegeneracy of the situation. Suppose we have introduced ray coordinates (a’, cy2, . . . , c.rrn‘: r) (see Section 1) and let al, a2,. . . , amr, r(~l,. . , Ql) be a point on the ray where J = 0 (that is, a common point of the ray and S). Then r = ~(a’, . , cP‘, 7(d,...,Cyml )) (where z = x(0,7) is t,he formula realizing the transition from ray coordinates to Cartesian coordinates) is a parametric equation for the caustic. Our first assumption is that ~(a’, 02,. . , cPP1) is a smooth functionand (c$,c~~,...,cP~) are regular coordinates on the caustic S, that is, the vectors dr/dai = dx/daj + dx/dr . dr/daj, j = 1, . . . , m  1, are linearly independent. We shall now prove the following lemma: Lemma.
The rays are tangent
to S
’ Instead of writing “the series is the asymptotic wording “the expansion holds.”
expansion”
we shall use the shorter
V. M. Babich
106
To prove this it suffices to establish that the tangent vector to the ray 8x/& is expressed as a linear combination of the dr/da~ , for which it suffices to prove the linear dependence of the vectors &/da and dx/daJ (since the dr/daJ are linearly independent). We shall first prove that the vectors dx(ai’, . . . , cP‘, r)/&rj are linearly dependent. In fact: the equality J = 0 is equivalent to the linear dependence of 8x/&j and 8x/&, that is, the existence of numbers dj such that m1
c
djdx/@
+ d,ax/th
= 0,
j=l
c
,dj, > 0.
(29)
j=l
Taking the scalar product of equality (29) with dx/dr and using the fact that axjaolj I ar, axjar f 0,we find that d, = 0 and thus m1
m1
C djax/aaj = 0, C ldjj > 0,
(30)
j=l
j=l
that is, the ax/da3 are linearly dependent. We shall now prove the linear dependence of &/a& dependence is a consequence of the obvious equality
and ax/dr. This linear
m1
C
dZiarlaa'
(31)
+
j=l
where the dj , j = 1, . . , m  1, are the sameas in (30). The following (and last) nondegeneracy assumption is that the tangency of the ray and the caustic S is of first order. 3.2. The Analytic Character of an Eikonal Close to a Caustic. It is natural to expect that close to a caustic S which is the envelope of a field of rays, the eikonal r will no longer be uniquely defined. (Two rays intersect at the point x. On each ray the value of the eikonal r = s ds/c will be distinct. See Figure 3.)
Fig. 3 The ray coordinates cease to be regular. We shall prove that in a neighborhood of a caustic there exist smooth functions r(x) and p(x) of x such that
III. The HigherDimensional
WKB
Method
or Ray Method
107
(32) (p is positive in the domain illuminated by the rays). To prove this we note that it follows from (31) that C,“<’ dj&/daJ # 0, so that at least one of the derivatives dr/daj is different from zero and we can take the value of r on S as one of the coordinates on S. We denote this coordinate by cyi. Thus, in ray coordinates the equation of the caustic has the form 7 = a?. We now introduce coordinates crl, 02,. . . , crm‘, n close to S, where n is the distance along the normal from S, which is taken with a + sign (respectively, with a  sign) in the domain illuminated by the rays (respectively, in the unilluminated domain). We write the eikonal equation (7) in the coordinate system o?, . . , aml, n. The canonical system of equations that the rays . satisfy has the form: dn =c2pn,..., dr
$f=g(~~(V~)~i)...,
(we will not write down The geometric meaning
where
(p,g)
(33)
the other equations). On the caustic we have p, = 0. of n and the equations (33) give us:
where (35) (where l/p is the curvature of a normal section of S along a ray). of a ray and the caustic is Inequality (35) is valid because the tangency of first order and, moreover, n > 0 in the illuminated domain. The quantity P is called the effective radius of curvature [Babich and Buldyrev (1972)]. Inequality (35) is equivalent to the positivity of the curvature of a normal section of S along a ray if the curvature is understood as the curvature of a surface in a space with the Riemannian metric E(dxj)2/c). It follows from (33))(35) that n = (r  a1)2A, where A is a function that is positive close to S. Taking the square roots of both sides of this last equality, we will find q = (T  c+ll, Formula theorem
n,=a>o,
7)=ffi.
(36)
(36), the inequality &((T  a’)Ai) > 0, and the implicit imply that we can write down the following representation r=F(c2,...,
(F is a smooth
function
dl,?&
of all its arguments),
7/=*&i, from
which
we get
function for r:
108
V. M. Babich F(al,
7=
. . . ) al,
+ F(d,
r)) + F(c2,.
. . ) aml,
q)
2  F(d,. . . ,cPl, 7) 2 nz1,q2) + ?7F&21,. . . ,am1,r/2),
. . . ,cPl,~)
=F&2,...,a
(37) 77 =
&/ii.
Because dr/dn 1s = 0 and because of formula (37) we obviously have: dFlldn(n=o = 0 and F21nzo = 0, so that F2 = nF3, where F3 is a smooth function of ol, . . , o?l, n that is positive for n = 0. The last of these is proved using the eikonal equation. Recalling that the coordinate system a’, . . , am‘, n is regular for small n, and passing to Cartesian coordinates, we obtain the desired formula (32), starting from (37), where
< = F~(d(cr~),n(x)), Using
the eikonal
equation,
+ip3/2
= Gzn3/2F3(aj(z),n(x)).
it is not hard
to show that
(W2 + P.(W2 = f > VEVP = 0,
(38)
where
(39) 3.3. Asymptotics of a Solution. We can look for an asymptotic solution of equation (3), describing a wave field in a neighborhood of a caustic, in the form u = Av(W2/3p) + i~l/“Bv’(~~/~ p)] eiw~w6. (40) Here b = const,
A = 2 Aj(z)(Gj, j=o
B = F B,(z)(hj, j=o
(41)
u(c) is the Airy function, satisfying Airy’s equation ‘u”cc21 = 0 and having the asymptotics V(C) N $<1/4 exp(  $c3j2) as C + co (see Babich and Buldyrev R e pl acing the function in formula (40) (1972, Appendix) and Fock (1970)). by its asymptotic behavior for ,LL> 0, w + oo, we obtain instead of (40) the sum of two ray expansions u 
lw6l/2eai/4
 BOp1’4) +  UT (iw)
(AOpd4
2
{[ l/4
+
J30p1/4)
+
&
The first (respectively, the second) ray expansion with the eikonal r
+
. . .]
+ .
1
. &A3?)]
&,(EtibW]}.
of these expansions corresponds = <  $p3i2 (respectively, with
to the r+ =
III. The HigherDimensional J+$p3/‘). The gradient Vr(respectively, away from the whose gradients are directed +cc the Airy function tends shadow. Substituting the ansatz in
L,A, where
Llf
WKB
Method
or Ray Method
109
(respectively, VT+) is directed toward the caustic caustic). Every point corresponds to two eikonals along the rays (see Fig. 3). For p < 0, w + exponentially to zero; this is the zone of caustic equation
(3), we arrive
+ L3Bj = AAjl,
at the recursive
equations:
CIA, + L2Bj = ABj_,,
(42)
A1 = 0, B1 = 0, j = O,l,. . ., = 2VpVf
+ fAp,
Cd = 2VEV.f + fat,
Lsf = pLlf
+ f(Vp)‘.
(43) The coefficients Aj and Bj can be found by integrating over the rays. We set j = 0 in the formulas (42). If we multiply the first of the equations of (42) by the results, we arrive at two P 114, the second by p1/4 and add and substract equations analogous to the transport equations of the ray method: 2VrV@;
+ @iAr
= 0,
2V+V@of
+ @,+Ar+ = 0,
(44)
where r+ Using
formula
= 5 f p,
@t = Aop1’4’f
Bop1’4.
(45)
(16) we obtain (46)
from
which
In order
we get
to have Bo bounded
it suffices to require
that
In order to find the diffraction coefficient $0 it suffices to define the ray expansion of a wave going to the caustic, that is, a wave corresponding to the eikonal c . The coefficients Aj and Bj lie on the same path as A0 and Bo. If a wave going to the caustic is defined, then Aj andBj are uniquely determined. It is possible to show that Aj and Bj, j > 0, are smooth. However, the mathematical apparatus described in this section does not determine either t.he
V. M. Babich
110
functions [, domain not expansions the caustic.
,LAor the coefficients Aj and Bj in the shadow illuminated by rays (see Fig. 3). Here we have of <, 7, Aj and Bj in asymptotic power series n plays the role of a small parameter in these
zone, that is, in the to be satisfied with in the distance n to expansions.
Chapter 2 Some Modifications of the Ray and Caustic Expansions 51. Asymptotics
of Vibrations
of Whispering
Gallery
Type
1.1. The Ray Method in the Small. The theory of whispering gallery waves is closely connected with the theory of caustics. We consider the following ray picture (Fig. 4): close to the reflecting surface S there is a caustic, and the family of rays after grazing the caustic, reaching S and being reflected, is transformed to itself. If we were able to associate a wave field to this family, then there would exist a “surface wave”, that is, a vibration that is essentially different from zero only close to the surface. Such vibrations are called vibrations of whispering gallery type. If we could construct a continuous family of caustics that coincide in the limit with the reflecting surface S and possess the above properties, then we would have to consider a whole set of whispering gallery waves. The existence of a family of caustics such as described above is a rare exception. However, such families can be constructed approximately, in the form of formal power series [Babich and Buldyrev (1972), Ludwig (1966)]. Following Buldyrev (see Babich and Buldyrev (1972)) we shall call the technique of constructing such approximate families of rays the ray method in the small. For simplicity we restruct ourselves to the plane case.
Fig. 4 We shall look for a pair of functions equations (see Chapter I, $3) (W2 and the boundary
+ PL(W2
5 and p that
= 1/c2,
vL$vp
satisfy = 0
the system
of
(1)
condition
< and p are assumed
to be power
I*ls = 7; series in y:
(2)
III. The HigherDimensional
WKB
Method
or Ray Method
111
In a neighborhood of 5’ we introduce coordinates s and n, where s is arc length, measured from a fixed point on S, and n is distance along a normal from 5’. The coefficients & and ,uj are in turn assumed to be power series in n: cc co
(4) The velocity c 
Substituting
c has a Maclaurin co + cl72 +
c2n2 + . . , where
cj = @c/i%
expansion I,=,(j!)‘.
(5)
(3) in (1) we find (vEo)2
2VJow1+Pl(VPo)2
series as its asymptotic
+ Po(vpo)2
+2povpovp1
= 1/c2, =
0,
v
= 0,
V
+
E &g
+
VEOVPl
(6) = 2
Granting
that
~~01~ = 0 (see (2))
and
where p = p(s) is the radius of curvature (6) (assuming that ~10 # 0) that
VaVb
of 5’ at s, we will
(
1+ %
> easily find
0.
(7)
p$b s s, from
(8)
Assuming for definiteness that dfo/ds is positive, we will find <ec up to a constant term that is left undetermined, the diffraction coefficient. For the partial sum of our expansions appearing as an argument in the Airy function to tend to +co for fixed n > 0 and w + +oo, it is necessary that the effective radius of curvature P = (l/pcl /CO)1 be positive (a concentration condition for a whispering gallery wave close to S. For P < 0 it is not possible to construct whispering gallery waves). From the formulas in (7) and the condition ~01 = 1 (see (2)) we will find dEo1 ds Thus <ei has been found up to a constant term, the diffraction coefficient. The construction of the remaining functions & proceeds analogously. All the coefficients pij and &, i > 0, are found uniquely, and the <ej are found up to constants of integration. After assuming that the series (3) and (4) converge, we obtain a continuous family of caustics p(n, s, y) = 0. Condition (2), the formulas rh = < f $p3j2
112
V. M. Babich
and the inequalities d~/dnl~=e > 0 and d 0 ensure that the angles of incidence and reflection are equal for waves going to the caustic and leaving it. However, it should be noted that the series (3) and (4) diverge as a rule. In particular, it is possible to prove that if there exists a continuous family of caustics close to S and S is a closed curve, then this family can be destroyed by an arbitrarily small analytic perturbation of the contour S (due to D. Sh. Mogilevskij). 1.2. Boundary Conditions and Transport Equations. An asymptotic solution called a wave of whispering gallery type has the form of Equation (I.40)3, where < and p were found in 1.1 above, and Aj and Bj are in turn the expansions:
Aj = 9
Bj = 2
Akjyk,
Bkjyk,
(10)
Bkj = c Blkj(s)nz. l=O
(11)
k=O f&j
=
E
k=O Al,&)&
l=O
We shall consider the case of the Dirichlet boundary condition uIs = 0. Substituting an ansatz of the shape of Equation (1.40) in the boundary condition u 1s = 0 and considering condition (2)) we find
Alsu(~2’3y) For
this
+ i~~‘~Bl~z~‘(~~‘~y)
= 0.
(12)
equality to hold for sufficiently small y, it is necessary that = 0. It is impossible to set Al, = 0, since it can be shown leads to a trivial (zero) asymptotic solution. Hence, it is necessary
Alsv(~2/3y) that this to set
y = l&2/3 (where cl, In all the rameter and (13) formula
(13)
1 = 1,2,. . ., are the roots of the Airy function (zi(cl) = 0)). remaining constructions we shall treat y as an independent paonly in the final formulas replace y by expression (13). In view of (12) leads to the equality BI, = 0, or, equivalently (see Equation
(I.411 1
BjInzO=O, Using
Equations
(1.42),
(1.43)
2%
and (lo)(14)
.+ 2~10A100
3 Translator’s note: The notation X. Thus (1.40) means Equation
j=O,1,2
,.... above,
(14) we find the equalities
+ AoooA~o~,=o = 0,
(15)
+ AoooA~oI~=~
(16)
= 0.
(X.y) here and later means Equation 40 of Chapter I.
y of Chapter
III. The HigherDimensional
WKB
Method
or Ray Method
113
By analogy with Equation (1.6), it is natural to call (15) a transport equation. This is an ordinary differential equation along S. Computations show that a solution of it has the form
Aooo = Coo (39
l/6
> co = clnzo
(17)
Here P is the effective radius of curvature of 5’ and COO = const is the diffraction coefficient. We find Aloo from Equation (16). Differentiating Equations (1.42) and (1.43) with respect to n. and setting n = 0, we find Biec (obviously Bcee = 0) and A 2oo. Continuing this process, we can find all the coefficients Aloo and Bloc, that is, the expansions for A00 and Boo. Then we find in sucfinding cession &IO, &IO, ADO, BZZO (I = (41, . ..I and thus A0 and Bo. After A0 and Bo, we proceed to find Al, B1, etc.
1.3. Analogues and Generalizations. The case of a Neumann boundary condition is considered completely analogously. Here; though, there is a complication: in the expansions (10) the summation must begin with j. Because of formula (1.41) and the negativity of
=
wq
+
alwq
1’3
+
o(w;1’3),
(18)
where
In the case of a Dirichlet (respectively, Neumann) boundary condition <, is a root of the Airy function, u(&p) = 0 (respectively, of the derivative of the Airy function, ~‘(6) = 0). The parameters p and q are integers > 0 and q + 00, p = O(1). The last condition can be weakened. Formula (19) (more precisely, an analogue of it) gives the possibility of estimating the width of lacunas in the spectrum of the Laplace operator [Lazutkin (1979)] from above. Passing from the twodimensional problem to the case of arbitrary dimension requires the construction of a ray field on S that corresponds to the velocity c, that is, the construction of a field of extremals of the functional s ds/c, where the level curves lie on S. There is significant interest in considering the ansatz (1.40) in the case of a negative effective radius of curvature. In this case it is appropriate to set
114
V. M. Babich u = (Awl(
w~/~~U)
+ i~“~Bzui
( w2i3p))
exp iw<,
(20)
where WI(<) is a solution of Airy’s equation, having the asymptotics  (<)1/4eiR/4exp $(<)3/2i as < + co. As before A = xA,(iw)j; j=o Ai = x
AI&“,
B = c B+iw)j, j=o
(21)
Bj = c
(22)
k=O
Bkjyk,
k=O
(23) j=O
j=O
Here, however, Aki, Bkj,
Fig. 5
52. Surface
Wave
Propagated
Along
an Impedance
Surface
2.1. Initial Statements. Ansatz. Suppose that Equation (1.3) holds close to some surface S on which the socalled impedance boundary condition
g+whu=O
(24)
is given. Here d/dn is the normal derivative directed to the side where the vibrations are considered and h is a positive function on S. Here we shall consider an asymptotic solution of Equation (1.3), concentrated close to the
III. The HigherDimensional
WKB Method or Ray Method
115
surface: its partial sums tend exponentially to zero as w + co outside of strips close to 5’ of width of order greater than l/w. The previously constructed expansions are completely analogous to the wellknown Rayleigh waves, propagated along the surface of an inhomogeneous elastic body of arbitrary shape, to Stoneley waves, and to some types of surface waves in acoustics and electrodynamics. The corresponding ansatz is formally the same as the ray expansion (1.4) and r and uj satisfy the same equations (1.5) and (1.6). The main difference from Chapter I is the assumption that 7 and ZL~are formal power series in the distance to S. Let ol, o?, . . , aml be an arbitrary regular coordinate system on S and n the distance from S; then (Y’, 02,. . . , o?‘, n define a regular coordinate system close to S. Let n be the second small parameter for this problem and let (25)
Uj =~Ulj(a’,...,am‘)n’.
(26)
z=o For the expressions for wr and Cuj/(iw)j to be asymptotic expansions it is sufficient that n = O(wc), where Eis any positive fixed number. 2.2.
Construction
of the
Asymptotics
of a Solution.
The boundary
condition leads to the following equations (27)
gI,=o = 0, where U =
C,“=, uj(iw)j+T
= C,“=, zlej (iw)J+Y.
(28)
For convenience we write the eikonal equation (1.5) and the transport equation (1.6) in the coordinates a’, . . . , cP‘, n. The square of the length element in these coordinates has the form Gklda’dcx’;
~2” = n, G,,
= 1, Gkrn = 0 for k < m, (29)
Gkl = gkl  hbkl + o(TL”).
Here gkl (respectively, bkl) are the coefficients of the first (respectively, the second) fundamental form of the surface S. The eikonal equation (1.5) and transport equation (1.6) then look like: Gkl
a?da”
2Gk1$g
+ s&
(v??GklA)
a7 dd
_
1 c2’
(G”‘) = (Gkl)‘,
= Au~~.
Setting n = 0 in (30) and using (27), we find the equation
(3’4 (31)
116
V. M. Babich /.“l al dUj g =&F=z+h2.
1
co = ClhrO’
(P)
= (9kV.
(32)
Equation (32) is the eikonal equation on the surface 5’. It is integrated using the rays, that is, the extremals of the integral s da/w (where da is the differential of an arc of a curve on S, the level curves lie on 5’ and ZI is the velocity of surface waves on S: l/u 2 = l/c: + h2), like the classical eikonal equation. Finding 70, differentiating (30) successively with respect to n and setting n = 0, we find the rj, j > 0. We turn to the transport equations. Here it is convenient to carry out the considerations using ray coordinates ~e,fl’, . . . , ,SmP2 on S corresponding to the eikonal re(cr?, . . . , cP’ ). Integrating Equation (1.6) leads in this case to successively finding the coefficients ulj in the expansion (26). Recursive equations are obtained for these coefficients if we successively differentiate relation (31). These equations are integrated in the same way as the transport equations of Section 1 of Chapter I. We shall give only the expression for uoo:
uoo=lLoo(P1,... , ,Fe2) x exp
fiexp
lrn
hiKv2dT
7o 1 T u2c1 ~ dr . exp dr. s o 2ihR s o 2icih
Here K is the mean cxurvature of S, l/R is the curvature of a normal section of S along a surface ray, cl = dc/dn~,=O, J is the geometric spreading of surface rays: J = ; 1Do”;;;
‘;.$;l!,
/.
It is interesting to note that ~00 is rebresented as a product of several factors, each of which characterizes the influence of some one factor on the principal term of the surface ray: $00 characterizes its initial form, v’% is the influence of the impedance, J 1/2 is the geometric spreading, etc. In the construction of the theory of Rayleigh and Stoneley waves there is an analogous decomposition of the principal term into factors.
Gaussian $1. Solutions
Chapter 3 Beams and Their
Concentrated
in a Neighborhood
1.1. Statement of the Problem. of the ray expansion eiwr(z)
Applications
2 j=o
Ansatz. uj(s)/(iw)j
of a Fixed
An interesting
Ray
modification
(1)
III. The HigherDimensional
WKB
Method
or Ray Method
117
if the eikonal is assumed to be complex, with Im r 2 0, and equality occurs only on some curve 2. One can prove that 1 must be a ray. We assume this from the start. The partial sums of (1) will then be exponentially small everywhere except for a small neighborhood of 1. The complex eikonal and the corresponding coefficients uj can be constructed as expansions, which we shall now describe. We introduce special coordinates [Babich and Buldyrev (1972)] in a neighborhood of 1. We shall describe them in the threedimensional case. Suppose that arclength, measured from a fixed point, characterizes the points on 1. Let n(s) and b(s) be the principal normal and binormal of 1 at a point s. We introduce basis vectors ei and e2 by the relations
is obtained
el = ncos6  bsin8,
e2 = nsin8 + bcos0.
Here 8 is the angle between e and n, satisfying the condition dO/ds = l/T, where T = T(s) is the radius of curvature. One can show that ej(s) is obtained from ej(sc) (where se is a fixed value of s) up to a positive multiple via parallel transport of ej along I, corresponding to the connection defined by the metric $((dx1)2 + (dx2)2 + (dx3)2). The points close to 1 will be characterized by the coordinates s, $, q2, whose relationship with the Cartesian coordinates xi, i = 1,2,3, is given by the formula n
T = TO(S)+ 2
qiei.
(2)
i=l
Here T is the radiusvector of the points x1, x2, x3 and T = Q(S) is the vector equation of the ray I. All the succeeding constructions will be done in the coordinates s, ql, q2. (We note that we could use the Fermat coordinates defined by the Riemannian metric $ C(dzi)2 with equal success,even in the case of an arbitrary number of variables.) We proceed to a description of the ansatz. We assume that the desired expansion is given by formula (l), where I and IL? are in turn given by the expansions: &4 )
T=
dh) =
i:
h=O
il,i2,...,ih=l
h=O
il,i2,...,ih=l
uj =
Til..&(s)qil
Ujili2...ih
(S)qi’
. . . qy
. . . qih.
(3)
(4
The coefficients of the polynomials do not change when ii,. . . , ih are permuted. It is assumedthat r(‘)(s), 71(s) and 7i(s) are real, and that
V. M. Babich
118
is a ositive definite quadratic form. The last requirements ensure that Im xF!c 7ch) > 0 for any fixed M and sufficiently small ql, q2, not simultaneously equal to zero. An asymptotic solution of equation (1.3) corresponding to the ansatz (l)(4) is called a Gaussian beam. 1.2. Determination of T(O), 7(l), and T(‘). The square of the length element and the eikonal in the coordinates s, ql, q2 have the respective forms [l  K(s)(ql cos0 + q2 sin0)]2ds2 + (c&l)’ + (&2)2, 1 (a>“+ [l  K(s)(ql cos0 + q2sint9)12
($)2+
(&)‘=
(5)
$7
(6)
where K(s) is the curvature of I at s. Substituting (4) in (6) an d considering the positive definiteness of Imr(2) and the equation of 1 in the coordinate system s, qi, q2, it is not hard to find
s s dslco(s),
4’) = const +
co = ClqzO’ T(l) E 0,
so
2 dd2) +(VqT co as
c2q2 + t (!Pq, q) = 0, 4 2 a&) co as
Here c77(dv7(2) = I;=,
+ 2(V#,
W2))
= ni.
4 = (4J’A2),
(7)
(8)
(9)
G$$
The function Ai is defined if r(i’) are defined for j’ < j. We shall look for a solution of equation (8) in the form
where r equation setting r have the
= (rzj) is a (2 x 2) symmetric matrix. We then find a matrix Riccati for r. We look for a solution, following a wellknown method, by = P&l, where P and Q are new desired matrices. Thus, ~(~1 will expression .c2) = ;(PQ‘q,
q).
(11)
The matrix r will satisfy its Riccati equation, and ~(~1will satisfy equation (8), if P and Q are solutions of the system of equations: dQ
cop, ds
dP = ds
(12)
III. The HigherDimensional
WKB
Method
or Ray Method
119
If 4 and p are vectors generating the jth column (j = 1,2) of the matrices Q and P, respectively, then the system of equations (12) is equivalent to the canonical system of equations dqi =ds
dHz
=
(13)
b’pi ’
It is not hard to seethat the system of equations (13) is also a characteristic system of equations for the HamiltonJacobi equation (8), and simultaneously a system of equations in the first approximation for rays close to I, that is, a system of Jacobi equations for the ray 1. We also require that the matrices P and Q satisfy the relations: QTP
 PTQ
= 0,
Q*P
 P*Q
= iI,
(14)
where I is the identity matrix, T denotes matrix transpose and * denotes the Hermitian conjugate. Since the lefthand sides of (14) are first integrals of the system (13), in order that the formulas (14) hold it suffices that they hold for some one value of the argument. The first of the relations (14) ensures that the matrix r is symmetric and the second ensures the positive definiteness of Im r. It follows from the second equality of (14) that the matrix r is not singular at any point of 1. To prove this, we shall1 show that det Q # 0 everywhere on the ray 1. Indeed, suppose that det Q = 0 at some point. Then there exists a vector c= cl IG + lC2l > 0 ( c2 > ’ such that QC = 0. Then (14) gives us (i)lC12
= (C, (Q*P
 P*Q)C)
= (QC, PC)
 (C, P*QC)
= 0,
which is impossible, since ICI2 > 0. 1.3. Determination of TiTil...ih)ujil...ih. After rc2) has been found, the determination of the remaining ril,,,i,, does not present great difficulties: if instead of q = (ql, q2) we introduce new variables (gl, g2), where
(g’,g2) = g =
Qh
(15)
then the determination of r(h) from the relation (9) leads in view of (11),(12) to a quadrature with respect to s. We pass to the problem of solving the transport equation. We consider only the equation for ue. We shall assume that uc $ 0. Let UP’ be the first term of the series C,“=, uF’ which is not identically equal to zero. The equation for u.(“I has the form:
120
V. M. Babich 2 dtp  ~ co as
+ 2~p)vup)
+ ,(yo)
$;+TrPQ’
where Tr denotes matrix trace. Using (Liouville’s) formula for the derivative $det and the first equation
22 co as Equation set equal
of (la),
(17) is obtained to zero. Equation
of a determinant:
Q = det Q. Tr(Q’Q‘) we find
(ho)+ 2VTW&p O
(16)
in place
of (16) that
= 0; gm) = ,p
J
det, C
from (9) if rch) there is replaced by v?“) (17) .is solved by the change of variables
$2. The Case of a Closed
(17) and Aj is (15).
Ray
2.1. Properties of the System of Jacobi Equations. Let 1 be a closed ray. In this case, under some additional assumptions there exists a sequence of nonzero asymptotic solutions of equation (1.3), which are exponentially small as w + 00 outside of 1 and are defined and unique in a neighborhood of 1 (socalled quasimodes). If to every point s E 1 (where 1 is a closed ray) we associate the set of vectors that are orthogonal to 1, then we are led in a natural way to a socalled normal bundle: a bundle space 0 whose base is the ray 1 and whose fiber is the plane C normal to 1 at the point s E 1. If we assume that s varies from co to +co, tUhSenwe have a fibration @ which is the universal cover of fi. If L is the length of l! then the vectors a(s) E .YS, which are invariant under translation by L, that is, a(s + L) = a(s), are defined and unique on the abovedefined normal bundle CS’. The Jacobi equations (12) for the closed ray 1 can be considered as an equation on @. The classical methods from the theory of systems with periodic coefficients are then applicable in their entirety to the equations (12). For later use it will be convenient for us to consider the natural complexification of the planes Es normal to 1. The solutions of (12) that satisfy the condition q(s + L) = yq(s);
p(s + L) = y&s),
where
y =const,
will be called Floquet solutions and the constant y is called a multiplier. If y is a multiplier, then l/y is also a multiplier. The ray 1 is said to be stable if any solution q(s), p(s) (oc < s < +oo) of the system (12) is bounded. In this case all the multipliers have the form efcvz (Im cx = 0). We shall assume that the closed rays under consideration are stable. The stability of a ray is one variant
III. The HigherDimensional
WKB
Method
121
or Ray Method
of a general position situation. The Floquet solutions form a fundamental system of solutions of (12). Let eial, eiaz, eei”l, ePiaz be multipliers, and y(j),p(j), j = 1,2,3,4, the corresponding Floquet solutions. It turns out that it is possible to number the multipliers and to normalize q(j),p(j) such that the matrices Q and P whose columns are q(l), q(‘) and p(‘),p(‘), respectively, will satisfy (14).
the set
2.2. Construction of Quasimodes in the Case First Approximation. In the first approximation u”, = eiWO[7(0)+d*)]C0
of a Closed Ray in to a quasimode we
&bl~a%2)a2~ /
($)Ql($),
(18)
Co=const#O.
The function ZL~ depends on ql, q2 and s, that is, we may assume that it is defined on the fibration @. We require that if s is translated by L (L =length of 1)) then U: is left invariant (u: (s + L) E U:(S)), that is, u”, can be considered as a function on the normal bundle R. Since the eigenfunctions of the operator c2(x)A must be defined and unique in a neighborhood of 1, it is natural to impose the condition $(s + L) = U:(S) on 71.51,which is used to represent the eigenfunction approximately. The quadratic form rc2) does not vary if s is replaced
by s + L, and the expression
J
&(gr
)“I (g2)“”
acquires
the
factor exp[$(oi + cua)  ialar  iazcrz]. This is a simple consequence of the definition of Floquet solutions and the fact that the columns of the matrices P and Q are composed of these solutions. By choosing w” suitably, we can obtain the periodicity of uz. We should obviously set
where The expressions
L
ds a1 +a2 ++ alal + u2a”i, 2 0 co(s) al, u2 = 0, 1,2,. . . , b > 0 and b is an integer.
w” = 2xb
IS
(18) and (19) define
a quasimode
(19)
in the first approximation.
2.3. Higher Approximations. In order to construct higher approximations we change the ansatz somewhat. We shall assume that the desired quasimode has the form of a ray expansion
where we is not the frequency in the equation, but the expression (19). The number we will be a large parameter for the problem. The frequency w in equation (1.3) is assumed to be expressed by
V. M. Babich
122
w = wo = s,/w, + s2/w; + . . . )
(21)
where the Sj are unknown numbers which we must find. We need an additional restriction on the multipliers: suppose that ol, cr2 and T = 3.14.. . are linearly independent over the ring of integers, that is, + h2Qr2+ h3n = 0 with
hiai
hi integers
=+ hl = 0, h2 = 0, h3 = 0.
(22)
We turn first to the determination of the rch). Again introducing (gl, g2) in place of (q1,q2), we reduce the problem of finding the coefficients to a successive computation of quadratures. For our purposes it is necessary that the ~(~1, h 2 2, be Lperiodic. For h = 2 this follows from the expression for ~(~1. Now suppose that the T ch’) have been constructed and are Lperiodic for h’ < h; then ~(~1 is found from the equation
(23) where
Ah is an Lperiodic
It follows
from
the expression
B,,,,(s Representing
and requiring
homogeneous
of g1 and g2, of degree
for gi and the periodicity
+ L) = exp(cial
h:
of Ah that
+ ~2cr~)iB,~,~(.s)
rch) as a sum of monomials:
$4= c AE,C, b)(g’N7”)”
that
r ch) be Lperiodic,
A,l,z(~ + L) = exp(tia’ Equation
polynomial
we arrive
at the equalities
+ qgx2)iAElEZ(s), with
er + ~2 = h.
(24)
(23) gives
2 d A co ds “”
= Be,,,
(25)
In view of (22) all the A,,,, In an analogous
are found uniquely from (24) and (25). way we can find uj = C;l”=, up). Here it is not the uy)
that must be Lperiodic, but the product exp iwsr (0) /up ) which when we take formula (19) for wc into account leads to a condition analogous to (24), from which all the coefficients in the expansion of uj (h) into monomials are found. However, there is one exception. Among the monomials that make up ~j there is, in particular, the following one: Jco.Cjalaz (s)(g1)a1(g2)az/m, where al and a2 are the same as in formula (19). The coefficient C,,,,, must be Lperiodic:
III. The HigherDimensional
WKB Method or Ray Method
CjcQaz (s+ L) = cja,,,(s). The equation
123 (26)
for Cjalaz has the form (27)
where Cs is the same number as in formula (18) and Ajalaz denotes an Lperiodic function that depends on ujr for j’ < j and on ~7’) for h’ < al + ~2. For an Lperiodic solution Cj,,,, of equation (27) to exist, the integral of the righthand side of (27) must be zero. This uniquely determines the Sj and completes the construction of quasimodes in the case of a closed ray. Along the same line one could consider the problem with reflections, that is, the problem of constructing quasimodes in a neighborhood of a ray that has been reflected successively from N mirrors. This problem is very essential in the theory of lasers.
$3. Summation
of Gaussian Beams
3.1. Basic Ideas of the Method (see Babich and Pankratova (1973) and Popov (1981)). Th e ray expansion (see Section 1.1) loses its asymptotic nature close to those points where the geometric spreading vanishes, that is, close to caustics. Caustics are far from exhausted by the case considered in Section 1.3. It would significantly extend the range of application of the ray method to be able to extend the ray expansion into a neighborhood of a caustic of arbitrary type. Of course, in such a neighborhood an asymptotic solution of Equation (1.3) will have a rather different form from the expansion (1.4). An important generalization of the ray method is the canonical operator method of Maslov, which gives the possibility of describing arbitrary caustics [Maslov and Fedoryuk (1976)]. The method of swnmation of Gaussian beams gives another approach to the difficult problem of describing a wave field in a neighborhood of an arbitrary caustic. The basis of the method is the property of a Gaussian beam that it does not have singularities (see Section 1 above). Summing Gaussian beams, one can obtain an asymptotic solution in the domain of regularity of the field of rays. This solution is equivalent to the ray expansion but does not have singularities even on caustics. Since Gaussian beams can be reflected and refracted (if both the incident and refracted rays are not tangent to the reflecting and refracting surfaces), then the summation of Gaussian beams is applicable with these restrictions and in the presence of boundaries. Here are the basic ideas of the method. We consider a field of rays corresponding to an eikonal 7. Let C be a level surface of the eikonal. For definitenessassumethat T = 0 on C. Close to C we introduce ray coordinates cyl, 02, r as described in $1.1. Suppose in addition that a ray solution of the form (1.4)
124
V. M. Babich
corresponds to this field of rays. To each ray CY~= const, j = 1,2,. . ., in this ray field we associate a Gaussian beam (28) Suppose that on the ray CY~= const, j = 1,2, in a neighborhood Gaussian beam is constructed, we have
uao(~)/ql+=o
=
\i
of which
the
co(x) Q’
and that the eikonal rol(x) vanishes on the surface C. We multiply the expansion (28) by a nonnegative cutoff function na(x) of class C”, equal to one for ~‘(4~)~ + (q2)” < E, and equal to zero for ,/(q’)2 + (q2)2 > 2~. Th e number E can be chosen so that the coordinates ql, q2, s (see Section 1) do not lose regularity for d(q1)2 + (q2)2 5 2~. We consider the integral
w=
J c
C~,Uada1da2,
(29)
where c
=
2 j=.
w1~2), (ibJp+J
6 = const,
(30)
and the Cj are smooth functions whose determination will be discussed below. The integral is understood as follows: the series in powers of l/w are formally multiplied out. The series obtained as a result of termbyterm integration of this expansion is W, by definition. Here we need to make precise the question of how to understand the integral of an individual term. We recall that r, and U,j are not functions in the expression (28), but expansions in homogeneous polynomials of q1 and q2. We note that if r,, Uaj are replaced by the sums C,” 7L’) and Co”? U$’ with A4 and Mj sufficiently large, and if E is sufficiently small, then the integral of the corresponding expression will already be meaningful. Replacing M and Mj by still larger numbers M’ and Mi leads to corrections of a higher order of smallness as M’ and Mi become larger. We shall assume that in the integrals in which we are interested M and Mj are replaced by sums of a sufficiently large number of terms that are homogeneous in q1 and q2. If we consider the choice of the functions C~(CX’, 02), then it turns out that close to C the series W will be asymptotically equivalent to the given ray expansion up to terms of arbitrarily high order of smallness. In extending the ray solution along rays we may encounter caustic singularities, but the integral expressions for the terms of U, do not become singular. W is the extension of the ray field into the caustic domain and beyond it. Since it is a superposition of formal asymptotic solutions of equation (1.3), W remains a formal asymptotic solution of this equation.
III. The HigherDimensional
WKB Method or Ray Method
125
3.2. Application of the Method of Stationary Phase. The absolute value of the exponential factor T,(X) in (28) (recall that 7, and Uaj are replaced by finite sums) for fixed z attains a maximum for al, a2 corresponding to a ray passing through Z. This follows from the positive definiteness of Irn7c2). Applying the method of stationary phase, we obtain an expansion of the form cc (31)
pT(z) c (i$+J+l j=.
close to C. A sufficiently large number of terms of this expansion must satisfy the recurrence equations of the ray method (1.5),(1.6), since (31) is asymptotically equal to W, which formally satisfies equation (1.3), and, hence, this expansion must likewise formally satisfy the same equation. Using the method of stationary phase we have for the principal term of (31) uo = C~(c2, a2)x(a1, a2)
J
;,
where x(cr’, CX’) is some nonzero function and J is the geometric spreading corresponding to the ray coordinates cyl, 02, T. For the principal terms of the expansions (31) and (1.4) to be equal it is sufficient that 6 + 1 = y and that ug = ZQ on C, that is, for T = 0. This can be done by choosing CO(& , a2) suitably. For succeeding approximations of expansions (31) and (1.4) to be equal one must consider higher terms of the expansion of W in powers of (&I. We will omit these constructions.
Chapter 4 On Other ShortWave Diffraction 51. The
Case of Smooth
Reflecting
Problems Boundaries
1.1. Fock’s Problem. Suppose that a wave defined by its ray expansion hits a convex body with nonzero curvature. We briefly consider the plane case. The asymptotic expression for the wave field is naturally partitioned (see Fig. 6) into an illuminated region and a shadow zone. One would naturally expect to have a rather complicated expression in the transition zone between the illuminated region and the shadow zone. The key region whose investigation leads to an explanation of the behavior of the wave field both in the penumbra and in deep shadow is a neighborhood of the point C, which is the point of tangency of the ray of the incident wave and the reflecting
126
V. M. Babich Rays of the
Rays of the incident wave
Fig.
6
surface 5’. This domain was first studied by V. A. Fock using the boundary layer method in the 1940s. Introducing the coordinates s and n that we used in Chapter 2 (the point C has coordinates 0,O) and assuming that y = 0 in the incident wave (see formula 1.4)), we can obtain the first term of the shortwave asymptotic expansion of the wave field in the form
(1)
To(S) =
s ’
ds ~ 0 co(s)’
J/3
cJ= q%#3(op2/3(~)
'
v =
J
2
3 P(s)c~(~)
'
where u and wi are Airy functions 4, P(0) is the effective radius of curvature of the curve at C (see Chapter II), and =t
1 d/l%
in the case of Dirichlet boundary conditions in the case of Neumann boundary conditions.
Fock’s formula (1) t urns into rather complicated penumbral asymptotics in a neighborhood of the limit ray CM (see Fig. 6), into geometric formulas in the illuminated domain and into the wave field corresponding to gliding rays with respect to a tangent that leave S in the shadow zone (see Chapter II, Section 1). For an arbitrary number of dimensions m the role of the point C is played by the (m  2)dimensional manifold of tangent points of S with the field of rays of the incident wave. This manifold is called the terminator.5 The wave field in a neighborhood of the terminator is expressed via Fock’s function (1). * For the definition of the function v(C) see Chapter I, $3.3, and for the definition of WI(<) see Chapter II, $1.3. 5 The terminator in astronomy is the boundary between the illuminated and unilluminated parts of the Moon or some planet by the Sun.
III. The HigherDimensional
WKB
Method
or Ray Method
127
An interesting approach to the construction of shortwave asymptotics in shadow and penumbral regions and in a neighborhood of the terminator is due to Ludwig (see Ludwig (1967) and Babich and Buldyrev (1972)). His formulas give a unified representation of a wave field in these domains as integrals over caustic solutions, analogous to those considered in 51 of Chapter II,. Unfortunately, Ludwig’s method is restricted to being applied only in the case of the Dirichlet problem. 1.2. Point Sources of Vibrations on the (see Babich (1979)) Babich and Buldyrev (1972), be a domain contained in R2 with a boundary of the problem of finding an asymptotic solution to
(A+w")u Eels
Boundary of a Domain and Buslaev (1975)). Let R positive curvature. We pose the problem
= 0,
= S(s  so), where
E’=lor&,and&($&zl)+Oasr+oo.
(2) In this case formal constructions allow us to construct an expansion for 21 close to C (s = 0). In the first approximation this expansion is equal to the solution of the problem for the Helmholtz equation in a halfplane with a source on the boundary. This solution is combined with the ray expansion corresponding to the central field of rays (the center is the point C). The penumbra close to C is described by an integral which is close in nature to the Fockfunction. This expression becomes the shadow asymptotics corresponding to the field of gliding rays shown in Fig. 7 (see Babich and Kirpichnikova (1974)). All these constructions can be generalized to the higherdimensional case. Buslaev (see Buslaev (1975)) was able to construct a unified expression for an asymptotic solution of the problem (2) which can be applied both close to the source and in the Fock zone. Using this expression he was able to give a justification for the shortwave asymptotics of the Green function for the Helmholtz equation outside a convex smooth domain, under some natural assumptions.
Gliding rays Fig. 7
V. M. Babich
128
In the case of a source of vibrations on a concave reflecting boundary no gliding wave arises, but there is an uncountable set of reflections. Waves corresponding to rays that exit from the source at an angle to the boundary less than w1/3 in order of magnitude lose their individuality. The total position effect of these waves ca.n be described by some special function (see Babich and Buldyrev (1972), Babich and Molotkov (1985), and Philippov (1981)).
$2. Various
Problems
2.1. The Problem of the Change of Sign of the Curvature (Babich and Smyshlyaev (1984), Popov (1979)). Suppose that a whispering gallery wave moves along a curve S on which a Dirichlet condition holds. Suppose that the effective radius of curvature changes its sign at a point C (see Fig. 8). Apparently, close to the ray r tangent at C to the curve S, the wave field will have a penumbral character and then the typical picture of a shadow wave field corresponding to gliding waves tangent to S arises. The boundary layer method reduces the problem on the behavior of a whispering gallery wave close to S to the solution of the following problem from mathematical physics:
ll$ll
I* Iq!~(s,v)l~dv = const < +oo, II+  &II + 0 as B i J $0 = (2a)l16exp(i<. 3/20. (2a)5/3)~((20)1/3y  <), where
=
u is the Airy
function
co,
(3)
and I is its root.
Fig. 8 Problem (3) is wellposed. There are a number of numerical results associated with it; however, no explicit solution for it has been found. In particular, an explicit analytic expression has not been found for the diffraction coefficients defining the gliding waves.
III. The HigherDimensional
WKB
Method
or Ray Method
129
2.2. Problems with Sharp Edges and Screens. This is a wide class of problems. The problem of the incidence of a wave defined by its ray expansion on a curvilinear screen can be solved by the method of matched asymptotic expansions [Philippov (19Sl)]. Close to the boundary of a screen the problem iu the first approximation is the same as the classical problem of Sommerfeld or! the diffraction of a plane wave by a halfline; then follows the Fock zone, where the asymptotics in the first approximation is described by functions that have much in common with Fock’s function (l), after which the wave field turns into gliding waves on one side of the screen and into vibrations of whispering gallery type on the opposite side. Many other problems associated with the diffraction of waves by screens and by dihedral angles with curvilinear edges are considered in the monograph by Borovikov and Kinber (1978).
Remarks on the Literature A rather detailed and elementary presentation of the ray method is contained in the monographs by Babich and Buldyrev (1972), Babich and Kirpichnikova (1974)) B orovikov and Kinber (1978), and Whitham (1974). See also the book by Kravtsov and Orlov (1980), which contains a rather large bibliography. The familiar canonical operator method [Maslov and Fedoryuk (1976)] and the complex germ method [Maslov (1977)] are closely connected with the ray method. The canonical operator method gives the possibility of constructing asymptotics of solutions “in the large”. These methods enable the solution of many important problems. See the book by Whitham (1974) for another development of the idea of the ray method in connection with linear and nonlinear problems of mechanics of a layered medium. This book contains many interesting ideas. For asymptotic formulas in a neighborhood of a source of vibrations see Babich (1965). These constructions are closely connected with Hadamard’s method for constructing a fundamental solution of the Cauchy problem. Problems with point sources on the boundary of the domain are very much more complicated than problems in which the influence of the boundary is not considered. See Babich (1979), Babich and Buldyrev (1972), and Buslaev (1975). Uniform asymptotic formulas in a neighborhood of a caustic were proposed by Kravtsov (1964) and studied in detail by Ludwig (1966) and Babich and Buldyrev (1972). Whispering gallery waves and gliding waves are discussed in many places in the monographs by Babich and Buldyrev (1972) and Babich and Kirpichnikova (1974), but our treatment is based on a later paper by Kirpichnikova (1979). The approach to surface waves ($2 of Chapter II) is different in essential ways from the approach of Molotkov (1970) and is due to the present author. The presentation of the theory of Gaussian beams based on a parabolic equation is carried out in Chapters 8 and 9 of the book by Babich and
130
V. M. Babich
Buldyrev (1972). The approach discussed in this paper is close to the complex germ method, but is more elementary. Fock’s problem was considered in detail by Fock himself [Fok (=Fock) (1970)]. For a presentation of Fock’s problem based on ideas of the boundary layer method see the book by Babich and Kirpichnikova (1974). The other works in the bibliography have been cited in various places in this paper and their themes are clear from the context.
References* Babich, V. M. (1965): On shortwave asymptotics of a solution of the point source problem in an inhomogeneous medium. Zh. Vychisl. Mat. Mat. Fiz. 5, No. 5, 9499 951 [English translation: USSR COmput. Math. Math. Phys. 5, No. 5, 247.2511 Zbl. 167,243 Babich, V. M. (1979): A point source of vibrations close to a concave mirror. Zap. Nauchn. Semin. Leningrad. Otd. Mat. Inst. Steklova 89,3313 [English translation: J. Sov. Math. 19 (1982), 1279912871 Zbl. 427.35006 Babich, V. M., and Buldyrev, V. S. (1972): Asymptotic Methods in ShortWave Diffraction Problems. Nauka, Moscow [English translation: Springer Series on Wave Phenomena 4. Springer, Berlin Heidelberg New York 19911 Zbl. 255.35002 Babich, V. M., Buldyrev, V. S., and Molotkov, I. A. (1985): The SpaceTime Ray Method. Izdat. Leningrad. Gos. Univ. (Leningrad State University Press), Leningrad (in Russian) Babich, V. M., and Kirpichnikova, N. Ya. (1974): The Boundary Layer Method in Diffraction Problems. Izdat. Leningrad. Gos. Univ. (Leningrad State University Press), Leningrad, Zbl 317.35001 (in Russian) Babich, V. M., and Pankratova, T. F. (1973): On discontinuities of the Green function for the wave equation with variable coefficients. Probl. Mat. Fiz. 6, 9927, Zbl. 277.35065 (in Russian) Babich, V. M., and Smyshlaev, V. P. (1984): On the scattering problem for the Schriidinger equation in the case of a potential linear in time and position. Zap. Nauchn. Semin. Leningrad. Otd. Mat. Inst. Steklova 140, 617 [English translation: J. Sov. Math. 32 (1986), 1033112], Zbl. 557.35024 Babich, V. M., and Ulin, V. V. (1980): Complex ray solutions. Zap. Nauchn. Semin. Leningrad. Otd. Mat. Inst. Steklova 104, 613 [English translation: J. Sov. Math. 20 (1982), 174917531, Zbl. 476.58030 Borovikov, V. A., and Kinber, B. E. (1978): Geometric Theory of Diffraction. Svyaz’, Moscow (in Russian) Buslaev, V. S. (1975): On the asymptotic behavior of spectral characteristics of exterior problems for the Schriidinger operator. Izv. Akad. Nauk SSSR, Ser. Mat. 39, 149235 [English translation: Math. USSR, Izv. 9 (1976), 1392231 Zbl. 311.35010 Courant, R. (1962): Partial Differential Equations. WileyInterscience, New York London. [German original: Springer 19371 Zbl. 17,397 * For the convenience of the reader, references to reviews in Zentralblatt fur Mathematik (Zbl.), compiled by means of the MATH database, and Jahrbuch iiber die Fortschritte der Mathematik (Jbuch) have, as far as possible, been included in this bibliography.
III.
The
HigherDimensional
WKB
Method
or Ray
Method
131
Fedoryuk, M. V. (1983): Asymptotic Methods for Linear Ordinary Differential Equations. Nauka, Moscow [English translation: Springer, Berlin Heidelberg New York, 19931 Zbl. 538.34001 Fok, V. A. (=Fock) (1970): Diffraction Problems and the Propagation of Electromagnetic Waves. Sovetskoe Radio, Moscow [English translation: Pergamon, 19651 Hadamard, J. (1932): Le Probleme de Cauchy et les Equations aux D&i&es Par.tielles Lineaires Hyperboliques. Hermann, Paris, Zbl. 6,205 Keller, J. B. (1956): Diffraction by a convex cylinder. Trans. IRE Antennas and Propagation 4, No. 3, 312321 Kirpichnikova, N. Ya. (1979): Uniform asymptotics of eigenfunctions of whispering gallery type. Zap. Nauchn. Semin. Leningrad. Otd. Mat. Inst. Steklova 89, 112119 [English translation: J. Sov. Math. 19 (1982), 136613721 Zbl. 427.35048 Kravtsov, Yu. A. (1984): On a modification of a method of geometric optics. Izv. VUZov, Radiofizika 7, No. 4, 6644673 Kravtsov, Yu. A., and Orlov, Yu. I. (1980): Geometric Optics of Inhomogeneous Media. Nauka, Moscow (in Russian) Lazutkin, V. F. (1979): Estimate for the width of lacunas in the spectrum of the Laplace operator. Dokl. Akad. Nauk SSSR 245, No. 1, 2023 [English translation: Sov. Math., Dokl. 20 (1979), 25662591 Zbl. 425.35074 Ludwig, D. (1966): Uniform asymptotic expansions at a caustic. Commun. Pure Appl. Math. 19, No. 2, 21525, Zbl. 163,137 Ludwig, D. (1967): Uniform asymptotic expansion of the field scattered by a convex object at high frequencies. Commun. Pure Appl. Math. 20, No. 1, 103138, Zbl. 154,128 Maslov, V. P. (1977): The Complex WKB Method in Nonlinear Equations. Nauka, Moscow [English translation: Birkhauser, 19941 Zbl. 449.58001 Maslov, V. P., and Fedoryuk, M. V. (1976): Semiclassical Approximation for the Equations of Quantum Mechanics. Nauka, Moscow [English translation: Reidel 19811 Zbl. 449.58002 Molotkov, I. A. (1970): Excitation of surface waves under differaction by an impedance contour. Zap. Nauchn. Semin. Leningrad. Otd. Mat. Inst. Steklova 17, 151167 [English translation: Semin. Math., V. A. Steklov Math. Inst., Leningrad 17 (1972), 833921 Nomofilov, V. E. (1981): Asymptotic solutions of secondorder equations, concentrated in a neighborhood of a ray. Zap. Nauchn. Semin. Leningrad. Otd. Mat. Inst. Steklova 104, 170179 [English translation: J. Sov. Math. 20 (1982), 185441860] Zbl. 476.58029 Philippov (= Filippov), V. B. (1981): Diffraction by a nonplanar screen. Wave Motion 3, 7180, Zbl. 514.73015 Popov, M. M. (1973): Wellposedness of the problem of whispering gallery waves in a neighborhood of a rectification point of the boundary. Zap. Nauchn. Semin. Leningrad. Otd. Mat. Inst. Steklova 89, 261269 Popov, M. M. (1981): A new method for computing wave fields in the shortwave approximation. Zap. Nauchn. Semin. Leningrad. Otd. Mat. Inst. Steklova 104, 195216, Zbl. 489.73039 [English translation: J. Sov. Math. 20 (1982), 186918821 Smirnov, V. I. (1981): A Course in Higher Mathematics. Vol. 4, Part 2, Nauka, Moscow, Zbl. 44,320 Whitham, G. B. (1974): Linear and Nonlinear Waves, Wiley, New York London, Zbl. 373.76001 Zayaev, A. B., and Filippov, V. B. (1984): On a rigorous justification of the FriedlanderKeller formulas. Zap. Nauchn. Semin. Leningrad. Otd. Mat. Inst. Steklova 140, 49960 [English translation: J. Sov. Math. 32 (1986), 1341431
IV.
Semiclassical Asymptotics of Eigenfunctions V. F. Lazutkin Translated
from the Russian by J. S. Joel
Contents 51. Introduction.. ........................................... ................................ $2. Quasimodes and Spectrum 2.1. A SelfAdjoint Operator Generated .......................... by a Differential Expression ...................... 2.2. Spectrum and Discrete Spectrum ........................................ 2.3. Quasimodes ............................. 2.4. A Family of Quasimodes 2.5. Remarks.. ......................................... 53. A Classical Dynamical System and Principles for Constructing Quasimodes .............................. 3.1. A Classical Dynamical System Associated ........................ with the Schrodinger Equation 3.2. Phase Space and Dynamics in the Presence of a Boundary ................ 3.3. How Does One Construct Quasimodes? 3.4. A Completely Integrable Hamiltonian System ........... 3.5. Integrability on a Cantor Set ......................... 54. Quasimodes Corresponding to Kolmogorov Tori .............. 4.1. The Maslov Index and the Boundary Index ............. 4.2. Quantization Conditions. Estimate of the Number of Quasimodes .............................. 4.3. The Maslov Canonical Operator. ...................... 4.4. Formulas for Quasimodes. ............................
. 135 137 ,138 ,138 .139 . 140 . 141 .141 141 . 143 146 ,147 148 ,149 . 150 ,151 . 152 . 155
V. F. Lazutkin
134 4.5.
Choice of a Normalization Constant and Behavior of a Quasimode in a Neighborhood of a Point That Is Regular Relative to Projection . .. . . .. . . .. .. . . 4.6. Orthogonality of Quasimodes. Estimate of the Total Multiplicity of the Approximating Spectrum. .. . . . $5. History of the Problem and Some of the Problems That Have Been Studied . . . . . . .. . .. . . . .. . . .. . .. . .. . . 5.1. History of the Problem. ... . .. . . . .. .. . . .. . . . .. . .. . . 5.2. Problems with Separated Variables . . .. . .. . .. . . .. . . 5.3. Small Perturbations of Problems with Separated Variables. 5.4. A Neighborhood of a Closed Trajectory . .. . . . . .. . .. . . 5.5. Quasimodes Concentrated Close to the Boundary of a Domain .. . .. . . . . .. . .. . .. . 5.6. Quasimodes Corresponding to the Complement of Kolmogorov Tori . . . . . . . . . . . . . . . . . References . .. . . . .. . .. . . .. . . . .. . .. . . . .. . .. . . .. . . . .. . .. . . .. . . ..
157 158 159 159 159 160 163 165 167 168
IV. Semiclassical
Asymptotics
of Eigenfunctions
135
5 1. Introduction One of the most important problems of mathematical physics is the study of the spectrum of selfadjoint operators associated with differential equations and boundary value problems for them. The spectrum in such problems, depending on the configuration of the domain in which the equation is considered and on the properties of the coefficients of the equation, turns out to be both discrete and continuous. A situation is possible in which both components occur. In a number of cases we have to deal with an operator containing a small parameter in the highest derivatives (for example, Planck’s constant in the Schrodinger equation), or the problem arises of the asymptotic behavior of the discrete spectrum of the operator as the eigenvalue tends to infinity. In such situations a natural method for finding the asymptotic behavior is the semiclassical method, or WKB method, well known in its application to onedimensional problems [Heading (1962)]. S erious difficulties arise in trying to apply this method in the higherdimensional case. The nature of these difficulties lies in the complexity of the apparatus corresponding to the “classical” dynamical system in terms of which the WKB asymptotics are expressed. This paper contains a presentation of the semiclassical method for obtaining the asymptotic behavior of the discrete spectrum in the multidimensional case. A more detailed exposition can be found in the author’s book [Lazutkin (1993)]. The following are typical examples. I. The Schrodinger equation of nonrelativistic quantum mechanics [Landau and Lifshits (1963)] ;A$ Here A = Cz=,
&
+ V(x)$
is the Laplace
operator
= E$J. in R”,
(1) tr. is a small
positive
pa
rameter (Planck’s cknstant), V( z ) .is a real function (the potential), and E is the spectral parameter (the energy). We,seek a function $J : R” + C (a wave function) satisfying equation (1) and possessing regular behavior as 1x1 f oo, where 1x1 = xy + x; + . . . + xi. Those values of E for which equation (1) has a solution 1c,belonging to &(R”), that is, ]]$]]’ = JRn ]$(z)12dz < co, and is not identically equal to zero, are called eigenvalues of the SchrGdinger operator, while the corresponding solutions are called eigenfunctions. In physics these eigenfunctions are interpreted as bound states of a quantummechanical system described by equation (I), and the eigenvalues are interpreted as the corresponding energy levels. II. The wave equation [Babich and Buldyrev (1972)]:
Au+
W2 u=o. C2(X)
Here A is the Laplace operator in ndimensional space and w is a real parameter, called the frequency. Equation (2) is considered in a domain X c R”
136
V. F. Lazutkin
whose boundary dX is assumed to be smooth, that is, it is of class C”. necessary to add boundary conditions to equation (2). We shall consider Dirichlet boundary conditions upx
= 0.
It is the
(3)
Equation (2) describes oscillating processes in the domain X. If X is a bounded domain in R” with a smooth boundary and c2(x) is a positive smooth function, then problem (2) together with the boundary conditions (3) has a discrete spectrum, and the eigenvalues wk are interpreted as the eigenfrequencies of a resonator in the domain X filled with a medium of variable density which is proportional to c2(Z). III. The LaplaceBeltrami operator on a Riemannian manifold. Let X be a closed smooth ndimensional manifold provided with a smooth Riemannian metric ?I ds2 =
c
gi&)dzidz”.
(4
i,k=l
Here xi, i = l,..., n, are local coordinates. The spectrum of the LaplaceBeltrami operator presents a rich collection of invariants of the Riemannian structure on X. This operator is defined in local coordinates by the following expression
where g is the determinant of the matrix {gik} and {gi”} is the inverse matrix to {gik}. The spectrum of this operator consists of a sequence {Xk, k E N} of eigenvalues, that is, numbers X for which the equation &u = Au has a nontrivial solution. Many geometric characteristics of a Riemannian manifold, in particular the Riemannian volume and the lengths of closed geodesics, are expressed by means of the spectrum of the LaplaceBeltrami operator [Colin de Verdi&e (1973)], [D ms’ t ermaat and Guillemin (1975)]. We shall consider the following problem, of which the preceding examples are special cases. Let X be a smooth ndimensional oriented manifold with boundary dX or without boundary, compact or noncompact, and suppose a Riemannian metric (4) with smooth coefficients gik(x) is given on X. We consider the differential equation
;A2u
+ V(x,
h)u = Eu
(6)
together with the Dirichlet boundary condition (3). In (6) A2 is the LaplaceBeltrami operator (5), fi is a small positive parameter, and V(z, ri) is a smooth real function defined on X x [0, ha]. The variable E in (6) plays the role of spectral parameter. The eigenvalues of the problem are the values of E for which (6) has smooth solutions, the eigenfunctions, which are not identically
IV. Semiclassical
Asymptotics
of Eigenfunctions
137
zero and satisfy the boundary condition (3) if dX # 0. If X is noncompact, then we must also require that the integral of the square of the modulus of the function U(Z) be finite with respect to the measure &dx. The problem for the wave equation reduces to the problem just stated if we take gik(5) = C~(X)& as the metric tensor, where &k is the Kronecker symbol, Sik = 0 if i # Ic and &k = 1 if i = k. In this case it is convenient to introduce the additional parameter fi > 0 and to set E = iw2h2. The wave equation (2) with these parameters is equivalent to equation (6) with the above metric and the potential
V(x, ix) =
[
 n(ng
2, (grad
c)2 + qc~c]
T
The main problem, considered later, is the construction of asymptotic formulas for the eigenvalues and eigenfunctions of the problem (6),(3) as /i $ 0. We shall give a partial solution of this problem in the following sense. Among all the eigenvalues there is a distinguished part, those corresponding to the Kolmogorov tori [Arnol’d, Kozlov, and Nejshtadt (1985)] of the corresponding classical dynamical system. In terms of the Maslov canonical operator [Maslov and Fcdoryuk (1976)] one writes down asymptotic formulas that approximate the eigenvalues belonging to this subset, and also approximate formulas for “eigenfunctions”, for which, however, one cannot assert that they approximate the true eigenfunctions (for more details see the following section). The part of the approximable asymptotics of the eigenvalues turns out to be proportional to the measure of a set related with the Kolmogorov tori: the Kolmogorov set. It follows from KolmogorovArnol’dMoser (KAM) theory [Arnol’d, Kozlov, and Nejshtadt (1985)] that this measure is positive if the classical system is close to completely integrable and tends to full measure if the parameter that describes the deviation of this system from being completely integrable tends to zero. As we have said, we shall assume that the manifold X itself, its boundary dX, and the potential V are smooth, and we restrict ourselves to the case of the Dirichlet condition (3). The constructions that follow below can be extended to the more general case in which the boundary and potential are piecewisesmooth, as well as to the case of other boundary conditions, and to wider classes of operators (l/tipseudodifferential operators; see Maslov and Fedoryuk (1976)). For the wave equation (2) and for the LaplaceBeltrami operator (5) we shall consider the problem of finding the asymptotics of the eigenfrequencies Wk and, respectively, the eigenvalues XI, as k + co, which corresponds to the asymptotics as /% + 0 after making the reduction to equation (6) indicated above.
32. Quasimodes
and Spectrum
In this section we consider the question of what can be said about the true eigenvalues and eigenfunctions if we know how to solve the problem of determining them approximately. It is convenient to state the results in the
138
V. F. Lazutkin
framework [Akhiezer
of the abstract theory of selfadjoint operators in Hilbert and Glazman (1966), Birman and Solomyak (1980)].
space
2.1. A SelfAdjoint Operator Generated by a Differential Expression. First we shall indicate how our problem can be stated in this framework. For our Hilbert space we take the space &(X, dp), whose elements are equivalence classes of measurable functions u : X + C such that
(7) Here &(z) = fidx, dx = dx’dx2 . . . dxn, is the measure in X generated by the Riemannian metric, measurability is defined relative to this measure, and two functions are equivalent if they differ only on a set of measure zero. The scalar product in L2(X, db) is defined by the formula
and the norm by formula (7). We denote by C,(X) the set of smooth complexvalued functions on X such that supp u is compact and does not intersect dX (supp u is the support of the function U, that is, the closure of the set of points of X at which u is different from zero). The set C,(X) is dense in L2(X, dp). We define an operator Lo : Cr + L2(X, dp) by the formula LOU = 512u The
operator
Lo is symmetric.
+ V(x,
It is semibounded inf
V(x,h)
h)u. from
(8) below
if
> DC).
XEX
In the last case Lo always has a selfadjoint extension, called the Friedrichs extension (see Reed and Simon (1975)), corresponding to the Dirichlet condition. If X = R” and gik = 6ik, then this extension is unique [Berezin and Shubin (1983)]. If dX # 0, then the operator Lo always has several selfadjoint extensions, and different extensions correspond to different boundary conditions. We shall assume that condition (9) holds. All the assertions that follow concern the Friedrichs extension L of the operator Lo. We note that every C2 function u that is equal to zero on dX and is such that Lou E L2(X, dp), where Lo is defined by formula (8), belongs to the domain D(L) of the operator L and Lu = LOU. 2.2. Spectrum and Discrete Spectrum. We give the definitions and basic information about the spectrum and discrete spectrum of a selfadjoint operator L in a Hilbert space 31. A complex number X is called a regular point
IV. Semiclassical
Asymptotics
of Eigenfunctions
139
of the operator L if the operator L  XI, where I is the identity operator, has a bounded inverse operator. This inverse Rx = (L  XI)’ is called the resolvent of the operator L. The complement to the set of regular points is called the spectrum of L and is denoted by c(L). The spectrum of a selfadjoint operator is a nonempty closed subset of the real axis. If X is an isolated point of a(L), then X must be an eigenvalue, that is, there exists a nontrivial solution u E D(L) of the equation Lu = Xu. This solution is called an eigenaector of the operator L corresponding to the eigenvalue X. The dimension of the subspace spanned by all the eigenvectors corresponding to X is called the multiplicity of the eigenvalue X. We shall say that X belongs to the discrete spectrum of L if X is an isolated eigenvalue of finite multiplicity. The discrete spectrum of L is denoted by the symbol ad(L). If the manifold X is compact, then the whole spectrum of the operator L defined by expression (8) is discrete. If X = R” with the Euclidean metric, then the spectrum of L that lies to the left of the point lim inf/zl+oo V(x, h) is discrete, and the corresponding eigenfunctions decrease ‘exponentially as 2 + cc [Berezin and Shubin (1983)].
with
2.3. Quasimodes. domain D(L),
Let L be a selfadjoint and let E be a nonnegative
operator number.
in a Hilbert
space ‘Ft
Definition. A quasimode’ with remainder E for the operator L is a pair (u,X), where u E D(L), /lull = 1, and X E R, such that IILu  Xull < E (where II 11is the norm in Z). Thus, a quasimode with remainder 0 is a pair (u, X), where X is an eigenvalue of L and u is a normalized eigenvector corresponding to X. We assume that we have succeeded in finding a quasimode with remainder t instead of an exact eigenvalue and eigenvector. What can we deduce about the spectrum of L? The following theorem gives and answer to this question. Theorem 1. Let (u, A) be a quasimode with remainder c > 0 for the operaL and suppose it is known that the spectrum of L is discrete in the interval [A  E, X + t]. Then this interval contains an eigenvalue A* of the operator L. tor
The proof of Theorem 1 is so simple that we shall give it here. Suppose that X $ a(L). Th e d is t ante dx from the point X to the spectrum of L is computed (and estimated) by th e f ormula (see Birman and Solomyak (1980)): dxl
= ll(L  XI)1ll
’ The term “quasimode” guish this approximate L. The difference can
= N;
ll(L ;“l;‘“‘II 21
was introduced solution u from be very significant.
by V. I. Arnol’d a true eigenvector
> II(L  Jwl~ll ll4l
’
(1972) in order to distin(mode) of the operator
140
V. F. Lazutkin
where 6 is an arbitrary vector of X. Substituting V = (L  X1)u in the righthand side of this inequality, we will obtain dx < E, from which the assertion of the theorem follows. We note that we cannot assert anything at all about the relative closeness is true. Let of the quasimode u to a true eigenvector u*. Only the following E(n) be the spectral projection of the operator L (see Birman and Solomyak (1980)) that corresponds to the interval A = [X p, X + ~1, where p > 0. Then for a quasimode (u, X) with remainder E the following inequality is true:
IIE(A)u
 u/I L tp?
(10)
In particular, if in the interval A there is exactly one eigenvalue p* of L of multiplicity one with normalized eigenvector U* (that is, (Iu* I( = l), then we can assert that for some cy E R.
1121  eiau*II < 2ql.
(11)
The estimate (10) is a simple consequence of the spectral theorem and the definition of a quasimode, while estimate (11) follows immediately from (10) if E(A) = (u, u*)u*. Remark. If the interval A contains several eigenvalues of L and does not contain the continuous spectrum (the continuous spectrum of an operator L is by definition a(L)\ad(L)), th en f rom (10) it follows only that u approximates some linear combination of true eigenvectors. In semiclassical asymptotics this phenomenon actually occurs, as was apparently first noticed by Arnol’d (1972). 2.4. A Family of Quasimodes. In the problems considered in this paper we shall encounter the situation when the semiclassical method gives many different quasimodes, and we must estimate what part of the true spectrum of the operator L is approximated by these quasimodes. In this situation an important characteristic, which allows us to estimate the dimension of the approximating spectral subspace of L, is the degree of closeness of quasimodes to being an orthonormal system. Let E and S be nonnegative numbers. Definition. A family of quasimodes with remainder E and deviation from orthogonality 6 will be a family {(uI, Xl), 1 E A}, where n is a finite set of indices, UI E D(L), Xl E R, such that the following holds: for all 1, Ic E A
1141= 1, llw  hUllI I 6, ll(wk, w)  &II Here bkl is the Kronecker
symbol.
I 6.
(12)
IV. Semiclassical
Asymptotics
of Eigenfunctions
141
We assume that there is a family of quasimodes and that the set UIEn [Xl E, Xl + F] contains only the discrete spectrum of L. Then it follows from Theorem 1 that in each interval [Xl  E, Xl + E] there is at least one eigenvalue Xl of the operator L. We fix a positive number ,LL. Let Efi = UICn[Xl  p, Xl + ,u], and let E(E,) be ,the spectral projection of L corresponding to the set &,, ,!Z{vl, 1 E A} the linear span of the system of vectors (~1, 1 E A}. We denote by N*(Z@) the dimension of the subspace E(E,)l{ ~1, 1 E A}. It is natural to call the number N*(E,) the total multiplicity of the part of the spectrum of L in the set &, approximated by the family of quasimodes { (ul, Xl)}. We denote the number of elements of the set A by IAl. It is useful to explain the conditions under which the total multiplicity of the part of the spectrum approximated by quasimodes is equal to the number of quasimodes, that is, to IAl. . Theorem 2. Let { (ul, Xl), 1 E A} b e a f amily of quasimodes E and deviation from orthogonality b for the operator L. If ql then N*(E,) = IAl. The proof of Theorem different formulation.
2 can be found
in Lazutkin
(1981)
of remainder
+ 6 < [All,
in a somewhat
2.5. Remarks. Theorem 1 has been used by many authors. It apparently occurs first in the works of V. P. Maslov (1965a, 196513). Pankratova (1984, 1986) generalized Theorem 1 to the case when there is a family of quasimodes with close numbers {Xl, 1 E A}. In this case we can assert that the spectrum of the matrix {(Luk, UL), Ic, 1 E A} approximates the spectrum of L up to order 6’.
$3. A Classical Dynamical for Constructing
System and Principles Quasimodes
One of the possible ways of finding the semiclassical asymptotics of eigenvalues is the construction of quasimodes. How does one obtain formulas for them? Equation (6) 1s . nothing but a slightly generalized Schrgdinger equation, which is the main equation of nonrelativistic quantum mechanics. It is said that as fi + 0 quantum mechanics becomes classical mechanics. This means that the asymptotics as fi + 0 of solutions of equation (6) should be expressed in terms of a classical dynamical system corresponding to the given quantum problem. 3.1. A Classical Dynamical System Associated with the dinger Equation. For simplicity we assume first that the manifold no boundary. In this case the phase space of the classical dynamical
SchrGX has system
142
V. F. Lazutkin
associated with our problem is the total space T*X of the cotangent bundle 7~: T*X + X of X. If (U, Q ) is a chart in X, that is, U is an open subset of X and (Y : U + R” is a diffeomorphism onto o(U), a(z) = (x1,x2,. . . ,xn), space then the differentials &ri, i = 1,. . . , n, form a basis of the cotangent T,*X at x E U. A general cotangent vector p E TZX is written as p = &dxi
(13)
i=l
in this basis. A chart (U, cy) in X corresponds to a chart in T*X which is defined on r‘(U) with coordinate mapping (x,p) H (x1,x2,. . . ,xn,pl,p2, pi, i = 1, . . , n, are called the momenta dual to the . . . >p,). The quantities coordinates x1, x2,. . . , xn. On T*X there is a canonical symplectic structure [Arnol’d (1974)], which is defined as follows. Let p E T,*X. A tangent vector ‘u E Tc~,~)(T*X) can be projected onto T,X via the mapping 7riT*: T(T*X) + TX, tangent to the projection 7r : T*X + X. One can apply the cotangent vector p E T,*X to the vector ~T*U E T,X. Considering p(n,v) as a function of U, we will obtain a linear form on the space Tc~,~) (T*X), that is, an element of TTz,p, (T’X). We denote the resulting smooth section of the bundle T*T*X + T*X by the letter 0. The lform 8 in the local coordinates on T*X defined above has a shape that agrees with the righthand side of (13). The differential of the form B w = &I
=
2
dxi A dpi
(14)
i=l
which defines a is a nondegenerate closed (i.e., dw = 0) 2form on T*X, symplectic structure on this manifold in a natural way. We define a Hamiltonian H(x,p) corresponding to the operator Lo, by making the formal change of variables (ifi)‘d/dxi + pi in Lo and replacing V(x, fi) by V(x, 0). In the local coordinates (x,p) this function has the form H(x,p)
= f
k
gik(x)l%pl,
+ V(x,
0).
i,kl
The function H(x,p) is defined on T*X and its values do not depend on the choice of the chart (U, a). It follows from the nondegeneracy of w that it generates an isomorphism of vector bundles over T*X # : T*T’X
+ TT*X
(see Arnol’d (1974)). The H amiltonian H defines a vector will be denoted by dldt, according to the formula f
= #(dH).
field on T*X,
which
(15)
IV. Semiclassical The integral curves of this 1,2, . . . , n) are the solutions
@ixF
Asymptotics
of Eigenfunctions
vector field in the local coordinates of the Hamiltonian system
dH xz ’ yg
i = 1,2,.
143 (z’,pi,
. ) 7?,.
If the vector field d/dt is complete2, which happens, for example, when compact, then it generates a dynamics, that is, a Hamiltonian flow ft : T*T
+ T*X,
i =
X is
t E R.
We recall that a flow on a manifold M is a oneparameter group of diffeomorphisms ft : M + M, t E R; a flow is generated by vector field u if, for any smooth function ‘p defined on M, s~,~o f” It=a = uu’p. Vector fields are identified with firstorder linear differential operators defined on smooth functions. Not all vector fields, also including Hamiltonian fields, are complete. For our purposes, however, completeness is not necessary. We shall call a pair consisting of the cotangent space T*X together with its canonical symplectic structure and the vector field d/dt on it defined by formula (15) the classical dynamical system associated with equation (6) in the case when dX = 8. 3.2. Phase Space and Dynamics in the Presence of a Boundary. If dX # 0, then the total space T*X of the cotangent bundle is a manifold with boundary equal to n‘(8X). We consider the reflection mapping r : 7c’(L?X) + r ‘(ax), which associates a covector p’ E T,*X, x E dX, to a covector p E T,*X so that the following conditions hold IPI = IP’I, (p  p’) I 8X.
06) (17)
The length Ipl of the covector p = (pi, 1 < i < n) is computed according to the formula Ipl = (Cy,=, gi”(x)pipk) II2 . We introduce a scalar product corresponding to the quadratic form IpI2 on T,*X. It allows us to identify T,*X and T,X, so that we may assume that T,iaX c T,*X. The orthogonality in (17) can be understood in the sense of this scalar product. If p E ZY~aX, then we must have p’ = p; if p $ T,*aX, then we require that p’ # p. The mapping r is uniquely determined by these conditions. The physical meaning of condition (16) is the conservation of energy while that of condition (17) is the conservation of the tangential component of the momentum. The formula for the mapping T looks particularly simple in the local coordinates (xl,... , xn, pl , . . , p,), in which the boundary is rectified and is defined by the equation x ’ = 0 , while the interior points of X correspond to positive values of the coordinate x1: 2 A vector field is said to be complete boundedly, that is, if they are defined
if its integral
curves can be extended
for all
of the
values
parameter
t.
un
V. F. Lazutkin
144
Pi = P:,
i=2,...,n;
vl=(ul)‘,
(18)
where 1~~ = Cz=, gl”(x)pk and (‘u’)’ = CL=, gl”ph. The conditions (18) mean that the tangential component of the momentum is conserved under reflection while the component of the velocity normal to the boundary changes sign. We glue the manifold T*X along the boundary via the reflection mapping T, that is, we identify each point p E BT*X with its reflection p’ = r(p). As a result we obtain a 2ndimensional topological manifold 2 without boundary. The previous boundary rl (8X) after gluing becomes a (2n  l)dimensional submanifold I7 of 2, and 17 has the boundary C = T*dX. It is not hard to equip Z\C with a smooth symplectic structure, whose restriction to Z\I7 = T*X\aT*X coincides with w and in which the vector field d/dt extends to a smooth Hamiltonian field on Z\C. According to Darboux’s theorem [Arnol’d (1974)], a symplectic structure can be defined via socalled symplectic charts, i.e., coordinate systems in which the symplectic form has a standard form equal to the righthand side of (14). If z E Z\n, then, in a neighborhood of z that is disjoint from 17, we take one of the previous coordinate systems (xl,. . , C,pl, . . . ,p,) as a symplectic chart. If z E n\E we proceed as follows. In a neighborhood of the point 20 E dX into which .z is projected, we introduce a coordinate system (x1,. . . ,x7’) in which dX is rectified, and x1 > 0 corresponds to the interior points of X. We denote the momenta dual to the coordinates (xi,. . , x”) by (pi,. . ,pn). Suppose that p” and (p’)’ are two points of T,*X glued to a. single point Z, and that the corresponding point ~0 on dX has coordinates (0, xi,. . . , xg). We consider a subset fi of fl consisting of the points (x,p) at which the condition H(z,p) = H(xe,pO) holds. The Hamiltonian vector fields corresponding to the Hamiltonians H and t, where t denotes time along the trajectories of the Hamiltonian flow corresponding to the function H, equal to zero on fl, are defined in neighborhoods Q and KY of the points p” and (PO)’ in T*X, are transversal to I7, and commute. Using the local flows generated by these vector fields we can extend the symplectic coordinates (x2,. . . , xn,p2,. . . ,pn) from l? to some neighborhoods G and Q’ of the points p” and (p’)’ .m T*X. We denote these new coordinates by (2,. . ,P,&, . . . , &). Gluing the neighborhoods Q and R’ via r into a single neighborhood Q U, Q’ of the point z in 2 (see Fig. l), we obtain a symplectic chart (t, Z2, . . ,5” , H,&, . . ,&) on this neighborhood (cf. Arnol’d (1974, $43)). Thus, we have provided the open subset Z\C of the topological manifold 2 with a smooth symplectic structure, and in this structure the function H will be smooth and the vector field d/dt corresponding to it is defined on Z\C. The projection 2 + X, which arises from the projection YT : T*X + X, will be denoted by the same letter 7r in what follows. The pair consisting of the topological manifold 2 together with the smooth symplectic structure on Z\C and the vector field d/dt on Z\C will be called the classical dynamical system associated with equation (6) in the case
8Xf0.
IV. Semiclassical
Asymptotics
of Eigenfunctions
145
Fig. 1 Remark. The vector field d/dt that we have constructed may not be complete, even if X is compact. This is true, for example, for the Sinai billiard (see Fig. 2a). The projection of the trajectory corresponding to a motion with positive velocity along the line abc in Fig. 2a is tangent to dX at the point b, and the corresponding point of phase space belongs to C. For the opposite case of a billiard in a strictly convex domain on the plane Halpern (1977) proved the completeness of the vector field d/dt if the boundary of the domain is of class C3. He also gave an example of a strictly convex domain on the plane with boundary that has derivatives of order up to three (unbounded) with an incomplete field d/dt (after a finite time some trajectory reaches the set C). In the book by Kornfel’d, Sinai and Fomin (1980) there is a proof of the fact that, in the general case, if some set N of measure zero is excluded, then on the remainder of phase space there exists a uniquely determined dynamics ft : Z\N
+
Z\N,
(4 Fig. 2
t E R.
0))
146
V. F. Lazutkin
3.3. How Does One Construct Quasimodes? Since in the limit as fi + 0 a quantum problem should become a classical one, it is reasonable to expect that formulas for quasimodes can be found in terms of the classical dynamical system described in Sections 3.1 and 3.2. The analysis of a number of problems in which a satisfactory solution of the problem has been obtained leads to the deduction that quasimodes should correspond to invariant sets of the classical dynamical system. The following approach seems possible: decompose the phase space into invariant sets, on each of which the dynamical system has been constructed rather simply, and relate to each such invariant set quasimodes concentrated on the projections of the given invariant set onto X. If the invariant sets exhaust the whole phase space, then, apparently, these quasimodes will approximate the whole spectrum of the operator L. In connection with this it is useful to give the following asymptotic formula as tz + 0 for the number N(a, b) of eigenvalues (including multiplicity) of the operator L that are contained in the interval (a, b):
NC% b) N
Vol{ (2, p) E 2 : a 5 H(z,
p) < b}
(27rti)n
Here Vol denotes the Liouville volume of the domain of phase space corresponding to the volume form wn = n dxi A dpi. One can expect that the number of eigenvalues that can be approximated by quasimodes corresponding to an invariant set & c 2 is expressed by the formula
Formulas (19) and (20) are an expression of the representation, widely spread among physicists, of “quantum cells” by a volume (27rti)n in phase space, each of which contains a single “quantum state” (see Landau and Lifshits (1963)). Formula (19) is valid for a wide class of problems. A discussion of the corresponding results can be found in the survey paper by Birman and Solomyak (1977). In the process of realizing our projected program there are great difficulties, related principally to the fact that general Hamiltonian systems have not been well studied (see Lichtenberg and Lieberman (1983)). It is not known how to construct phase trajectories of a generic Hamiltonian system. The two limiting cases, differing by the opposite nature of their motion, regular and stochastic, are an integrable system and an Anosov system [Kornfel’d, Sinai, and Fomin (1980)]. Geodesic flows on Riemannian manifolds of negative curvature are examples of the latter. In Anosov systems the only integral is the integral of the energy H(x,p). On connected components of the constant energy submanifolds H(z,p) = E the system is transitive. In the integrable case the dynamical system is not complicated to construct and the motion occurs along invariant tori. The quantum problems in which the variables are separated correspond in fact to integrable dynamical systems. A general
IV.
Semiclassical
Hamiltonian system is neither manifold (Markus and Meyer We consider the integrable
Asymptotics
of Eigenfunctions
integrable nor transitive (1974)). case in more detail.
147
on a constant
energy
3.4. A Completely Integrable Hamiltonian System. Let M be a smooth 2ndimensional manifold with symplectic form w, H : M + R a smooth Hamiltonian, d/dt = #(dH) the corresponding Hamiltonian vector field. We say that the dynamical system (M, d/dt) is completely integrable on an open invariant set U c M if there exist n smooth functions Zi : 2.4 + R, 1 5 i < n, possessing the properties: 1) Zi are integrals of the system, that is, (d/dt)Zi = 0, 1 < i 5 n; 2) the Poisson brackets of the functions Z, are equal to zero, that is, {Xi,&} = w(#(dZ,), #(d&)) = 0, i, k = 1,. . ,n; 3) the differentials {d&(m), 1 < i 2 n} form a linearly independent system in TAM at each point m E M. The functions 1, are called integrals of the dynamical system. It follows from condition 3) that the level sets of the system of integrals {Zi, i = 1, . . , n}, that is, sets of the form T Cl,CZI...rCn ={mEU:
&(m)=ci,
l
are smooth ndimensional submanifolds, invariant relative to local flows generated by the vector field d/dt, and it follows from condition 2) that T,.,,c,,,,,,c, are Lagrangian manifolds, that is ndimensional submanifolds such that the symplectic form w is equal to zero on tangent vectors to it. If Tc,,,,,,c, is a closed (compact without boundary) manifold, then it is a torus (see Arnol’d (1974)). W e sh a 11 ca 11 such tori Liouville tori, and a set equal to a union of Liouville tori is called a Liouville set. The system is very simple to construct on a Liouville set. As a model we can take the following standard system. The phase space of this system is the product of an ndimensional torus Tn = Rn/Zn by a domain G c R”. The symplectic structure is defined by the standard form 0 = C d@r\dIi, where the 13~are cyclic (with period 1) coordinates on the torus Tn, I = (11, . . . , In) E G. The vector field
f =~wi(Il,...,In)~
(21)
i=l
is Hamiltonian, where wi : G + R, 1 5 i 5 n, are smooth functions, called frequencies. If the domain G is simply connected (for example, a cube), then there exists a Hamiltonian Ho(I) such that (22) The
quasiperiodic
flow fi : T”
x G + Tn
x G, defined
by the formulas
e”(t) = eg + Wi(I)& Ii(t)
= I,“,
1 5 i 5 71,
(23)
148
V. F. Lazutkin
is Hamiltonian. The phase space of the standard integrable system is foliated by invariant Lagrangian tori T” x {I}, I E G. A completely integrable Hamiltonian system on the set U is locally isomorphic to the standard one in the following sense [Arnol’d (1974), Markus and c U is an invariant torus, then there exists a cube Meyer (1974)l. If Tc,,...,c, A c R” and a smooth symplectic embedding $ : T” x A + U conjugating the fields (15) and (21), where the wi(l) are expressed by formula (22), where He = H o +. From this, in particular, it follows that the restriction of the vector field (15) to the set +(T” x A) is complete and defines a flow f” : $(T” x A) + $(T” x A) conjugate via 1c,to the quasiperiodic flow (23). The mapping $’ can be considered as a coordinate system in 2; the corresponding coordinates (0, I) are called actionangle variables. There are not many problems which admit an integration of this form. Moreover, the property of a system’s being completely integrable is not stable relative to small perturbations. A generic Hamiltonian dynamical system is not completely integrable even on some nonempty open set. The reason for this is that the standard system $, : T” x G + T” x G with a Hamiltonian Ho(I) of general form has a dense set of invariant tori with rationally dependent frequencies, that is, frequencies satisfying relations of the form Cniwi(I) = 0, w h ere the ni E Z and are not all zero. Motion along a torus with such frequencies is not ergodic, and if wi (I) = miwe, where mi E Z, then all the trajectories on the torus are periodic with period l/we. This circumstance leads to the poor solvability of the socalled homological equation (40). The vanishing of the integrals $ G o ftdt over all closed trajectories, which is difficult to satisfy, enters into the condition for the solvability of this equation.
Remark. spread opinion, modes.
The last circumstance also leads to the fact that, despite widea Liouville set is a bad basic set for the construction of quasi
3.5. Integrability on a Cantor Set. In problems connected generic Hamiltonian function it is necessary to deal with systems completely integrable on a Cantor set. We shall describe a standard which is a model of this type of integrability. We shall fix numbers u > n  1 and K > 0. We consider the set 0 consisting of all those w E Rn for which the inequality
I(m,w)I2 &
with a that are system, = flu,&
(24)
holds for all nonzero integer vectors m = (ml,. . . , m,) E Z” c R”. Here (., .) is the standard scalar product in R”, [ml = dm. For small 6 the Lebesgue measure of the set A\Q, where A is a cube in R”, is of the order of K. The set R is homeomorphic to a Cantor set. Let Ho(I) be a smooth real function on the domain G, with the property: the mapping w : G + R” defined by formula (22) is an embedding. Let Z be the inverse image of 6’
IV. Semiclassical Asymptotics
of Eigenfunctions
149
under this embedding. The standard dynamical system on a Cantor set is the restriction to T” x Z of the standard dynamical system (23). We shall also denote it by {fi, t E R}. We shall say that the system (M, d/c&) is completely integrable on a Cantor set if there exist a standard dynamical system fi : T” x Z + Tn x Z and a smooth symplectic embedding + : Tn x Z + M such that $* maps the vector field (21) to d/dt on M. In this case the set K: = +(T” x Z) will be invariant relative to the vector field d/dt, the restriction of d/dt to K will be complete, and the mapping r+G conjugates the flow fi and the flow f” on K: obtained via this process. A smooth embedding of the set Tn x Z into a manifold of dimension 2n is understood to be a mapping that can be extended to a smooth embedding of some open neighborhood of the set T” x Z into T”xR”.
The set K: = $(Tn x 1) admits a natural partition into invariant tori, which are the images of the tori Tn x {I} under $. We shall call these tori Kolmogorov tori, and their union will be called the Kolmogorov set. If a system is completely integrable on an open set, then it will not necessarily be completely integrable on a Cantor set. For the latter to happen, we need that the matrix {a2Ho/d&d1k} of partial derivatives of the Hamiltonian of the standard system be nondegenerate. In this case some (but not all) of the Liouville tori will be Kolmogorov tori. Using consequencesof the KAM theory [Poeschel (1982), Lazutkin (1981)] one can prove that, in contrast to integrability on an open set, the property that a system be completely integrable on a Cantor set is stable relative to small perturbations and that a number of interesting systems have this property, in particular a billiard on a strictly convex domain on the plane [Lazutkin (1981), Douady (1982)] and an arbitrary Hamiltonian system having a stable periodic trajectory of general elliptic type (see 55, Section 5.4). The Kolmogorov sets in systems admitting integrals on a Cantor set are the best adapted for a foundational role for the construction of quasimodes. It is even possible to claim that at this time we have no other methods of constructing quasimodes besides those methods relating quasimodes to Kolmogorov tori.
$4. Quasimodes
Corresponding
to Kolmogorov
Tori
Suppose that the Hamiltonian vector field d/dt on the phase space Z\C (Sections 3.13.2) is completely integrable in the senseof Section 3.5, that is, there exists a symplectic embedding II, : T” x Z + Z\C that conjugates the vector field (21) and the restriction of d/dt to the image of G, which is the Kolmogorov set. In this section we shall describe a construction that allows us to construct quasimodes for the operator L from these data.
150
V. F. Lazutkin
4.1. The Maslov Index and the Boundary Index. Here we define indexes that occur in the quantization conditions. Using the Riemannian metric on X and the bundle n : T*X + X, with Lagrangian fibers, the tangent space Tp(T*X) can be canonically endowed at each point p E T’X with a complex structure and a Hermitian scalar product whose imaginary part is equal to the form w [Guillemin and Sternberg (1977)]. Under the gluing at points projected onto dX described in Section 3.2, the tangent spaces together with the indicated structure at the glued points p and p’ are identified in a natural way, and we will obtain an analogous construction in the tangent spacesto the manifold Z\C. We fix a torus Tn x {I} from Tn x 1. For each 8 E Tn x {I} the tangent plane Lo to $(T” x {I}) at G(0) will be a Lagrangian subspace of T+(e)(Z\C), that is, a subspace of dimension n on whose vectors the symplectic form of Z\C is equal to zero. There exists [Guillemin and Sternberg (1977)] a unitary transformation Ue (not uniquely defined) that maps a Lagrangian subspace that is vertical relative to the projection into Le. It turns out that det’U0 does not depend on the arbitrariness in the choice of Ue and therefore defines a smooth mapping cp : T” x {I} + S1, where S1 = {< E C : ]<] = 1) is the unit circle on the complex plane. On S1 we consider the real form &/27ri<. The Muslow index [Arnol’d (1967)] of a closed oriented curve y on T” x {I} is defined by the formula
d”)[y] = q*&. I7 It is clear from the definition that the Maslov index is an integer, since it coincides with the topological degree of the mapping y + S’, where S1 is oriented via the form &/27ri<. A more intuitive definition of the Maslov index can be formulated in the case when the embedding of the torus T” x (1) is in general position relative to the projection 7r. In this casethe singular set of points on $(T” x {I}) relative to 7r is a smooth (n 1)dimensional submanifold r, embedded in $(T” x {I}) in a twosided fashion. Indeed, the set +(Tn x {I})\r is partitioned into two disjoint nonempty (if r # 8) op en subsets U+ and U corresponding to the sign of the fraction x(e) = (7r 0 ?Jl)*dzl dO1 A..
A. ‘. A dz” . A de”
’
If the curve y intersects G”(r) transversally (which can be done by making a small perturbation of y), then the Maslov index vc”)[y] can be obtained by summing the contributions of the intersections of y with $l(r), and the passageof y from U to U+ gives a contribution of +l to the sum, while the passagein the opposite direction contributes a 1. In the case when the image of the torus Tn x {I} under the mapping II, intersects I7 = 7r i (ax), the corresponding intersection index, denoted by v(a)[y], enters the asymptotic formulas. In our case the mapping $ is always
IV. Semiclassical
Asymptotics
of Eigenfunctions
151
transversal to n, since the invariant torus +(T” x {I}) contains a vector of the vector field d/c& transversal to II. The set S = $(T” x {I}) n I7 is a submanifold of dimension n  1 twosidedly embedded into $(T” x {I}). For an oriented closed curve y c T”(I) which intersects +l(S) transversally the boundary index ~(~)[y] is by defini tion equal to the sum of the contributions of the intersections of y with +l(S), and the contribution is equal to fl depending on whether $l(S) is in a positive or negative direction. The direction is positive if it coincides with the direction of the vector field d/dt.
4.2. Quantization Conditions. Estimate of the Number modes. We define two integer vectorvalued functions: yc”)
: Z + Z” and uCa) : Z +
of Quasi
Zn
by the formulas V(~)(T) = {I, 1 < i < n}, Q = M, d, where pi : R/Z + Tn x (1) is the ith basis cycle on the torus T” x {I}, defined by the equations 0, = 0,
j # i,
Oi =
t,
t
E R/Z.
We shall formulate the semiclassical quantization conditions. With the Kolmogorov set we associate a subset A of the integer lattice Zn. For this we shall fix numbers c > 0 and cy > 1. An integer point m = (ml,. . . , m,) belongs to A if and only if there exists an I E 1, for which II  m27rfi  ~&“)(I) For each point m E A we choose (25). The vector S,=
1 I,2Tmu tr.
 7rfiJa)(I)f a vector
I,
< c/i”.
E Z, satisfying
7r ‘M’(Im) 2
is called a discrepancy of the quantum conditions. 6, = o(1) uniformly in m E A as fi + 0.
(25) the inequality
 7rzP)(I,) It follows
from
(25) that
Remark 1. In view of the discontinuity of the Kolmogorov set, we cannot satisfy the quantum conditions exactly, which are usually formulated in problems with separated variables. It is necessary to introduce the discrepancy, which will enter into the later formulas for quasimodes. In the later constructions we shall assume that the Kolmogorov set K: is compact. If this is not so, then we replace Z and Z’ = In A, where d c G is a finite union of closed cubes, and we set K: = $(T” x 2’). From geometric considerations we obtain the following estimate, connecting the number of points IAl of the set A with the Liouville measure Vol(lc) of the set K: Vol( K) IAl  (2~tz)” L: const /c~+‘~&. (27)
152
V. F. Lazutkin
In (27) the constant const depends only on the mapping A, /F, and v. If the number v occurring in the definition Section 3.5) satisfies the inequality
w : G t R”, on of the set fl,,, (see
v>n2+n
(28)
and the parameter (Y is chosen in the interval ]I, (U  n)nP2[, then the righthand side of (27) has order less than fiP, which gives the possibility of asserting the validity of an estimate of the form (20) for a family of quasimodes. Remark 2. For the case of the Kolmogorov set of the geodesic flow on a compact Riemannian manifold without boundary Colin de Verdi&e (1977) proved the asymptotic equality of ]A] and V0l(lC)(27r/%~ as h + 0 without the additional restriction (28). In the definition of the set A Colin de Verdiitre assumes that a l ]0,1[. 4.3. The Maslov Canonical Operator. The general construction of the Maslov canonical operator [Maslov (1965a, 1965b), Maslov and Fedoryuk (1976), Mishchenko, Shatalov and Sternin (1978); see also the papers by Vainberg (paper II) and Fedoryuk (paper I) in this volume] contains an arbitrariness: in the choice of a canonical atlas, partition of unity, etc., which we limit to some extent for our purposes. Moreover, the presence of a discrepancy in the quantum conditions and a number of other peculiarities leads to some modification of the canonical operator itself. Here, therefore, we give a construction of the Maslov canonical operator in a form adapted to the problem under consideration. The proofs of the properties of the operator defined below are the same as in the references we have cited. We cover the Kolmogorov set K: by a finite collection of domains {.f&} diffeomorphic to the 2ndimensional open ball so that the following conditions hold: 1) Each Lagrangian submanifold of the family LF) = $(T” x {I}) n f&, I E 1, is simply connected. 2) If 0% does not intersect 111, then n(Q) is contained in the domain of definition K of some coordinate system (x1,. . . , P) in X, and the Vi do not intersect dX. In this connection there exists a partition {a%,&} of the set { 1,2, . . , n} into two classes such that each submanifold Lf), 1 E 1, is projected diffeomorphically onto the Lagrangian subspace {z?, Ic E oi, ~1, 1 E Zi} along {z’, k E i&, pl, 1 E CY~} in the natural symplectic trivialization of T*X n F’(V~) associated with the coordinate system (x1,. . , z:“). We shall fix this partition {ai,&} and the coordinate system {~(~~),p(,~)} These coordinates will be called focal coordinates. 3) If fii has a nonempty intersection with fl, then ~(0%) is contained domain of definition Vi of the coordinate system (x1,. . . , z”) on X, equation of 8X in these coordinates has the form x1 = 0 and x1 2 points of Vi. Here there exists a partition {cri, &} of the set {1,2,. .
on Lp). in and 0 at , n}
the the the into
IV. Semiclassical
Asymptotics
of Eigenfunctions
153
two classes with properties analogous to those considered in the previous case. We require in addition Here we shall also fix focal coordinates {z (“t),p(,%)}. that 1 E ai. It is not hard to satisfy this last requirement, since the projection of the tangent space to Ly) contains the vector n,(d/dt) transversal to dX. In this case we shall assume that 0% is divided by the surface J7 into two nonempty simply connected pieces 0: and fi:, and the vector field d/dt is directed from tic7 into 0%’ at each point of Q fl 17. The focal coordinates {z(“~),p(~~)} are smooth in 0%: and Q+ separately. 4) Suppose
Lf)
# 0. We denote focal coordinates in Lp) by 0 ne can prove that the difin L,(‘I by {Ic(“~),lj(cy3~}. of the symmetric matrices {%i?“/@l, Ic, 1 E Zj} and
fJ Lp)
{z(“t),p(,%)} and those ference of the signatures
{&r:“/~Ypl, Ic,l E @} is locally constant on Ly) 0 Ly’. (The signature sign A of a symmetric matrix A is equal to the number of positive eigenvalues minus the number of negative eigenvalues.) We shall assume that the domains {Qi} are so small that this difference is constant on .(2i n f2j n K. We set k
s(f&,
fZj) = sign
C dz
k,l
.
EziTj
35 ’
(29)
We note that in correspondence with the reflection formulas (18) the righthand side of (29) is the same for the pairs fit+, 6’; and Q,T, C 3 in the case when Qi n II # 0 and C?, n 17 # 0. For each triple {Q, fIj,l}, where C&, flj are members of the covering and 1 is a smooth path lying entirely within $J(T” x {I}), starting in fii and ending in Gj, we define an integer [Qi, fij, l] in the following way. We choose a sequence {Q, , 1 5 IC 5 s} from the cover, covering 1 such that ie = i, i, = j and Mini, n flik+l n 1 # 0. We set sl [G, flj, ll = C
s(.ni,)
f&k+1 1.
(30)
k=l
One can prove that the number [fi2,, 6’,, 11 does not depend on the choice of the sequence of domains {&}. The Maslov index of a closed path y on the torus T” x (1) is connected with (30) by the relation 2v(“)[y] = [Go, fle, 71, where 0, is an arbitrary member of the cover that has nonempty intersection with y (see Mishchenko, Shatalov, and Sternin (1978)). Let 1 be a smooth path whose beginning and end do not belong to n. The symbol [l, n] denotes the integer intersection index of 1 with the submanifold l7: If the path 1 is transversal to 17, then [I, n] = c fl, where the sum is taken over all points of the intersection of 1with II, where the “+” sign corresponds to passing from one to the other in the direction equal to the direction of d/d& and the “” sign corresponds to the opposite direction. In Z\C we fix a “partition of unity” {ei} for K that is subordinate to the cover {Q}, that is, a system of smooth nonnegative functions subject to the
V. F. Lazutkin
154
conditions: for all i, supp ei c K&, C ei[lC = 1. We also construct a system of smooth functions {Ei} on X which has the properties: 1) supp & C Vi, where {Vi} are the domains of definition of coordinate systems in X that occur in the preceding constructions; 2) Ei(z) = 1 for 5 E 7r( Gi). We fix a family of points ze(1) E +(T” x {I})\n, which depends smoothly on a parameter I E 1. For each domain Ri that is disjoint from II we distinguish a family zi(I) E $(T” x {I}) n Ri; f or a domain K& that intersects II we distinguish two families of points z’(1) E $(T” x {I}) I (@\n) (see Condition 3) in the definition of the Qi). We fix families of paths Zi (I) and 1’ (I), lying wholly on $(T” x {I}), starting at ze(l) and ending at &(I) and z’(I) respectively. For a variable point z E Lf) = +(Tn path joining .ze(l) and z, lying wholly
x {I}) in $(T”
n
Gi we denote by Zi(I < 2) a {I}) and obtained from li(1)
x
or l’(1) by adding a piece lying wholly in Ly). After these preparations we can write down a formula for the Maslov canonical operator. We introduce the spaces in which this operator will act. As usual, the symbol C”(X) denotes the space of smooth complexvalued functions on X, C”(T” x Z, n) denotes the space of complexvalued functions on T” x Z that are smooth on the set Tn x Z\$‘(n), and it is assumed that all the derivatives of all of these functions have smooth limit values from each side on the set T” x Z n tip1 (n). The symbol C’, (Tn x Z, n) denotes the subspace of C(Tn x Z, n) consisting of the continuous functions. For I E Z and b E R” the canonical operator K = K1,6 : C==(R” is defined
x 1,n)
+ C”(X)
by the formula
(31) where
in case Qi
n II = 0
&((p)(2)= ,iq[no,n,,l,lia[l,,n]1 m d(@,. . )en) l/2 x 3ll/fi,p(,i)+z(“%) (P(~%+),P(,,))) m(zt),
x
(32)
z(9
i
exp {
i
[.i
1 (I ) (P’, dz’) z,z
 (wq
( J%(cy,)),P(Z,))
]
 (
6, L
,@}
.
In (32) we assume that supp 4 o $’ c LF), and on this set we use the focal coordinates (~(“t),p(~~)). Th e point z in (32) has coordinates (z(“%),p(,%)), where ~(~2) are the corresponding coordinates of the point 2. The symbol denotes the inverse l/&Fourier transform with respect to p(z): 3l l,tL,p~,)+z(a)
IV. Semiclassical
Asymptotics
of Eigenfunctions
155
In case C& n 17 # 0 the formula for &(c$) has the form of a sum of two terms of the shape (32), corresponding to the domains @ and tic, in which we have 1’ in place of li, and instead of cp o $’ we have the restriction of this function to the set $(T” x {I}) n 0:. Let ‘p E Ccw(Tn x Z, n). Then for z E dX the two terms described above that correspond to Q,’ and 0%: cancel out in view of the fact that they differ only by sign: [lt, n] = [Zi, n] + 1 (mod 2). The terms in (31) corresponding to the domains Qi that do not intersect U are equal to zero on dX in view of the properties of gi(x). Therefore the righthand side of (31) satisfies the Dirichlet boundary condition (3) if we just have cp E CF(T” x 1, II). It follows from the definition of the discrepancy (26) that for S = 6,, I = 1, the expression in the righthand side of (31) does not depend on the arbitrariness in the choice of the paths li, 1’. From the remaining elements of the construction (the functions ei and &) the dependence is all that, remains, but this dependence only manifests itself in the terms following the principal terms of the asymptotic expansion in powers of fL. A commutation formula holds for the canonical operator and the operator L; in our case this formula has the following form. There exists a sequence {&}~=e of linear differential operators Rk:
CDO(TnxZ,~),COO(TnxZ,17)
of order k with coefficients belonging to C(T” x 1, II) such that S = 6, with error estimated in the Lz(X) norm, for any natural uniformly in m E /l
for I = I,, number N
N LKp
=
C(ifL)kRkei(6.0)p
Kei(‘>‘)
+
o(fiNfl).
(33)
k=O
Ro is the operator of multiplication by the scalar function He(l), =  CT=z=,wi(I)b/86Ji. W e note that the coefficients of the operator Rk are defined for all I E Z and do not depend on 6. The commutation
In particular,
RI = d/dt
formula (33), however, holds only for values of I and b that satisfy the quantization condition (25),(26), that is, for I = I,, S = 6, for some m E A. The operator Rk contains differentiations only with respect to the variables @I,. . . ) P). 4.4. follows
Formulas
I = I,,
for
Quasimodes.
We fix m E A. In the formulas
that
6 = 6,. We set
%n(z) = c,K
[
j&%)*$&y k=O
N
Em = j+h)k&, k=O
1 ,
(34)
(35)
156
V. F. Lazutkin
where xk = &(I, 6) are polynomials in 6 with coefficients from c(z), pk = (Pk(e, I,@ are polynomials in 6 with coefficients from Cca(Tn x 1, n), and c, is a normalization constant. Applying the commutation formula (33) to (34) and using (35), we will find that (L  E,)um
= c,Ke+‘) +
&ih)‘l n=o
{ c
(&  **)iiops}
k+s=n
(36)
O(C,hN+l).
We shall require that the expressions in the curly brackets in (36) vanish. Then (u,, E,) will be a quasimode with remainder E = O(c,tiN+l). For n = 0 we obtain the equation
[Ho(I)  Xo(QlPo= 0. We set cpo= 1, The equation for
n
X0(I) = ffo(I).
(37)
= 1 has the form eo)
= 0,
from which we will find that Xl = i(S, w(I)). Considering (37) and (38) and cancelling by ei(sle), the equation for can be reduced to the form
(38) n 2
2
(39) Equation (39) is the socalled homological equation. Lemma
(on the solution of the homological equation). Consider the
equation &=g:
(40)
where g E C”(T” x Z, II). For equation (40) to have a solution cpE Ccw(Tn x 1, II) it is necessary and suficient that for each I E Z the mean value of g over the torus Tn x {I} be equal to zero, that is, g(0, I)d& . . . d8, = 0 for all I E Z. s T”XZ
(41)
The general solution of equation (40) is obtained by adding an arbitrary smooth function of the variable I E Z to some partial solution.
IV. Semiclassical Asymptotics
of Eigenfunctions
157
The assertion of the lemma is particularly simple to obtain in caseK: n 17 = 8. In this case a partial solution is expressed by the formula
where gm(l) are the Fourier coefficients of the function g. From condition (41) it follows that go(I) = 0. The denominators in the remaining terms of the sum (42) are estimated via (24). The proof in the general case is different only in technical details. Applying the lemma to equation (39), we obtain as a consequence of the solvability condition (41) the following expression for X,: A,
=
s [
&wR
T”x{I}
xd&...d&.
n2 n
,a@
+
c j=l
,w4~,j
+
~,.)ew9pj
1 (43)
Solving the system of equations (39) successively, we will obtain (Pi, n = 1 . . >N  1, and X,, n = 2,. . . , N. Here it turns out that the X, are polynomials in S of degree 2n  2 and the (Pi are polynomials of degree 2n. 4.5. Choice of a Normalization Constant and Behavior of a Quasimode in a Neighborhood of a Point That Is Regular Relative to Projection. The constant c, in (34) is chosen from condition (12). Here it is important for us to see that c, = O(1) as fL 4 0 uniformly for m E A. In order to obtain this estimate we impose the following nondegeneracy condition relative to projection on the Kolmogorov set Ic: (*) for each 1 E Z there exists a point zr* = z*(I), belonging to X\aX and which is a regular value of the projection ~1 = 7rI+(Tn x {I}) of the torus $(T” x {I}) onto X. In this case there exists a simply connected open neighborhood V of the point 5*, consisting entirely of regular values of the mapping ~1. The inverse image of the neighborhood V relative to ~1 is a finite union of pairwise disjoint simply connected open sets L%‘j c $(T” x {I}). For z E V, I = Im, the expression (34) for u,(z) can be transformed to the form
by the method of stationary phase, where &(z) is a path lying entirely in Wj starting at .rrl’(z*) and ending at 7r11(z), and the Aj(x) are smooth functions satisfying the estimate IAj(x)I > const > 0, where the constant const does not depend on fi (we have considered the value (37) for cpc). We see that in a neighborhood of a point z* E X that is regular relative to projection a quasimode looks like the sum of a finite number of plane waves.
V. F. Lazutkin
158
The neighborhood V and the constant const are suitable not only for the given Im, but for any I,, belonging to some neighborhood of I, in 1. Multiplying the square of the modulus of the righthand side of (44) by a smooth cutoff function concentrated in the neighborhood V, integrating and applying the stationary phase method to the resulting expression, it is not hard to obtain the estimate 1 = ]]u~]]~ > ]c,12 const, where const does not depend on fL and on m E A. 4.6. Orthogonality of Quasimodes. Estimate of the Total Multiplicity of the Approximating Spectrum. Let N > 4. One can prove [Lazutkin (1981)] that the quasimodes that have been constructed satisfy a condition of approximate orthogonality: for m, 1 E A and m # 1 I(%, ul)l 5 const AT]l.
(45)
We note that in the proof of the estimate (45) essential use is made of the fact that the family of Kolmogorov tori depends smoothly on the vector of frequencies w (transversal smoothness). The results of the constructions of this section, taking Theorems 1 and 2 of Section 2 into consideration, can be formulated in the following theorem. Theorem 3. Suppose that the classical dynamical system associated with equation (6) has compact Kolmogorov set Ic, for which the nondegeneracy condition (*) relative to projection holds. Then there exist two sequences of funcsuch that the family of quasimodes {u,, E,, m E A} tions ‘ok (0, I, 61, h (I, 6) with remainder O(hN+’ ) and deviation from orthogonality
is defined by formulas (34) and (35) f or every N 2 4. The set A is defined from the quantization conditions (25) with (Y > 1, and the functions (PO,& and X1 are defined by formulas (37) and (38). The distance from any number E,, m E A, to the spectrum of the operator L is of order O(hN+‘). Suppose that the parameter u in the de5nition of the Kolmogorov set satisfies inequality (28), th e number (Y in the definition of the set A is chosen from the interval 11, (V  n)nP2[, N > 2n + 2, p < N + 1  n, p = I?‘, EP = Umen[Em  pu, E, + ~1. Then the total multiplicity of the spectrum of the operator L in the set &P, approximated by the family of quasimodes with the indicated parameters constructed from the Kolmogorov set K, is equal to
N*(&,)
Vol(lc) = (2~~i)n + 0
hn+lag
>
.
(46)
IV. Semiclassical
and Some
Asymptotics
of Eigenfunctions
159
$5. History of the Problem of the Problems That Have Been Studied
5.1. History of the Problem. The quantization conditions from which the first approximations for the eigenvalues are determined were formulated by physicists in the first stage of the development of quantum mechanics, before the problem was precisely stated. The basic equation of quantum mechanics (1) was written down by Schrijdinger in 1926 while the quantization conditions in their simplest form were proposed by Niels Bohr in 1913 in order to explain the spectrum of the hydrogen atom. Bohr started from the fact that all the trajectories of a particle in a Coulomb field are periodic, and he introduced one quantum number by equating the angular momentum to an integer multiple of 27rFi. In 1916 these conditions were generalized by Sommerfeld and Epstein to the case when the HamiltonJacobi equation admits a separation of variables. The number of quantum conditions was increased to the number of degrees of freedom. In an almost contemporary form, but without indices, the quantization conditions for Liouville tori were formulated by Albert Einstein in 1917 [Einstein (1917)]. Keller (see Berry (1983), Pankratova (1986)) included halfinteger corrections in the quantization conditions, taking the index into account. A number of problems with separated variables in which the quantization conditions contain these corrections were analyzed by Keller and Rubinow [Keller and Rubinow (1960)]. A global construction for the asymptotics of equations of solutions of equations of SchrGdinger type appeared in 1965 in the work of Maslov [Maslov (1965a, 1965b)]. In particular, he introduced an index which now bears the name of the Maslov index. The geometric meaning of the Maslov index was studied by Arnol’d [Arnol’d (1967)]. Q uasimodes associated to Kolmogorov tori and estimates for the total multiplicity of the approximating spectrum in the case n. = 2 were obtained by the author [Lazutkin (1973, 1981)]. Krakhnov [Krakhnov (1975a)] considered qussimodes corresponding to an individual Kolmogorov torus in the ndimensional case. The construction of quasimodes corresponding to a Kolmogorov set of positive measure in the higherdimensional case is given in a paper by Colin de VerdGre [Colin de Verdi&e (1977)]. 5.2. Problems with Separated Variables. A number of problems associated with an equation of the type (6) admit a separation of the variables, and, thus, their study reduces to the study of solutions of ordinary differential equations. Among the quantummechanical problems of this type are the problems of the spectrum of a hydrogenlike atom and the spectrum of a linear oscillator [Landau and Lifshits (1963)]. A mong other problems Keller and Rubinow (1960) studied the asymptotics of the eigenvibrations of an elliptical membrane (that is, eigenfunctions of the Laplace operator in an ellipse with a Dirichlet condition on the boundary). They found that two types of vibrations
160
V. F. Lazutkin
0))
(4 Fig. 3
are possible in an ellipse: “whispering gallery” type (Fig. 3a) and “bouncing ball” type (Fig. 3b). The corresponding eigenfunctions oscillate in a domain contained in the first case between the boundary and a confocal ellipse, and in the second case between the branches of a confocal hyperbola, and they decrease exponentially as the ordinal number of the eigenvalue increases outside the indicated domains. In a disk there are only vibrations of whispering gallery type. A detailed presentation of these results can be found in the monograph by Babich and Buldyrev (1972). A n intermediate type of vibrations in an ellipse, bounded between vibrations of bouncing ball type and whispering gallery type, was studied by Osmolovskij (1974). Bykov (1965) and Vainshtein (1965) considered the problem of vibrations inside a threedimensional ellipsoid. In this case there are four types of vibrations with different geometry of the domain of concentration.
5.3. Small Perturbations
of Problems
with
Separated
Variables.
Problems with separated variables correspond to a classical dynamical system that is completely integrable. Under small perturbations of the problem the classical dynamical system is also subjected to a small perturbation. If the perturbation is of order E, then it follows from KAM theory [Arnol’d, Kozlov, and Nejshtadt (1985)] that in the phase space of the perturbed classical system there exists a Kolmogorov set Ic, the measure of whose complement is of order &. The construction of Section 4 allows us to construct quasimodes with respect to Ic. As follows from the presentation given above (see §3.3), the part formed by the approximated eigenvalues in relation to the set of all eigenvalues for these problems has order 1  O(&. As an example we consider the problem of the construction of quasimodes for the LaplaceBeltrami operator on a twodimensional torus with a metric that is slightly different from the Euclidean metric. Let X = R2/Z2 be the twodimensional torus obtained from the plane R2 by identifying the points whose coordinates differ by integers. On X we consider the Riemannian metric coordinates defined by the expression (4), where n = 2, zl, x2 are Cartesian on R2 and gik(x
1
>2
2
) =
&k
+
‘%k(xl
,x2),
i,k = 1,2,
IV. Semiclassical
Asymptotics
of Eigenfunctions
161
f is a small positive parameter, {uik(zl, x2)} is a symmetric matrix whose elements are smooth real functions, periodic of period 1 in z1 and x2, and 6,k is the Kronecker symbol. For t = 0 the geodesic flow on X generated by this metric is completely integrable. The phase space T*X = (R2/Z2) x R2 is foliated by invariant tori (pi,px) = const E R2, uniquely projected onto X = R2/Z2. There are explicit formulas for the eigenfunctions and eigenvalues: u(o) + m2x2)}, wL~?n* = exp{i2n(mrz1 x(O) m1m2 = 47r2(mf + 77x3, (rnl,rn2)
(47)
E z2. I
For small t, according to KAM theory, the phase space T*X contains a large set of Kolmogorov tori, slightly different from the tori (pi,pz) = const. Because of the homogeneity in p of the Hamiltonian H(z,p) = 1 su ces to consider the classical dynamical system on ; c:&=l si”(+w ‘t fi the constant energy submanifold H(z,p) = i. The equations mentioned above for Kolmogorov tori lying on this submanifold can be written in the form p1 =p1(x1,x2$)
=cose+o(J;),
p2 =p2(x1,x2,H)
= sinQ+O(J;).
(48) I
Here 8 is a parameter that runs through some Cantor set 0 on the circle S1 = R/27rZ, whose Lebesgue measure satisfies the estimate Mes 0 > 2rconstfi. The dependence of the functions (48) and other functions on t is not explicitly indicated in the notations. We introduce the phase function
S(xl, x2, a) =
(2’2) I (04)
pl(xl, x2, @ix1 + p&l, x2)dx2.
(49)
As a function on the torus, S will not be singlevalued or, in other words, the righthand side of (49) is not a periodic function of (z’, x2). The following relations hold: S(x1+n1,x2+n2,0)
= S(x1,x2,B)+n1P1(B)+n2P2(t3)
for every
(n1,n2)
E Z2,
where Pi(Q), i = 1,2, are smooth functions of the variable 6’ E 0 that admit estimates of the form (48) for small E. We fix a natural number M, a number CY~10, :U  l[, where v > 3 is the parameter in the definition (24) of the set R,,, for a given Kolmogorov set, and a positive number K. We define an infinite subset A of the twodimensional integer lattice Z2 in the following way: (ml, m2) E A if and only if there exists a 19= 8(mi, m2) E 0 for which
lmlPz(O) 
mzP1(6)1
162
V. F. Lazutkin
The
formulas
Here
(mr,mz)
for the quasimodes
are the following:
runs over the set A and 0 = B(mr,mz),
@0(x1,x2, e) = S(xl, x2, e) 
P@)xl

P2(8)x2,
@k and ok, 1 5 k 5 M + 1, are polynomials in the variables ,Bi, i = 1,2, of degree at most Ic with coefficients that depend smoothly on the remaining variables, and the @k are periodic functions of z1 and x2 of period 1. They are defined successively as solutions of homological equations of the form (39). From the estimates (48) it is not hard to see that the expressions (50) are different from the corresponding expressions (47) by correction terms that tend to zero as E + 0 with speed 0(,/Z). Theorem 4. 1) For a suitable pressions (50) satisfy the following E ~10, el[, where tl is a suficiently
+ Llmzwn,m,
l142Gn1m2
choice of the functions @k and ak the exestimates uniformly in (ml, mz) E A and small positive number:
II&(X)
lbmm2 II&q 2)
If
(ml,4
I(k,,,
#
(mi,mL),
= 0
((J&Z").
= O(l).
then
, urn; ,m;) I I const min
1 ml i rnz’
(mi)2
: (mb)2 1 .
3) Let N(X) be th e number of eigenvalues, including multiplicities, of the operator A2 that do not exceed A, and let N*(X) be the number of such eigenvalues that are approximated by the numbers A,,,,, , (ml, m2) E A, in the sense of Section 2. Then for X > 0 we have
IV. Semiclassical
Asymptotics
N*(X)/N(X) The constants
in all these estimates
of Eigenfunctions
163
> 1  constJ;. depend
only on {Q(X)},
(51) M,
K and v.
It follows from the statement of the theorem that for small E the majority of the eigenvalues and eigenfunctions are only slightly different from the eigenvalues and eigenfunctions (47) of the unperturbed problem. For eigenfunctions it is of course necessary to make the restriction that nearby eigenfunctions and some finite linear combinations of eigenfunctions with very close eigenvalues are not the same. Estimate (51) shows that the part of the unapproximated asymptotics of eigenfunctions that we have constructed is O(G). This estimate cannot be improved, since in the gaps between the Kolmogorov tori under consideration (uniquely projected onto X) there are domains with phase volume (on the submanifold H = l/2) of order 4, partially filled with Kolmogorov tori of another type, projected onto X with singularities [Arnol’d, Kozlov, and Nejshtadt (1985)]. W e may assume that the consideration of all the Kolmogorov tori allows us to approximate a larger number of eigenfunctions, and here, in the case of a realanalytic perturbation of the Euclidean metric, we can replace fi by exp(constfi), const > 0, in the estimate (51). We note that the analogous problem for the case of the wave equation was studied by Lazutkin (1974), and for the LaplaceBeltrami operator by Maslov and Fedoryuk (1976) and by Mishchenko, Shatalov, and Sternin (1978). Svanidze (1980) extended the results presented in this subsection to higherdimensional tori. Lazutkin (1981) f ormulated analogous results for the Laplace operator with a Dirichlet condition in a plane domain that is slightly different from a disk. 5.4. A Neighborhood of a Closed Trajectory. Let 1 be a closed trajectory of a classical dynamical system, lying on the constant energy submanifold H(x,p) = E. We consider a (2n  2)dimensional area element S, transversal to 1, lying in this constant energy submanifold and having a single point of intersection ~0 with 1. The trajectories of the system that issue from any point .Z E S that is sufficiently close to ~0 again intersect S. We denote by f : U + S, where U is a sufficiently small neighborhood of ze in S, the mapping that associates to a point z E U the point f(z) of first intersection of this trajectory with the area element (see Fig. 4). The set S is called the surface of section and the mapping f is called the Poincare’ map. The restriction of the symplectic form w to S turns S into a symplectic manifold, and the Poincare map turns out to be a symplectic diffeomorphism with fixed point ~0. The trajectory 1 is said to be elliptic if all the eigenvalues of the complexification of the linear part T,,f : T,,S + T,,S of the mapping f at ~0 lie on the unit circle in the complex plane and are not real. In this case they have the form X: = e*iak, 0 < ok < YT, 1 5 /C < n  1. If these eigenvalues satisfy the noresonance condition
V. F. Lazutkin
164
Fig. 4
At1
....X>pm:
# 1 for ]lcr] +...+ 0 #
(h,.
. .
) l&l)
]I+11 E zl,
then in a neighborhood of zo the transformation Birkhoff normal form (see Arnol’d (1974, Appendix Pp via a symplectic change of variables:
f can be reduced to the 7)) up to terms of order
(cp> T) ++ (‘p + TbL(~L 7)
(cp,~) E R”l/Z+’
x [O,blnl,
p = Ji;;i,
where Q(T) is a polynomial of degree at most [p/2]  1 and 4(O) = (or, . . , onr). A closed trajectory is called a closed trajectory of general elliptic type if it is elliptic, the noresonance condition holds for some p > 4 and g’(O) is an invertible matrix. Applying KAM theory, it is not hard to prove the existence of a Kolmogorov set lying in a neighborhood of 1, whose density tends to 1 faster as p increases. Lazutkin (1981) gave a detailed presentation of these questions for the case n = 2. Any compact piece of this Kolmogorov set can be taken as a basis for the construction of quasimodes. We will obtain a family of quasimodes corresponding to a closed trajectory of general elliptic type. We point out the following characteristics of these quasimodes: 1) they are concentrated in a neighborhood of the projection of the trajectory 1 onto X; 2) among the quantum numbers there is a distinguished “large” one m,, corresponding to the coordinate varying along 1, while the remaining ones ml,... ,m,1 are “small”; the latter, however, vary between the bounds qfil 2 rni 5 ezfi‘, 1 5 i < n  1, where ~1 and ~2 are small constants. Using other methods it is possible to obtain quasimodes corresponding in the limit to narrow Kolmogorov tori close to 1, for which the quantum numbers rni,l 5 i < n  1, are suitably small, that is, which satisfy the inequalities
IV. Semiclassical Asymptotics
of Eigenfunctions
165
0 5 rni 5 const (see Babich and Buldyrev (1972), and Babich (1988)paper III of this volume). Lazutkin and Terman (1981) showed in a special case that the formulas for the eigenvalues obtained by these methods retain their asymptotic character if 0 5 mi 5 constfi“, where Q is any fixed number belonging to the interval [0, l[. Krakhnov (197513) considered the problem of the construction of quasimodes corresponding to conditionally periodic motion on an sdimensional torus, where 1 5 s < n. 5.5. Quasimodes Concentrated Close to the Boundary of a Domain. Let X be a strictly convex domain in R2 with smooth boundary dX. We consider a billiard in X, that is, a classical dynamical system in X in the senseof Section 3 generated by the Hamiltonian H(z,p) = i ]p12.In order to study this system it is convenient to pass to a dynamical system with discrete time, taking as surface of section a set S equal to the intersection of the manifold n = r‘(aX) (see $3.2) with the constant energy submanifold H(x,p) = ;. Th e surface S is homeomorphic to the cylinder S1 x [0, I] and has a twocomponent boundary r+ U r = Sn Z. The trajectories close to r+ correspond to rays that extend along dX (see Fig. 5a) and go in the positive direction. The projections of the trajectories close to R go in the opposite direction. Since the system is symmetric relative to a transformation consisting of a change of sign in the momenta, the behavior of trajectories close to r is similar to the trajectories that close to them, close to r+. Therefore in what follows we shall restrict consideration to a small neighborhood of r+. The classical dynamical system cuts out the symplectic diffeomorphism f (the Poincare map) on S. Using the Euclidean metric we identify the tangent and cotangent bundles. Under this identification the points on the surface S are nothing but the vectors of unit length indicating the interior of X, and the points of r+ are the unit tangent vectors to dX indicating the positive direction. On S we introduce the coordinate system (s, 0), which associates to a unit vector t at a point z E dX a number s equal to the length of the arc dX, counting from some fixed point on dX in a positive direction up to the point x, and an angle 19,0 5 19< n, formed by the tangent at x and the direction of the vector t (see Fig. 5a). In a neighborhood of r+ we pass to the coordinates (2, y) according to the formulas
x = c1 sP2’3(s’)ds’,y = s
~c$/~(s)
sin i,
0
where c = s,” pe213(s’)ds’, L: is the length of dX, p(s) is the radius of curvature of dX at the point with coordinate s. In the coordinates (z, y) the mapping f has the form (5,~) H (xi, yr), where xl =x+y+O(y3)
(mod l),
yi =y+O(y4),
and preserves the form w = ydx A dy, which degenerates at the points of r+. We can apply KAM theory to this transformation, considering the terms
V. F.
166
Lazutkin
(4
(b) Fig.
O(Y~)and WY”) as small
5
perturbations, and we can assert the existence of a family of Kolmogorov tori close to r+. The intersections of these tori with S form a system of invariant curves relative to f, close to y = const, a system homeomorphic to the product of a Cantor set by a circle. The density of the system of invariant curves relative to Lebesgue measure tends rapidly to 1 under approximation to r+. The projection onto X of the invariant torus T2 x {I} belongong to this Kolmogorov set has the shape of an annulus bounded by dX and a convex curve CT called a caustic (see Fig. 5b). The projections of the trajectories of the classical dynamical system lying on T2 x {I} are segments of the lines tangent to the caustics Cl. A caustic is characterized by the fact that the segment abc of any tangent to it, after reflection from the boundary according to the law “the angle of incidence equals the angle of reflection”, is mapped to another segment cde that is also tangent to the caustic. In an ellipse the caustics form a family that depends continuously on one parameter, which runs over the interval (see Fig. 3a). In an arbitrary strictly convex domain with a sufficiently smooth boundary there is a family of caustics CI which depend continuously on the parameter I, running through some Cantor set 2. Details of this can be found in the book by Lazutkin (1981). The construction of $4 allows to associate to any compact piece of the resulting Kolmogorov set a system of quasimodes which, as it turns out, oscillate in a narrow strip around dX and are exponentially small in i? outside this strip. The nondegeneracy condition (*) is obviously valid in this case. In other words, without using KAM theory and the canonical operator, it is possible to construct quasimodes concentrated close to dX that oscillate in a strip of width of order fi2/3 (see Babich and Buldyrev (1972) and Babich (1988)). The part of the latter in comparison with all the quasimodes constructed above from the Kolmogorov set tends to zero as fi + 0. Babich and Buldyrev (1972) considered the case of the wave equation with variable velocity c(x). It is required that the boundary be geodesically convex.
IV. Semiclassical
Asymptotics
of Eigenfunctions
167
The author knows of two papers containing analogous results in dimensions n > 2. Svanidze (1978) considered a neighborhood of a closed geodesic 1 on the boundary dX of a threedimensional convex domain X. In the case when the corresponding periodic trajectory of the geodesic flow in dX is of general elliptic type, he proved the existence of a Kolmogorov set for a billiard in X. Here he required some additional genericity of the geodesic relative to the domain X. The Kolmogorov set is contained in a small neighborhood in 2 of a closed trajectory of the geodesic flow on dX, projected onto 1. For a domain X c Rn+’ Krakhnov (1982) constructed quasimodes concentrated in a neighborhood of dX of order fi2i3, taking as “ray foundation” an ndimensional Kolmogorov torus of the geodesic flow on C = T*BX. 5.6. Quasimodes Corresponding to the Complement of Kolmogorov Tori. As was already stated, under small perturbations of a nondegenerate completely integrable system the classical dynamical system has a Kolmogorov set filling up all of phase space except for a set of small measure [Arnol’d, Kozlov, and Nejshtadt (1985)]. Little is actually known at the present time about the structure of this additional set. It is not excluded that this set may contain ergodic components of positive measure (of (2n  1)dimensional measure on a constant energy manifold). In any case the following question arises: How can one construct quasimodes corresponding to a chaotic motion of a classical dynamical system if this is possible? At the present time the author does not know of a satisfactory answer to this question. There are a number of problems in which the classical system is completely ergodic, or, more precisely, is ergodic on every constant energy manifold. Included among such systems are the geodesic flows on manifolds of negative curvature, mentioned in 53, and some billiards, in particular, the ‘Sinai billiard” [Sinai (1970)], a billiard in the domain on the Euclidean plane obtained by removing a strictly convex domain, for example, a disk, from the unit square (see Fig. 2a), or, in another variant, a billiard on the torus with a flat metric from which a convex domain has been removed, and also the “stadium”, a billiard in the domain on the Euclidean plane (see Fig. 2b) formed from the rectangle abed to which two semidisks amb and dmc have been adjoined, of diameters equal to the lengths of the sides ab and cd [Bunimovich (1979)]. The corresponding quantum problem in these examples is the problem of the spectrum of the LaplaceBeltrami operator with Dirichlet boundary conditions. How does one construct eigenfunctions and what are the eigenvalues for these problems? Shnirel’man (1974) showed that eigenfunctions for problems with an ergodic dynamical system on a constant energy manifold are in a definite sense uniformly “spread” with respect to the corresponding eigenvalue of the energy hypersurface. The eigenvalues XI, of the LaplaceBeltrami operator on a compact boundaryless surface of constant negative curvature can be computed from the formula &  sk(l  Sk) in terms of the zeros Sk of the Selberg zeta function (see Venkov, Kalinin and Faddeev (1973), Hejhal (1976, 1983)):
168
V. F. Lazutkin
Z(s)
= n
fi v
(1
(&~~)
.
n=O
The outer product here extends over all simple closed geodesics on the surface and the {Zv} are the lengths of the geodesics. The eigenvalues of the Laplacian with Dirichlet boundary condition for the Sinai billiard was studied numerically by Berry (1981). The paper by McDonald and Kaufman (1979) contains an interesting picture of the nodes of an eigenfunction of the Laplacian with Dirichlet boundary condition for the “stadium” billiard, demonstrating the chaotic nature of the distribution of the wave vector. We complete our brief description of this topic by referring to a more complete survey article [Berry (1983)].
References* Arnol’d, V. I. (1967): A characteristic class entering into quantization conditions. Funkts. Anal. Prilzhen. 1, no. 1, l14 [English translation: Funct. Anal. Appl. 1, no. 1, l13 (1967)] Zbl. 175,203 Arnol’d, V. I. (1972): Modes and quasimodes. Funkts. Anal. Prilzhen. 6, no. 2, 1220 [English translation: Funct. Anal. Appl. 6, no. 2, 94101 (1972)] Zbl. 251.70012 Arnol’d, V. I. (1974): Mathematical Methods of Classical Mechanics. Nauka, Moscow, 431 pp. [English translation: Springer, Berlin (1978)] Zbl. 386.70001, Zbl. 647.70001 Arnol’d, V. I., Kozlov, V. V., and Nejshtadt, A. I. (1985): Dynamical Systems. III: Mathematical aspects of classical and celestial mechanics, in: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 3, 5304 [English translation: Encycl. Math. Sci. 3, 1291, Springer, Berlin (1988)] Zbl. 612.70002 Akhiezer, N. I., and Glazman, I. M. (1966): The Theory of Linear Operators in Hilbert Space. 2nd edition, Nauka, Moscow, 544 pp. Zbl. 143,365 Babich, V. M. (1988): The multidimensional WKB method or ray method. Its analogues and generalizations, in: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 34, 93134 [English translation: Encycl. Math. Sci. 34, Springer, Berlin (199x)] (Paper III of this volume) Zbl. 657.35119 Babich, V. M., and Buldyrev, V. S. (1972): Asymptotic Methods in ShortWave Diffraction Problems. Nauka, Moscow, 456 pp. [English translation: Springer Series on Wave Phenomena 4. Springer, Berlin Heidelberg New York (1991)] Zbl. 255.35002 Berezin, F. A., and Shubin, M. A. (1983): The SchrGdinger Equation. Izdat. MGU (Moscow State University Press), Moscow, 392 pp. [English translation: Math. Appl., Sov. Ser. 66, 555 pp., (1991)] Zbl. 546.35002 Berry, M. V. (1981): Quantizing a classically ergodic system: Sinai’s billiard and the KKR method. Ann. Phys. 131, 163216 * For the convenience of the reader, references to reviews in Zentralblatt fiir Mathematik (Zbl.), compiled using the MATH database, and Jahrbuch iiber die Fortschritte der Mathematik (Jbuch) have, as far as possible, been included in this bibliography
IV. Semiclassical
Asymptotics
of Eigenfunctions
169
Berry, M. V. (1983): Semiclassical mechanics of regular and irregular motion, in: Chaotic Behavior of Deterministic Systems (Les Houches Fr. 1981, Sess. 36), 171I 217. Zbl. 571.70018 Birman, M. Sh., and Solomyak, M. Z. (1977): Asymptotics of the spectrum of differential equations, in: Itogi Nauki Tekh., Ser. Mat. Anal. 14, 5558 [English translation: J. Sov. Math. 12, 247283 (1979)] Zbl. 417.35061 Birman, M. Sh., and Solomyak, M. Z. (1980): Spectral Theory of Selfadjoint Operators in Hilbert Space. Izdat LGU (Leningrad State University Press), Leningrad, 264 pp. [English translation: Math. Appl., Sov. Ser. 5 (1987: Zbl. 744.470170] Bunimovich, L. A. (1979): On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65, no. 2, 295312. Zbl. 421.58017 Bykov, V. P. (1965): Geometric optics of threedimensional vibrations in open resonators, in: HighPower Electronics (Elektronika bol’shikh moshchnostej) Vol. 4, Nauka, Moscow, pp. 6692 (in Russian) Colin de Verdi&e, Y. (1973): Spectre du laplacien et longueurs des geodksiques fermees. II. Compos. Math. 27, no. 2, 159184. Zbl. 281.53036 Invent. Colin de Verdi&e, Y. (1977): Q uasimodes sur les variktes riemanniennes. Math. 43, no. 1, 1552. Zbl. 449.53040 Douady, R. (1982): Applications du theoreme des tores invariants. These du 3eme Cycle, Univ. ParisVII, Paris, 102 pp. Duistermaat, J. J., and Guillemin, V. (1975): The spectrum of positive elliptic operators and periodic geodesics. Invent. Math. 29, no. 1, 39979. Zbl. 307.35071 Einstein, A. (1917): Zum Quantensatz von Sommerfeld und Epstein. Verh. Dtsch. Phys. Ges. 19, 8292 Math. Surv. No. Guillemin, V., and Sternberg, S. (1977): G eometric Asymptotics. 14, Am. Math. Sot., Providence, RI, 492 pp., Zbl. 364.53011 Halpern, B. (1977): Strange billiard tables. Trans. Am. Math. Sot. 232, 297305. Zbl. 374.53001 Heading, J. (1962): An Introduction to PhaseIntegral Methods. Methuen, London John Wiley, New York. Zbl. 115,71 Hejhal, D. A. (1976): The Selberg Trace Formula for PSL(2,R). Vol. I: Lect. Notes Math. Vol. 548, Springer, Berlin, 561 pp. Zbl. 347.10018 Hejhal, D. A. (1983): The Selberg Trace Formula for PSL(2,R). Vol. II: Lect. Notes Math. Vol. 1001, Springer, Berlin, 806 pp. Zbl. 543.10020 Keller, J. B., and Rubinow, S. (1960): Asymptotic solution of eigenvalue problems. Ann. Phys. 9, no. 1, 24475. Zbl. 87,430 Kornfel’d, I. P., Sinai, Ya. G., and Fomin, S. V. (1980): Ergodic Theory. Nauka, Moscow, 384 pp. [English translation: Grundl. math. Wiss. 245, Springer, Berlin (1982)] Zbl. 508.28008, Zbl. 498.28007 Krakhnov, A. D. (1975a): Construction of the asymptotics of eigenvalues of the Laplace operator, corresponding to a nondegenerate invariant torus of a geodesic flow, in: Methods of the Qualitative Theory of Differential Equations (Metody Kachestvennoj Teorii Differentsial’nykh Uravnenij) Vol. 1, Izdat. GGU (Gor’kij State University Press), Gor’kij, 66674 (in Russian) Krakhnov, A. D. (1975b): Eigenfunctions concentrated in a neighborhood of a conditionally peridic geodesic, in: Methods of the Qualitative Theory of Differential Equations (Metody Kachestvennoj Teorii Differentsial’nykh Uravnenij) Vol. 1, Izdat. GGU (Gor’kij State University Press), Gor’kij, 7587 (in Russian) Krakhnov, A. D. (1982): Quasimodes concentrated in a neighborhood of the boundary of a manifold, in: Methods of the Qualitative Theory of Differential Equations (Metody Kachestvennoj Teorii Differentsial’nykh Uravnenij), Izdat. GGU (Gor’kij State University Press), Gor’kij, 933100 (in Russian)
170
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Landau, L. D., and Lifshits, E. M. (1963): Q uantum Mechanics. Nonrelativistic Theory. Fizmatgiz, Moscow, 704 pp. [English translation: Pergamon Press, London (1958)] Zbl. 81,222 Lazutkin, V. F. (1973): Asymptotics of eigenfunctions of the Laplacian and quasimodes. The series of quasimodes corresponding to a system of caustics close to the boundary of a domain. Izv. Akad. Nauk SSSR 37, no. 2, 437465 [English translation: Math. USSR, Izv. 7 (1973), 439466 (1974)] Zbl. 255.35074 Lazutkin, V. F. (1974): Asymptotics of eigenfunctions of the Laplacian on the torus, in: Questions of the Dynamic Theory of Seismic Wave Propagation (Voprosy Dinamicheskoj Teorii Rasprostaneniya Sejsmicheskikh Voln), Vol. 14, Nauka, Leningrad, pp. 944108 (in Russian) Lazutkin, V. F. (1981): Convex billiard and eigenfunctions of the Laplacian. Izdat. LGU (Leningrad State University Press), Leningrad, 196 pp. Zbl. 532.58031 (in Russian) Lazutkin, V. F. (1993): KAM Theory and Semiclassical Approximations to Eigenfunctions. Erg. Math. 24, Springer, Berlin, 1993 Lazutkin, V. F., and Terman, D. Ya. (1981): On the number of quasimodes of “bouncing ball” type, in: Mathematical Questions of the Theory of Wave Propagation (Matematicheskie Voprosy Teorii Rasprostaneniya Voln). Zap. Nauchn. Semin. LOMI 117, 1722182 [English translation: J. Sov. Math. 24, 373379 (1984)] Zbl. 477.35070 Lichtenberg, A. G., and Lieberman, M. A. (1983): Regular and Stochastic Motion. Springer, Berlin, 521 pp. Zbl. 506.70016 dynamical systems are Markus, L., and Meyer, K. R. (1974): G eneric Hamiltonian neither integrable nor ergodic. Mem. Am. Math. Sot. 144, 52 pp. Zbl. 291.58009 Maslov, V. P. (1965a): Perturbation Theory and Asymptotic Methods. Izdat. MGU (Moscow State University Press), Moscow, 550 pp. Zbl. 653.35002 (in Russian) Maslov, V. P. (196513): The WKB method in the multidimensional case. Appendix to Russian translation of Heading (1962), pp. 1777237. Zbl. 131,84 Maslov, V. P., and Fedoryuk, M. V. (1976): Semiclassical Approximation for the Equations of Quantum Mechanics. Nauka, Moscow, 1976, 296 pp. [English translation: Math. Phys. Appl. Math. 7 (1981)] Zbl. 449.58002 McDonald, S. V., and Kaufman, A. N. (1979): Spectrum and eigenfunctions for a Hamiltonian with stochastic trajectories. Phys. Rev. Lett. 42, no. 18, 11891191. Mishchenko, A. S., Shatalov, V. E., and Sternin, B. Yu. (1978): Lagrangian Manifolds and the Maslov Operator. Nauka, Moscow, 352 pp. [English translation: Springer, Berlin, 19901 Zbl. 727.58001 Osmolovskij, V. G. (1974): On the asymptotics of eigenvibrations of an elliptic membrane. Z. Vuchisl. Mat. Mat. Fiz. 14, no. 2, 3655378 [English translation: U.S.S.R. Comput. Math. Math. Phys. 14 (1974), no. 2, 91103 (1975)] Zbl. 293.35024 Pankratova, T. F. (1984): Quasimodes and splitting of eigenvalues. Dokl. Akad. Nauk SSSR 276, no. 4, 795799 [English translation: Sov. Math. Dokl. 29, 597601 (1984)] Zbl. 592.34012 Pankratova, T. F. (1986): Quasimodes and exponential splitting of eigenvalues. Probl. Mat. Fiz. 11, 1677177. Zbl. 599.47033 (in Russian) Poeschel, J. (1982): Integrability of Hamiltonian systems on Cantor sets. Commun. Pure Appl. Math. 35, no. 5, 653696. Zbl. 542. 58015 Reed, M., and Simon, B. (1975): Method of Modern Mathematical Physics. Vol. 2: Fourier Analysis, Selfadjointness. Academic Press, New York, 379 pp. Zbl. 308.47002 Shnirel’man, A. I. (1974): Statistical properties of eigenfunctions, in: Materials of the AllUnion School on Differential Equations with an Infinite Number of Independent Variables and Dynamical Systems with an Infinite Number of Degrees of Freedom (Materialy Vsesoyuznoj Shkoly po Differentsial’nym Uravneniyam
IV.
Semiclassical
Asymptotics
of Eigenfunctions
171
s Beskonechnym Chislom Nezavisimykh Peremennykh i po Dinamicheskim Sistemam s Beskonechnym Chislom Stepenej Svobody), Izdat. Akad. Nauk Armen. SSR, Erevan, pp. 267278 (in Russian) Sinai, Ya. G. (1970): Dynamical systems with elastic reflections. Usp. Mat. Nauk 25, no. 2, 141192 [English translation: Russ. Math. Surv. 25, no. 2, 1733189 (1970)] Zbl. 252.58005 Svanidze, N. V. (1978): Existence of invariant tori for a threedimensional billiard, concentrated in a neighborhood of a closed geodesic on the boundary of a domain. Usp. Mat. Nauk 33, no. 4, 2255226 [English translation: Russ. Math. Surv. 33, no. 4, 267268 (1978)] Zbl. 396.58030 Svanidze, N. V. (1980): Asymptotics of eigenvalues and eigenfunctions of the LaplaceBeltrami operator an an ndimensional torus. Materials for the Fifth Research Conference of Young Researchers of the MechanicsMathematics Facults and Mechanics Scientific Research Institute of Gor’kij State University (Materialy Pyat’ej Nauchnoj Konferentsii Molodykh Uchenykh Mekh.Mat. Fakul’teta i NII Mekhaniki GGU), Gor’kij, 2829 April 1980, VINITI Deposit No. 183981, 15 pp. R. Zh. Mat. 813495 (1981) (in Russian) Vainshtein, L. A. (1965): Ray currents in a triaxial ellipsoid, in: HighPower Electronics (Elektronika Bol’shikh Moshchnostej), Vol. 4, Nauka, Moscow, pp. 93105 (in Russian) Venkov, A. B., Kalinin, V. L., and Faddeev, L. D. (1973): A nonarithmetic derivation of the Selberg trace formula, in: Differential Geometry, Lie Groups and Mechanics (Differentsial’naya Geometriya, Gruppy Li i Mekhanika), Zap. Nauchn. Semin. LOMI 37, 5542. [English translation: J. Sov. Math. 8, 171I199 (1977)] Zbl. 345.43011
V. The
Boundary
Layer
A. M. Il’in Translated
from the Russian by J. S. Joel
Contents $1. The Exponential Boundary Layer ....................... 1.1. Introduction. Examples of Boundary Value Problems ................. for Ordinary Differential Equations 1.2. Partial Differential Equations ...................... ......... $2. The Method of Matching Asymptotic Expansions. 2.1. A Boundary Value Problem for an Ordinary Differential Equation ............... 2.2. Uniform Asymptotic Expansion .................... 2.3. A Boundary Value Problem .................. for a Partial Differential Equation 53. An Elliptic Equation with a Small Parameter .............................. in the Highest Derivatives .......... $4. Singular Perturbations of the Domain Boundary. ....................... $5. A Quasilinear Parabolic Equation Comments on the Literature. ............................... References ...............................................
174 . 174 178 . 182 .183 186 ,188 ,190 . 194 ,199 ,204 ,208
174
A. M. Il’in
31. The
Exponential
Boundary
Layer
1.1. Introduction. Examples of Boundary Value Problems for Ordinary Differential Equations. The aim of this paper is to present the basic methods for studying the asymptotic behavior of solutions of some boundary value problems for partial differential equations. The problems considered here depend on a small parameter c > 0. Their solutions are not rapidly oscillating (as was considered by Fedoryuk in the first paper in this volume) and vary smoothly everywhere in the domain except for some small sets. As a rule the sets are neighborhoods of some point, curve, or, more generally, of a manifold of smaller dimension. The width of the neighborhood has order 67, y > 0. Most often such a set is a neighborhood of a part of the boundary of the domain and therefore is called the boundary layer. However this name has also been extended to sets of sharp variation of a solution which lie inside the domain. Thus the boundary layer can be treated as the boundary of subdomains of smooth variation of a solution of the problem. The functions that describe the asymptotics of a solution within the limits of the boundary layer are called functions of the boundary layer, or, more briefly, boundary layer functions. The most characteristic features of these problems can be most conveniently illustrated using the example of boundary value problems for ordinary differential equations. Example 1. The boundary value problem e2zL’ U = 1, 0 < z < 1, ~(0, E) = ~(1, t) = 0, has the following unique solution: u(x, t) = 1 + [cosh(2t)1]1cosh[e1(n: = 1 + exp(xcl)
+ exp((z
 l/2)]  1)~~l)
+ O(P)
as t + 0 for all N.
Here the boundary layer functions are exp(xc‘) and exp((z  1)~~‘). They are essentially different from zero only in neighborhoods of the points 0 and 1 respectively. Example 2. The boundary value problem 6~” + U’ = z (0 5 x 5 l), ~(0, E) = ~(1, t) = 0, has the unique solution U(X,E)
= x2/2
 tx  (l/2
= x2/2
 tx  l/2
 ~)[l  exp(xeCl)][l + E + (l/2
 c) exp(x6l)
 exp(el)]l + O(P)
as E 4 0 for all N. Here the boundary layer function is (l/2  e) exp(xc‘). It is essentially different from zero only in a neighborhood of the origin. The symbol 0 in these examples and throughout what follows means an estimate that is uniform relative to all the remaining variables (in this case relative to x).
V. The Boundary Example
3. The boundary
value
z,u E 2u”
Layer
175
problem
 q(z)u
= f(z),
0 < n: 2 1,
(1)
u(0, E) = U(1, E) = 0,
(2)
where q, f E C” [0, 11, q(x) 2 Const > 0. By analogy with Example 1, we shall look for a solution in the form sum of series (everywhere in this paper the series are asymptotic)
k=O
k=O
of a
k=O (3)
where < = XE‘, 77 = (1z)t‘. H ere we shall assume that all of the functions uk and wk are exponentially decreasing at infinity. Since for values of x lying inside the interval [0,1] the coefficients in the series V and W are exponentially small as E 4 0, it is natural to assume that U satisfies equation (l), and V and W satisfy the homogeneous equations 1,V = 0 and 1,W = 0. Formally we substitute the series U in equation (1) and equate the coefficients for like powers of E. We obtain a recursive system of equations: dx)uo(x) q(x)?&(x)
= f(x), = ulel(x)
(4) I Hence all the uk(x) G Cm[0, l] are uniquely determined. Further we consider only the series V since W is completely analogous to it. The equation for V in < has the form d2V/dE2  q(t<)V = 0. Let x + co, where qo > 0 by assumption. Substituting the q(x) = c~Oqkzk, series V in the equation, we also obtain a recursive system: for k > 1.
II;  qowo= 0, w[  qowk = eqi
From
the boundary
condition vZk(o)
=
uk(o),
(5)
for k > 1. I
(2) it follows t&+1(0)
that = 0 for k 2 0.
(6)
There exist unique solutions of the problem (5), (6) and these solutions tend exponentially to zero as [ + co. The functions wk(n) are also constructed analogously. Just as in previous articles in this volume a formal asymptotic solution (f.a.s.) of an equation as t + 0 is understood to be a series whose partial sums satisfy the equation up to O(P), where m + co as the number of terms in the partial sum tends to infinity. A f.a.s. of a boundary value problem is understood analogously: here in addition to the equation it is understood that the boundary conditions are satisfied approximately.
176
A. M. Il’in
Thus, the sum of the above series U, V and W is a f.a.s. of the problem (I), (2). It is easily seen that in this example the f.a.s. is the correct asymptotic expansion of the solution U(Z, t). Indeed, suppose that U,, V, and IVn are the partial sums of the series constructed above, and let z&(x,
t) = u(x, E)  Un(X,E)
 vqxel,
t)  Wn((l
 3$i,
6).
Since ,&(O,E) = O(P), &(l,c) = O(P) and Z,&(Z,E) = O(P), where ‘rn + cc as n + 00, we have .&(~,e) = O(P). Summarizing these results, we note that the series V and W do not essentially influence the asymptotics of the solution far from the boundary. There the asymptotics of the solution is determined by the series U. It is called the outer asymptotic expansion or, more briefly, the outer expansion. The series V and W are called inner expansion+ since in a sum with U they describe the asymptotics of the solution U(Z, t) inside the boundary layer. (These terms owe their origins to problems in fluid mechanics. For more details about this see the “Comments on the Literature” below.) Figure 1 shows an approximate graph of the function ~(2, E).
Fig. 1 Example
4. 1,u = 62
+ a(x)2
= f(x),
0 5 x < 1,
(7)
u(0, E)= $0, E) = 0,
(8)
U(l,E)
(9)
= E(l,t)
= 0,
a, f E c[o, 11,
u(x)
Here we shall also look for asymptotics expansion U(x, e) and inner expansions
> Const
> 0.
in the form of the sum of an outer describe
V([, 6) and W(q, E), which
V. The Boundary
Layer
177
the behavior of the solution close to the lefthand and righthand ends of the interval. Here < = ZE~, 7 = (1  z)Fc~~, where cy > 0 and p > 0 are still to be found. The equation 1,V = 0 in < has the form: ~~~~d~V/d[~ + ~~u(P<)dV/d< = 0. It is natural to choose cy so that, both terms in the lefthand side of the equation have the same order. Since a(O) > 0, we t,herefore have 1  4a = (Y + (Y = l/3. In an analogous way we choose p = l/3. (In this case the choice of scale of the boundary layer is trivial. As a rule, it is also easy to choose it in other problems for ordinary differential equations. However, for partial differential equations the choice of correct scales for the boundary layer with respect to different independent variables is often a rather difficult and unformalized task.) Thus, we set U = CrZ”=ot’“/3~k(~), V(<,c) = C,“=, ~“/~vk(<), W(q,t) = Cp=“=, E”/~w~(v), where t = ~~~~~~ and 7 = (1  CC)E~/~. As in the previous example, we obtain a recursive system of equations for the w,k(l~): 444
= f(z),
~‘1 = 0, ~‘2 = 0, and U(X)Z&
= u??~
However, now the solution of this system is not unique. must turn to the boundary layer functions V(<,t) and Example 3, for uk(<) we obtain the system
for Ic > 3.
(10)
To determine W(Y), E). Just
it we as in
(11) where the gk are linear coefficients. Analogously,
combinations
of the
dvi/d<
for i < k with
power
(12) After substituting the sums U and V in the boundary conditions the sums U and W in the boundary conditions (9), we will obtain vk(O) + ~(0)
= 0 for k > 0,
(13)
v;(O) = 0, z&(O) + us_, wk(O) w;(O)
+ uk(l)
= 0, and w;(O)
(8), and
= 0 for Ic > 1,
(14)
= 0 for Ic > 0,  &(l)
= 0 for k > 1.
(15)
Since U(Z) > 0, the equation vc4) + a(O)v’ = 0 has only one linearly independent solution, which tends to zero as < + 00, and the equation wc4)  u(l)w’ = 0 has two such linearly independent solutions. Thus, on the righthand end the function W’(q, C) can “correct” both boundary conditions, while on the lefthand end the outer expansion U(X, t) must itself partly assume responsibility for the validity of the boundary conditions. Therefore the order of determining the functions is as follows.
A. M. Il’in
178
Since u;(O) = 0, ve(<) is a solution of the homogeneous equation and ve([) + 0 as [ + 00, then ve(<) = 0. From condition (13) ue(0) = 0 and the function UO(X) is determined from equation (10). Condition (14) ZJ~(0) = u&(O) together with equation (11) for vi uniquely determines a solution vi(<) that is exponentially decreasing at infinity. Condition (13) determine the function ui (x), and ui(0) = TJ~(O) and equation (10) uniquely so on. Also the functions wk(v), which tend exponentially to zero at infinity, are uniquely determined by the equations (12) and the boundary conditions (15). The construction of a f.a.s. of the problem (7)(g) is complete. It is justified exactly as in the previous example. Indeed, by construction the residuals in the equation and the boundary conditions (that is, the values of the function uu,V,  IV, and its derivative at the ends of the interval, and also the operator 1, of this function) are small and the question is reduced simply to estimating the norm of the inverse operator. This norm must not exceed ME~, where r is some fixed number. In this example it is not difficult to obtain an estimate of this type (for r = 0) and we shall not go further into this. Thus, in this example, as in the three preceding ones, the asymptotics of a solution is the sum of an outer expansion and inner expansions. These inner expansions consist of functions that tend exponentially to zero outside of the boundary layer. Therefore this layer is called an exponential boundary layer. This concludes the examples of ordinary differential equations for now. Remarks concerning other, more complicated, cases and bibliographic comments for this section as well as for all the later sections are collected separately at the end of this paper. 1.2. Partial Differential Equations. First we consider an example of a problem where the behavior of the solution is essentially the same as in Examples 1 and 3, and the technique of constructing a solution is the same as in Example 3. Example 5. L,u
E c2Au
 q(x)u
= f(x)
for 2 E D c R1,
where D is a bounded domain, S = dD E C”, q(x) > 0 and E << 1. The boundary condition
and
q, f
(16) E C”(D)
~(2, E) = 0 for 2 E S. We look for an outer
expansion
where
(17)
in the form
u(x,6)= c 62ku&),
(18)
k=O
and, as in Example uo(x) = f(~)[9(x)11,
3, we find uk(z) from the recursive system and uk = [q(x)]‘Auk1 for k > 1.
of equations:
V. The Boundary
Layer
179
Fig. 2 In order to compensate for the residual in the boundary condition, it is necessary to construct a boundary layer function along the whole boundary S. In view of the smoothness of S, in a neighborhood of S we can introduce a coordinate system y, Z, where y is the coordinate on the smooth manifold S and z is the distance from a point J: E D to S (Fig. 2). The same arguments as in Examples 3 and 4 lead to the change of variables z = EC and to the following form of an inner expansion: (19) k=O
The function V satisfies the homogeneous the following form in the variables y, <:
equations
&V
= 0, which
has
(20) Here L1 and L2 are differential operators containing only yderivatives, having smooth coefficients of y and Z, that is, of y and t<. Let q(x) be equal to qo(y) on the boundary S. Expanding the coefficients in (20) in Taylor series with respect to z = EC and substituting the series (19) in this equation, we will obtain a recursive system of differential equations:
where the righthand sides Fk depend linearly on the previous Vi and their derivatives, and depend polynomially on (‘ and smoothly on y. The system (21) is essentially the same system of ordinary differential equations (5) from Example 3, except for the parametric dependence on the variable y. It follows from the boundary condition (17) that uzk(y, 0) = uk(~)I~, vzk+l(y, 0) = 0. There exist unique functions uk(y, <) satisfying these conditions, equations (21) and tending exponentially to zero as c + 00. Thus we have constructed the inner expansion (19). In view of this, it is true that the change of variables 5 tf (y, Z) is a diffeomorphism only in a small neighborhood cr (of fixed thickness not depending
180
A. M. Il’in
on e) of the boundary S; then, strictly speaking, we should consider the series (19) only in this neighborhood (Fig. 2). H owever, this is only a formal obstruction. Multiplying all the functions vk(y/, EC~Z) by a smooth function X(X) (not depending on t) that is equal to one close to S and to zero in D\a, we define the inner expansion (19) everywhere in D. Since vk(y, C) tends exponentially to zero as C + co, then ~k(y,<) and x(~)~k(y, <) and all their derivatives differ by quantities of order O(?) for all N. The shaded area in Figure 2 is a boundary layer of width O(c). The justification of the asymptotics is carried out just as in the case of Example 3. If U, and Vz, are partial sums of the series (18) and (19), then the difference 2, = U(Z, 6)  U,(z, E)  x(cE)V~~(~, <, E) is equal to zero on the everywhere in D, where m + 00 as n + co. boundary S and L,Z, = O(P) From this and from the estimate for the inverse operator (for example, from the maximum principle) it follows that the sum of the series U and x(z)V is a uniform asymptotic expansion of the solution U(Z, e) in D. A complication of the method described above arises in the case when the boundary of the domain is not smooth. We illustrate this type of problem with an example. Example 6. We consider the problem (16))( 17) in a plane domain D whose boundary is a curve S that is smooth everywhere except for one point. Suppose that this point is the origin 0 and that in a neighborhood of the origin the boundary S coincides with the positive 21,x2 halfaxes (Fig. 3).
&
Xl
Fig.
3
The outer expansion has exactly the same form as in the previous example. The inner expansion is as in Example 5 everywhere outside a neighborhood of the corner point. In a neighborhood of the origin the inner expansion is the sum of two functions: M
v(l)(Xl)c1x2, c)= c EkZll,k(X1) c574 k=O
and
V. The Boundary Layer
vC2) (elx1 ( 52, E) =
181
5k?J2.k(tlsl.22). k=O
Each of these functions is locally constructed just like the function V(y, <, E) in Example 5. In particular, the boundary conditions for the coefficients of these series are as follows: u1,2k(Zl,
0) = uk(xl,
Vl,2k+l(~l,O)
0); ~2,2k(o,
= o;V2,2k+l(o,Z2)
x2)
=
uk(o,
x2);
= 0.
Outside a neighborhood of 0 the partial sum Un+x(z)V2TL satisfies the boundary condition (17). In a neighborhood of 0 this sum is now nonzero on S:
w2 + V2n)zz=0 =
2
EkV2,k(E%,
O),
k=O
@hL + V2n)zl=0 = F fk~l,k(o, E%2). k=O
In order to liquidate the presence of residuals in the boundary conditions again, an additional corner boundary layer is introduced and to the series previously constructed we add the series (22) where E = E~z~ and 7 = ~~52. The boundary conditions for Wk(<,‘n) must obviously be: wk(<,o)
=
~Z,k(‘t,o),
The equation L,W
t > 0;
wk(o,v)
= %,k(O,v),
7 > 0.
(23)
= 0 in [ and 7 has the form
@W + d2W  Q(EJ, Erj)W = 0. a<2 dry After substituting the series (22) in this equation we obtain a recursive system of partial differential equations for functions in the quadrant < 2 0, q > 0: &,wo
 qowo = 0;
Agwk
 qOwk
= Fk(<,Q)
for
,t >
1.
(24)
Here qo = q(O,O) > 0 and Fk is the same as in the previous examples. It is not hard to show that there exist solutions of the problems (23)(24) that decrease exponentially at infinity and, thus, are the desired boundary layer functions. Hence, a complete f.a.s. of the problem (16)(17) in this example is equal to U(x, e) + V(y, tTz,E)X(X)
+ W(Elxl,
EC1X2,E),
182
A. M. Il’in
where U, V and x are constructed in the same way as in Example 5, W is the series (22), and its coefficients are solutions of the problems (24), (23). The domain of essential variation of the functions Wk(<, 77)(that is, the corner boundary layer) is shown in Figure 3 by the double shading. This f.a.s. is the correct asymptotic expansion of the solution ~(2, E), uniform in the domain D. This follows from the estimate for the inverse operator (for example, from the maximum principle).
32. The
Method
of Matching
Asymptotic
Expansions
The asymptotics of solutions of the problems considered in Section 1 has a rather simple structure in general. In most of the domain the solution can be represented by the outer expansion U(Z,E), which is the usual perturbationtheoretic power serieswith respect to the small parameter. In the whole closed domain the uniform asymptotics of a solution is the sum of U(Z, E) and the inner expansions V(<, E). The inner expansion coefficients up(c) depend on the “stretched” variables 6 and tend exponentially to zero at infinity. Therefore outside of boundary layers of width eY the inner expansions have no contributions to the asymptotics. However for partial differential equations this statement is more the exception than the rule. Often the functions ‘r&(x), the coefficients of the series U(X, E), have singularities at some points of the boundary or even inside the domain. Moreover, the order of the singularities increases as the number Ic increases.It is clear that in this casea uniform asymptotic expansion cannot be obtained as a sum of U(X, E) and some serieswith bounded coefficients. Since closeto such unpleasant points the outer expansion is already not a f.a.s. of the equation, it is natural to look for an inner expansion as a solution of the full equation (but this is not a homogeneous equation as in the examples of Section 1). Here the coefficients of V, the functions uk([), do not have exponential asymptotics at infinity, but power asymptotics, that is, they are expanded in infinite asymptotic power series in negative powers of E as < + oo. Therefore the influence of the boundary layer functions 7/k([), strictly speaking, cannot be restricted only to a narrow boundary layer of width O(cT). Problems of this type, perhaps without full justification, are often called problems with a power boundary layer. There is no other established, widespread term that characterizes the problems considered later in this chapter. All the problems considered in this section (and also in some other sections) are usually called singular problems, or problems with a singular perturbation. This term only emphasizes that the asymptotic expansion of a solution is not a regular power series in E, but more delicate properties of the asymptotic expansion are not made precise. For the sake of brevity in what follows we shall use the term bisingular problems to refer to singular problems for which the coefficients of the outer expansion have singularities that increase as the ordinal number of the coefficient increases.
V. The
Boundary
Layer
183
The characteristic difficulties that arise in bisingular problems and the methods for studying these problems are again conveniently discussed using the example of ordinary differential equations. 2.1. A Boundary Value Problem Equation. At first glance the following to the problem from Example 3: l,u
z
c3u”

q(z)u
=
for an boundary
f(z),
0 I
U(O,E) = U(l,E) with
q, f E Cm[0,
I]. But instead q(2)
z I
Ordinary Differential value problem is similar
1,
t <
1,
= 0,
of the positivity
(25)
(26) of q(x) we now assume
> 0 for 2 > 0; q(0) = 0, q'(O) = 1.
The assumption that q(z) vanishes only at one point change in the asymptotics of the solution of the problem.
that (27)
creates an essential The outer expansion
cc U(z,
E)
=
Jq
63kuk(2)
(28)
k=O
has the same form as in Example 3, and the recursive system of equations is the system (4). However, now ?&(z) E C” only for II: > 0,and for z = 0, generally speaking, they all have singularities. It is easy to see that
(29) Meanwhile the solution of the problem (25)(26) is bounded for each fixed E and belongs to C” [0, 11. It is also easy to deduce a uniform estimate in t for the solutions U(Z, E). In fact, let Z,z = ‘p. After the change of variables z = (2  z2)y we obtain the following equation for the function y(z, t):
E3Y”It follows
2&X(2 from
 k7z2)ly’  [q(2) + 2&2
the maximum
principle
 “2)l]y
= (2  Lz?‘(p(~)
that
IY/(T~11I Iv(O,f)I + l~/(L~)l+ E3maxlv(z)l. Thus,
l4T 6115 ~f3(140, c)l + l4L E)I+ IlL.4lc) and, in particular, ]u(x, E)] < iWieP3. In order to describe the asymptotics of z = 0 correctly, we make a stretching Then equation (25) takes the form
(30)
of a solution U(Z, E) in a neighborhood of the independent variable: IL: = eat.
A. M. Il’in
184
,32a$
 q(PE)v = f(E”l$),
where v(<, e) E u(E”<, 6). Since q(x) 5asz+O,itisnaturaltoset32o= ct*a=l. Thus, near the origin we look for a solution ~(2, E) in the form expansion V(<,E)
=
g
fkVk(C),
of an inner
(32)
k=I
where series
< = zrl. We introduce notations for the coefficients of the Taylor of the functions q(x) and f(z): q(x) = z + Cp=“=, qkx" and f(x) = x + 0. After substituting the series (32) in (31) we obtain a c,“=, fkxk, system of equations:
(33)
The
first of the equalities
(26) generates “k(0)
= 0,
the boundary
conditions
k > 1.
(34)
h as a unique solution for < > 0 in the class of funcThe system (33)(34) tions that grow no faster than some power of I. The asymptotics of the functions vk([) at infinity is also easy to find. It has the form
vk(t) = <” 9 hk,i<3i,
5 + oc), k > 1.
(35)
i=o The inner expansion near the other endpoint x = 1 has the same form as in Example 3. Therefore, in order to keep the notation simple, we shall only consider x < l/2 in what follows. Thus, we have obtained two asymptotic series. By construction and in view of the estimates (29) the series (28) is a f.a.s. of the problem (25)(26) for ~7 < x 5 l/2, for all y < 1. By construction and in view of the estimates (35) the series (32) is a f.a.s. of the same problem for x < ~7 for all y > 0. One can prove that the series (28) and (32) are in fact correct asymptotics of the solution of the problem (25)(26) in the indicated domains. Here there exists an extensive domain of overlaps of these asymptotics, where the solution can be well approximated both by the series (28) and by the series (32). This is the domain cyl < < < eY2, where yr and 72 are arbitrary numbers such that 0 < 72 < yr < 1. Clearly it is not convenient to operate with such vague boundaries with arbitrary yi. There exists a simpler method of comparing the outer and inner expansions. For this we must rewrite one of them in other variables and compare it with the second expansion. For example, substituting (29) in (28) an d replacing II: by E<, we obtain formally
V. The Boundary
U(x,
6) =
g
,1<3k1
k=O
Layer
2
185
Ck,#.
i=o
If we compare this with the series (32), where the Vk(<) are replaced by the series (35), then a necessary and sufficient condition for the formal equality of these series U and V is that the equalities ck,i = hil,l, hold. These equalities in fact hold. Hence, U(x, E) = V((, E). (36) This equality is understood as an equality of formal series obtained by the method indicated above. It is often convenient to write this equality in another, equivalent form. Let U, and V, denote the partial sums of the series (28) and (32), containing all we replace each term ea’~k(z) the terms of order ey, where v < n. In addition in the sum lJ,(x, t) by its asymptotic series (29) as x 4 0, substitute x = t< and in the resulting infinite sum we leave only the terms that have order EP for p 5 m. This finite sum is denoted by U,,, (2, e). We carry out the analogous procedure with I&([, E) and obtain V,,,(<, 6). Equality (36) is equivalent to the equalities Un,m(~, E) = Vm,n(<,~) for all m, n E N. (37) It is natural to interpret these equalities as a compatibility (or matching) condition for the asymptotic expansions (28) and (32): the common parts of the asymptotics (28) as x + 0 and the asymptotics (32) as < + CC coincide. This example is extremely simple. In fact, the coefficients ?&(x) and ?.&([) are uniquely determined in this case. But in the majority of bisingular problems the coefficients of one or both expansions are not uniquely determined from their boundary value problems. Such examples will be considered in later sections. In these examples every coefficient of one of the expansions (for example, the function uk(x)) depends on several arbitrary constants. Therefore the equalities (37) do not hold in general. But it turns out that these constants can be chosen so that the equalities (37) do hold. Such a choice of the constants, and thus, the coefficients of the outer and inner expansions, is called the method of matching asymptotic expansions. From the series (28) and (32) it is not hard to construct a uniform approximation of a solution of the problem (25))(26) everywhere on the interval [0,1/2]. Such an approximation is given by the sum: K
= Un(x,
6) + Vn(5,E)
 Un,n(x,
6).
(38)
By the matching of the series (28) and (32) and by construction of Y, this sum is a f.a.s. of the problem (25)(26) on the interval [0,1/a] up to O(C), where u + co as n + co. If we add boundary layer functions for the endpoint x = 1 to the sum Y,, that is, if we add a partial sum of the series (3) and apply inequality (30), then we are led to the estimate ]u(x, t)  Y,(s, E)] 5 Mtv3 for 0 I x I l/2. (A more detailed analysis shows that we can set v = 3+n/4.) The approximation Y, is often called the composite asymptotic expansion. The graphs of the functions U(X, E) and Yi (x, t) are shown in Figure 4.
A. M. Il’in
186
Fig. 4 2.2. Uniform Asymptotic Expansion. In spite of the extreme simplicwe can use it to demonstrate another approach ity of the problem (25))(26) to the construction of f.a.s. Namely, as in Example 3, we can look for a f.a.s. of the problem (25)(26) on the interval [0,1/2] as a sum of the series (39)
k=O
and the series k=1
However in this, as in other bisingular problems, the task of finding the functions yk(z) and zk(<) is sigificantly more complicated than in the problems of Section 1. The heart of the problem is that the composite expansion can also be represented as a partial sum of the series 11 + 12, but here the choice of yk (z) and zk (5) is not unique. The method which we will discuss below does not admit a growth in the order of the singularities of yk(z) and zk(<) as the number Ic increases, as was admitted in $2.1. For example, it is reasonable here to assume that all these functions are bounded and smooth, and the functions zk([) = c,c, ak,,<‘, + 00. This is very convenient from different points of view, but the as E cost for this convenience is the special and somewhat artificial construction of boundary value problems for yk(z) and z&(l). In our example these problems are: Q(z)yk(z) Lzk
= zl(o)
z;(r)
=
0,
@k(t) zk(o)
=
.fk(x),
0 <
II: <
1,
=
Sk(t),
0 5
t <
00,
=
?&(o),
k 2
(41)
0.
For these problems to be solvable in the indicated classes of functions, is obviously necessary and sufficient that the following conditions hold: functions fk(z)z’ and gk([)(l + [)l belong to the same classes.
it the
V. The Boundary
Layer
187
After substituting the sum of the series (39) and (40) in equation (25) we obtain the following equality:
where F(z, I, e) is some series that is rather complicated to construct. The basic idea of the method is that in order to be able to represent this series in the form cr=“=, ekfk(z) + ~~=“=_, ek+’ gk(<), where fk and gk satisfy the conditions described above. Roughly speaking, it is necessary to sort the terms of the series F(z, <, e). Those that have a singularity E + co, for example, &cl for 1 > 0, must be represented in the form Ejlx’ and added to the function fjk(X), but, for example, the function &(x + E)‘, where 1 2 0, must be represented in the form &‘(J + 1)l and added to the function gj+i(<). We carry this procedure out in more detail. We introduce the notations q(x) = x + x2h2(4
= & qjx’ + x”+‘hk++),
(43)
j=l
where q1 = 1, hk E Cm[0, l] and hk(0) = qk; (43’) where Wk,n E Cm[0, CG) and Wk,+ = O(<l), Then in equality (42) we have
< + 00.
F(X,I,f)= f(x)  2
+ x2h2(x)
~“+~y;(x)
F
f’Zk(‘$).
(44)
k=1
k=O
To have fc(0) = 0, we must set fo(z) = f(x)  f(0) + x?(x), where f(x) is defined later. Thus, in order to compensate for the constant f(O), we must set is already uniquely determined from equation sl(E) = f(O). H ence, zi([) (41). Then we must transform the term x2h2(x)ce1z1(<). Using the equalities (43) and (43’), 1‘t IS . not hard to see that this term is equal to
x 2 f’hj+2(x)al,j+l j=o
+ E2 fj+1qj+2Wl,j+2(~), j=o
It is clear that the terms of the first seriesmust go to fj(x) and the terms of the second series must go to gj(<). In particular, fe(z) = f(x)f(O)+xal,lhz(x) and go(<) = q2&1,2(J) are now completely defined. After finding the functions ye(x) and ze(c) we can, as above, transform the term x22h2(x)zo([) in the sum (44). Then this process continues and in successsionwe find all the ?&(XC)and the i&(l). Here for k > 0 we have gk(t)
=
c & j=o
‘&+2wjl,kj+2(<)
+
&3(o),
A. M. Il’in
188
and j=o The sum of the series (39) and (40) gives a uniform asymptotic approximation to a solution of the problem (25))(26) on the interval [0,6] for any 6 < 1. In order to obtain a uniform approximation on the whole interval [0, l] it suffices to add the series (3) as in Example 3. 2.3. A Boundary Value Problem for a Partial Differential Equation. To conclude this section we consider a simple problem for a partial differential equation, which is not very different to study than the problem treated above. Let D be a bounded domain in R1, containing the origin, and let 3D E C”. Let U(Z, E) be a solution of the problem c4Au  q(x)u = f(x)
for x E D
c
R1,
u(z, E) = 0 for 1cE dD,
(45)
(46)
where q(x) = [~1~h(z), with q, h E C”(D) and h(z) > 0 in D. The outer expansion in this problem is also easy to write down: 00 u = c E4kU&),
(47)
k=O
uk(z) = [q(z)]lAukl, k > 1. Residuals in where Q(Z) = f(x)[q(z)]l; the boundary condition (46) are corrected using the boundary layer functions uk(n, zeC2). Here y and z have the same meaning and the z’k are constructed by exactly the same method as in Example 5 (see Fig. 5). But it is more essential that the functions uk(z) in general have singularities at the origin. It is not hard to find that
Fig. 5
V. The Boundary
uk(x)
= IxI~~~
2
Layer
Pj,k(x)
189
as J: 4 0.
j=Zk
Here and throughout this section Pj, Qj, Pj,k, and Qj,k denote homogeneous polynomials of degree j. Thus, the problem (45)(46) is bisingular, and in a neighborhood of the origin the asymptotics of a solution are not described by the series (47). We consider the inner expansion w
=
2 k=2
(48)
fkWk(‘t),
where < = XC~. The choice of scale is carried out as in $2.1. The fact that the series (48) begins with a term N em2 .is easily seen from the principle of the matching of the series (47) and (48). A recursive system of equations for wk(t) is also obtained by substituting the series (48) in equation (45) and expanding q(x) and f(z) in Taylor series: A+2

hl<12w2
=
f(O),
=
Qk+2(6)
=
h(O,O)
k+4 A<%

h1‘$12wk
ho
+
>
c j=3
Pj(<)Wkjf2,
k 2
1,
0.
One can show that there exist solutions of these equations in the whole space RI, that these solutions are unique in the class of slowly increasing functions and for < + 00 these solutions have the power asymptotics: wk(‘$
=
itI2
2 j=o
ltl4JQk+2+j,k(t).
The series (47) and (48) turn out to be compatible in the sense that was explained in $2.1 (for example, the equalities (37) hold). The inner expansion (48) is the correct asymptotic expansion of the solution U(Z,E) in the neighborhood of the origin 1x1 < C, where yr is any positive number (see Fig. 5), and the outer expansion (47) possesses the same property on the set Gn {x : 1x1 > EYE}, where G is any interior compact set and y2 is any number less than 1. We can obtain an asymptotic expansion that is uniform in G by formula (38) from the series (47) and (48). This concludes the examples of bisingular problems that explain the method of constructing and matching the outer and inner expansions. In these examples the process of matching these expansions is essentially missing, since their coefficients were uniquely determined. It was only necessary to verify the matching of the series that were constructed. More complicated bisingular problems, in which the choice of the coefficients of the inner and outer expansions is not unique in the beginning and, in fact, is made by their matching, are described in the following three sections.
A. M. Il’in
190
$3. An Elliptic Equation with a Small in the Highest Derivatives
Parameter
We consider the boundary value problem
2(gg+$)
 ~x,Y): = f(x,y), for(x:,Y)ED, u(x,y,~)
= 0 for (~:,y) E dD,
(49) (50)
where D is the square {x, y : 0 < x < 1, 0 < y < l}, a,f E C”(D), u(2, y) 2 a > 0, E< 1. The outer expansion for this problem is written rather simply: (51) where C&(X,y)due/dy
= f(x,
y); a(~, y)duk/dy
= Aukr,
Uk(X, 0) = 0.
k > 1, (52)
The boundary condition (52) is obtained from the same considerations as in Examples 2 and 4: on the top side of the rectangle there is a residual in the boundary condition that can be neutralized easily by means of the exponential inner expansion w = &2kt&(x,
(1  y)62),
(53)
k=O
while this is not possible on the bottom side. Therefore we must keep the boundary condition (50) on the bottom side. The functions w~(x, 7) are constructed similarly to what was done in Examples 4 and 5, and we will not go into this. We turn now to the residuals that arise in the boundary condition on the sides x = 0 and x = 1. Since these sides are completely equivalent, in what follows we shall only consider the side z = 0. The series(51) is a f.a.s. of equation (49), but does not satisfy the boundary condition (50) for x = 0. Therefore a solution to the problem (49)(50) close to this side is not described by the series (51), but by the sum of (51) and an inner expansion. The new independent variables are obviously < = XC” and y. For the term e2d2u/dx2 to be compensated in these variables by the other term du/dy, it is clear that we must set (Y = 1. The inner expansion V close to the side x = 0 has the form v
=
2
ckuk(<,
Y),
OS<<9
OFYIl,
(54)
k=O
where < = XC‘. The coefficients vk are chosen so that the series V is a f.a.s. of the homogeneous equation (49) (since . the series U constructed above is a f.a.s. of the inhomogeneous equation). Moreover, the sum of the series U and
V. The Boundary
191
Layer
V must be equal to zero for v = 0 and for IC = < = 0. From these requirements we obtain a recursive system of boundary value problems for vk:
Lvo 3 c!!  u,(y)% = 0, Lvk~:Lg+g(y)++,
k>l 1
j=l
Here
c&(y)
c,“=,
ak(!&“,
(57)
uk(
0)
=
0,
7JZl(O,
Y)
=
W(O,Y),
are
the 2 +
Taylor
vzl+l(o,Y)
coefficients
=
of the
0.
function
(58)
a(~, y):
a(z,g)
=
0.
By hypothesis at(y) > 0; therefore (55))(58) are the usual problems for parabolic equations in the halfstrip 0 < C < co, 0 5 y 5 1. In view of the boundary condition (57) the solutions decrease exponentially as C + co, and at first glance the whole problem in the large is a usual singular problem, similar to the problems of $1.2. In reality the situation is more complicated. The disagreeable things are located not at infinity but at (0,O). Since, in general, %(O, 0) # 0, we have $$(O,O) # 0. Because of the incompatibility of the boundary conditions at the origin the function vs(<, y) is not sufficiently smooth at the origin. One sees easily that on the parabolas c2yP1 = Const the righthand side of the equations (56) for K > 2 has order y’‘. Correspondingly the solutions will have order y2“. Thus, the problem (49)(50) is indeed bisingular. But now it is not the coefficients of the series (51) that have increasing singularities, but the coefficients of the series (54). Close to the origin the inner expansion (54) is not a f.a.s. of the homogeneous equation (49) and we must introduce one more inner expansion. By the same considerations as above, the new variables have the form 6 z.z xc2 =
2 = 2 t2”zk(<, 7).
(59)
k=l
We should note that the series is inner relative to the series (51), but it is also an outer expansion relative to the series (59). In some problems it is possible to have several boundary layers embedded in each other and expansions corresponding to them. The boundary layer near the boundary x = 0, at which the asymptotics is described by the series (54) is often called a parabolic boundary layer, because equations (55)(56) are parabolic. We note one further peculiarity of the system (55)(58) that is new in comparison with the case of ordinary differential equations. How does one construct a solution of the parabolic equation (56) with a very strong singularity in the righthand side? In fact, the usual convolution of the righthand
192
A. M. Il’in
side with a Green’s function does not work here, because the integral diverges. But this obstruction is often merely technical. It is not very difficult to overcome this obstruction by one method or another, actually regularizing the divergent integrals. There is another main difficulty. Since the solutions of equation (56) for large Ic have strong singularities, then in the class of such functions the solution of the problems (56) (58) is not unique. In fact, let G(<, y) be a fundamental solution of the Cauchy problem for equation (55). Then arbitrary functions of the form $%G(C, are solutions of equation Y) (55), equal to zero for < = 0 and for y = 0. Any linear combination of these functions can be added to the solution of the problem (56)(58) and the question of what the correct solutions are remains open if we start only from the problems (56)(58). An answer to the question is given by the matching of the asymptotic expansions (54) and (59). Thus, we turn t.o the series (59). The boundary value problems for z~(C, 7) are obtained from the problem (49)(50) just as was done above:
Mzl
I AC,+1  a(O,O)
drl
= 0,
Here the Pj are homogeneous polynomials of degree j, which are obtained by expanding the coefficient a(~, y) in a Taylor series: a(~, y) = ~~=, Pj(x:, y), z + 0, y + 0. The problems (60) have unique solutions in the class of continuous slowly increasing functions. Now we have to study the asymptotics of these solutions as 17 + co. This, in general, is a very essential part of the method of matching asymptotic expansions. For the coefficients of the inner expansions we always obtain boundary value problems in unbounded domains, and we are always required to find the asymptotics of solutions of these problems at infinity. Moreover, if there is a need to obtain an asymptotic expansion of the basic problem up to an arbitrary (or sufficiently large) degree of the parameter E, then the asymptotics of the coefficients of the inner expansion must also be obtained up to arbitrary (or sufficiently large) negative powers of the space variables. One can show that for solutions of the problems (60) the following asymptotics hold: 2N1
z~([,v)
= 7’
C j=o
@k,j(t9)~j’~
+ O(qeN)
as 7 3 cc for all N,
where 0 = @1/2 and the functions &j(8) E Cm[0, co) and tends exponentially to zero as 0 + 0~). Here these previously defined functions sPk,j for
V. The Boundary
Layer
193
j > 21c also give the possibility of determining the unique solut,ions of the problems (55)(58) that generate the correct asymptotics of the solution of the problem (49)(50). Th is is done in the following way. We substitute the asymptotic series for zk in the series (59) and pass to the variables <, y:
One can show that there exist solutions z’k of the problems (55)(58), which as y + 0 have the asymptotics yk/2 c,“=, y’@j,k(
u+zNE2
NE
Fig. 6
194
A. M. Il’in
above that these boundaries are very vague and only of these boundary layers are shown in the figure.
34. Singular
Perturbations
of the Domain
the characteristic
sizes
Boundary
If the domain boundary in which the boundary value problem for an elliptic differential equation is considered is smoothly deformed, then the solution of the problem will also vary smoothly, in general. However, there is also interest in other variations of the boundary, which are called singular. Suppose, for example, that one of the components of the boundary of a domain D c sphere of radius E. How will a solution behave R’ is an (I  1)d’ imensional as e 4 0, that is, in the process of the disappearance of this part of the boundary? Instead of a sphere, we can, of course, consider a small surface diffeomorphic to it. Or, for example, we assume that the boundary of the limit domain De has singularities: corners, edges, cuts, conic points, etc., and the domain D, is obtained by smoothing these singularities. In all these cases the boundary value problems for elliptic equations are bisingular in the sense that was explained in Section 2, that is, the coefficients of the outer expansion have increased singularities and close to singular points of the boundary the outer expansion ceases to be a f.a.s. of the equation. The same situation is also observed for other types of equations, for example, for parabolic and hyperbolic equations. Very many of these problems can be studied by the method of matching asymptotic expansions. Here we shall consider the simplest example of the Laplace equation in a threedimensional domain with a small “hole”. We note that similar problems are connected with longwave asymptotics. This is explained by the fact that, for example, in the simplest case the oscillations of a medium in the exterior of a body w are described by the equations Au+v2u = 0, where v is the frequency of the oscillations. If the frequency is small (the wavelength is large), then one can make a change of variables %I, = xkv. Then the body w is mapped to a small body w,, and the equation no longer contains a parameter. Thus, the problem is reduced to the solution of a fixed equation outside of a small domain. Thus, we proceed to the precise statement. Let D c R3 be a bounded domain, dD E C” and the origin 0 E D. Suppose that the domain R possesses the same properties and, moreover, that the complement of G is connected. By G, we denote the domain obtained from R by a contraction with coefficient tl (that is, x E R ti EX E Q,), and we write D, = D\L‘,. The function U(X, e) E C”(Dc) is a solution of the problem Au = 0 for x E D,, u = p(x)
for x E dD,
u = 0 for x E dQn,, where
x = (x1,x2,x3)
and p(x)
E Cm(8D).
(61) (62) (63)
V. The Boundary The
outer
expansion
Layer
195
cxz u = c CkUk(X)
(64)
k=O
here is not as simple to write out as in the previous examples. It is obvious that UO(X) is a solution of the limit problem (61)(62), defined not only in D,, but also in 0, so that UO(Z) E c”(D). The remaining U&(x) are harmonic functions, equal to zero on dD. Clearly, they must have singularities at the origin, but what singularities? The answer to this question is given by a matching with the inner expansion
(65) k=O
where
< = x6l.
The
functions vk (<) are defined everywhere in R3\G, where = 0 for 5 E dfi. The functions Uk are obviously not uniquely determined and again the question of how to choose them arises. Thus, in the problem (61))(63) th ere is an arbitrariness in the choice of the coefficients of both the outer expansion and the inner expansion. The matching procedure allows us to uniquely determine both uk and ?&. This is done in the following way. First of all we explain the form of the functions uk(2) for Ic > 1, assuming that uk(z) = O(]X[~‘“) as x + 0. (Why this growth is in fact admitted is also not difficult to surmise, but we shall not go into this so as to keep the presentation simple.) It is easy to verify that the functions of the form Pj (x) lxl2j1 where Pj is a homogeneous harmonic polynomial, are harmonic. Hence, the linear combination U. = CTIi Pj(x)1x1f2je1 of these functions would fit the role of the function uk if U. were equal to zero on dD. But this residual on the boundary is easily corrected, substracting from U(x) a function U*(X) that is harmonic everywhere in D and such that U*(X) = G(X) on dD. In a neighborhood of the origin the function U*(X) can be expanded in a Taylor series, and the sum of terms of the Taylor series of fixed degree j is a homogeneous harmonic polynomial Qj(x). Thus, the asymptotics G(x)  U*(X) = C:ii Pj(x)1xm2Je1  c,“=oQj(x) holds as z $ 0 for the difference 6  u*. we Set ?& = 6  U* and, in order to distinguish the components of different functions uk, we equip the polynomials Pj and Qj with second subscripts:
Auk = 0 and ?&(r)
kl t&(x)
=
~Pj,,~x~2’1 j=o

2
Qj,k(x)
E&S X +
0.
(66)
j=o
Once again we emphasize that the Pj,k are arbitrary homogeneous polynomials of degree j and that the Qj,k are polynomials of the same type, but not arbitrary. They are uniquely determined as soon as all the Pj,k have been
A. M. Il’in
196
chosen. Thus, the problem of looking for the outer expansion (64) is reduced to the determination of the polynomials Pj,k. We also carry out similar preparatory work for the coefficients of the inner expansion (65). The principal part of the asymptotics of the functions ‘r&(c) as [ 4 cc will be sought in the form V(l) = C,“=, S,(t), where the S, are homogeneous harmonic polynomials. We “correct” this sum on the boundary dD via the harmonic function v*(t) in R3\fi, which tends to zero as < ) 0~). It is known that at infinity the function w* (E) is expanded in a series where the Tj are homogeneous harmonic polynomials of cj”=oMm2~1~ degree j. We set vr;([) = C(E) = v*(E). Thus, ?.&laD = 0,
vk(t)
=
5
Sj,k(‘$)

FTj,k(<),<,2’’
j=o
iLS < +
(67)
00,
j=o
where again the sj,k are arbitrary homogeneous harmonic polynomials of degree j, and the polynomials Tj,k of the same type are uniquely determined as soon as all the sj,k have been chosen. The final determination of all these polynomials and, thus, that of the functions uk(x) and uk(J) is obtained by matching the series (64) and (65). For this it suffices to substitute the asymptotics (66) and (67) in the series (64) and (65) and t o rewrite one of these series, say, (64), in other variables: u = uo(5)
+ 2
&&z)
=
i=l
= uo(Ic) + f&i
2
i=l
Here UO(X) is a known
Pj,&I),lc,+1
 g
j=o
harmonic
Qj,&)
=
j=o
function,
and therefore
finally
we have
where Qj,o(c) are known harmonic polynomials, and all the remaining nomials have to be found, by equating the series (68) and the series
v
=
2
tk
We will
obtain
&
s,,k(‘$)
j=o
k=O
the following

FT,;k(<),[,2’’
poly
.
j=o
equalities:
sj,, = Qj,lc3, 0 < j 5 ‘k
(69)
Pj,i = Tj,,jl,
(70)
0 5 j < i  1,
V. The Boundary
Layer
197
from which all the polynomials are defined in sequence. From the equalities and, in particular, Se,e = Qe,e = ue(0). The (69) we find Sj,j = Qj,e, constant Se,e uniquely determines the function ve(<) and, thus, all the polynomials j 2 0. It follows from the equalities (70) that Pj,j+i = T’,e, and, in particular, Pe,,i = Te,e. (By the way, we note that the constant Te,e is equal to the product of the capacity of the domain fi by 210(O).) The principal part Pe,ilz(’ of the asymptotics of ~1 (z) uniquely determines ~1 and, hence, all the poynomials Qj,i(z). It follows from the equalities (69) that Sj,j+i = Qj,i, and, in particular, S&J = Qe,i. From the polynomials Si,i(z) and Se,1 one can T’,i([). Further, this process is easily determine vi(<) and the polynomials extended by induction, which leads to the final determination of the outer and inner expansions (64), (65). The whole construction has been carried out formally, but we see that the series (64) is a f.a.s. of the problem (61))(63) for 1x1 >> t, and the series (65) is a f.a.s. for 111 < E ‘. In particular, they both have an asymptotic character for 1x1 = P, where o is an arbitrary number from the interval (0,l). In fact, in this intermediate domain the series (64) and (65) were actually equated, in order to determine their coefficients. One can prove that the series (64) as constructed is a true asymptotic expansion of the solution U(Z,E) for 1x1 > E”, and the series (65) is one for 1x1 < E”. Similar to what was done at the end of 52.1, it is easy to construct from these series the composite expansion (38), which uniformly approximates the solution U(Z, E) everywhere in D,. This completes the study of the problem (61)(63). If instead of the Laplace equation in (61)(63) we have a general secondorder elliptic equation with smooth coefficients and the limiting operator has an inverse, then the method presented above is a bit more complicated, but all the basic ideas are retained. Without loss of generality, we may assume that at the origin the operator coincides with the Laplacian. Instead of simple we must now take solutions of an equation solutions of the form P~(~)~czT~~~ with variable coefficients with sigularities at the origin. (We will continue to let Pj (CC) and Qj(z) denote harmonic polynomials of degree j.) One can show that the solutions of an equation that have power singularities at the origin possess the following asymptotics as n: + 0: C,“=, Z~+~~~(Z)IZI~‘~~~ + Tj,O(<),
w h ere Zj and 2, are homogeneous polynomials and Zk1 = C,“=, @d, 91. Here all the remaining Zk+sji(~c) for 0 < j 5 k  1 are defined up to P~~~~(~)~~~*~, where Pkj1(x) can be chosen arbitrarily. Analogous changes also happen in the study of the coefficients of the inner expansion. The functions vk(J) now satisfy the inhomogeneous equation Auk = fk: (ve, vi, . . , uk1) [). The solutions of these equations are again defined up to arbitrary &j(E) for j 5 Ic. The asymptotics of the solutions at infinity are easily studied and are similar to the asymptotics (67). A matching of the outer and inner expansions, just as in the case of the Laplacian,
198
A. M. Il’in
determines uk and vk uniquely and, thus, we will obtain the asymptotics of the solution U(Z, E). These results carry over without essential changes to other dimensions 1 > 3; however, the situation is significantly more complicated for the twodimensional problem. Attempts at obtaining asymptotics in the form of (64) and (65) are unsuccessful. Nothing is also obtained in the case when we take the factors C’ and In t as a gauge sequence; these factors often arise in problems like those discussed above. Although one can use the same method to obtain asymptotics up to 1In tlern for any m, this result is too weak. It turns out that the correct gauge sequence is the sequence ek ( 1 In EI+ X) j, for j < Ic + 1, where X is some constant depending only on the domain R and the coefficients of the operator at the origin. After introducing this sequence we can match the outer and inner expansions and construct an asymptotic approximate solution up to any power of E. The same method can also be applied to elliptic equations of higher order in the domain D,. One can also apply the method presented in s2.2 to all these problems. It is possible to create the impression that in any problem with a singular perturbation of the boundary one can construct asymptotics via these methods, only having to choose a suitable gauge sequence. In order to dispel this illusion, we consider a simple, but significant example. Example 7. Let G, be a cylinder with a cut along the axis of a small tube: for zr = (x~,cQ) E R2 and y E R1, G, = {zr,~ : 0 < y < YT, E < 1x1 < l}. The boundary value problem: Au = 0, x E G,;
u = 0 for y = 0, y = x and for 1x1 = E, u = f(y)
For simplicity suppose that f(y) consists f(y) = Cz=, c, sinmy. Then the solution following form: 4X,Y,E)
= 2 m=l
= 2 m=l
Gn
lo(mr)Ko(mc) lo(m)Ko(me)
cm[lO(m)]l
(71)
for 1x1 = 1.
of a finite number of harmonics: ~(2, y, E) of equation (71) has the

I0(mt)K0(mr)

lo(mf)Ko(m)
sin
b0(m7) + , ,Jc~i_‘~
my
=
] sinmy + O(E2), m
where (Pi = Ie(mrla(m)Kc(mr), h, = In $yw, T = (21. We see from this that even at an arbitrary fixed interior point of the cylinder (0 < 1x1 < 1, 0 < y < X} it is impossible to choose a single gauge sequence. More precisely: for any indicated point and any sequence Pi such that + 0 as E + 0 and PUN = O(F) for j > ja, there is a Pj+l(c)[PLj(e)ll boundary function such that the solution of the problem (71) is not expanded in a series with respect to Pi.
V. The Boundary
Layer
199
A similar situation arises infrequently (but nevertheless does occur) applied problems and requires a somewhat different approach than discussed in this paper.
$5. A Quasilinear
Parabolic
Equation
To conclude we give an example of a nonlinear problem that gular and can be studied by the method of matching asymptotic We consider the Cauchy problem for the equation %! + [cp(u)]z 4x,
= ~4%z,
in some the one
xsR1,
is also bisinexpansions.
t>to,c
(72)
to, E) = 1ci(z),
(73)
where $(x) E Cm(R1) and is bounded, cp E C”, P”(U) > 0. For the limiting equation (E = 0) a solution of this problem is smooth for times t close to to. Then, starting with some moment of time, a socalled “breaking of the wave” or “gradient catastrophe” occurs ~ a smooth solution no longer exists and the generalized solution is discontinuous. We assume that the solution u(t, Z, 0) for to < t 5 T has a unique such curve of discontinuity. Without loss of generality we may assume that it begins at the origin and to = 1. Also without loss of generality we may assume that $(O) = 0, ~(0) = p’(O) = 0. The characteristics of equation (72) for F = 0 are the lines 2 = Y + s(y)(l+ where y is a parameter (“Lagrangian” Fig. 7 below). The solution
coordinate)
U(G t, 0) = Uo(T has the curve of discontinuity s(0) = s’(O) = 0. The function ds z=
cp(uo(s(t)
t),
(74) and g(y)
= p/($(y))
t) = THY)
(75)
1 = {z,t : z = s(t), 0 < t < T}, s(t) satisfies the Hugoniot condition + 0, t))  cp(uo(s(t)
(see
where
 0, t))
uo(s(t) + 0, t)  uo(s(t)  0, t)
.
We introduce the notation: w(y, t) = dz/dy = l+g’(y)(l+t). By hypothesis the solution U(Z, t, 0) is smooth for t < 0 and a “gradient catastrophe” occurs at (0,O). Hence w(y, t) > 0 for t < 0, g’(0) = 1, g”(0) = 0, g”‘(0) > 0. Assume that g”‘(0) > 0. Then, without loss of generality, we may assume that g”‘(0) = 6. In addition, for simplicity we shall assume that ~“(0) = 1. The outer expansion of the solution of the problem (72))(73) is the series co u = c E4kU&, t), (76) k=l
where UO(Z, t) is the generalized solution of (72)(73) for t = 0. Outside of the curve I the function ~(5, t) is defined by formula (75). For the remaining functions ‘@(x, t) for /c > 1 we write down a recursive system of
200
A. M. Il’in
linear equations which are explicitly integrated in quadratures. All the functions ~k(~,t) E Cm(fl\l), where fi is the strip {z,t : 1 5 t 5 T, 5 E RI}. On the curve of discontinuity 1 the functions Q(z,~) in general have a discontinuity of the first kind, and at the origin, that is, at the point “where the wave breaks”, their asymptotics are rather complicated. One can show that, as x + 0 and t + 0,
m=O
id
where w(y, t) is defined above, @i,j E C”, x, y and t are connected by relation (74), and the inner sum is taken over all integers i and j for which 1 < i < 31c  m  1, j > 0, 2i  j < 41c  m  1. Thus, the singularities of the functions u~(z, t) at the origin grow as Ic f 00. The series (76) is not a f.a.s. of equation (72) in a neighborhood of the origin, and therefore the problem (72)(73) is bisingular. The choice of scales of the inner expansion is carried out in the following way. Let x = ~“5, t = tOr. It is also necessary to take into account that, as a consequence of the nonlinearity of the equation, the variation of scale of the function u also plays a role. At the origin, the solution is equal to zero. Let U(X, t, t) = eYzu(<, 7, t)  67. Then p(u)  u2/2 has order e2y. We choose the scales so that all the terms in equation (72) have the same order. For this we need
y/3=2fla=4+y22a.
(78)
Another equality is obtained from the characteristic equation (74). Since g’(0) = T)‘(O) = 1, in view of (75) U(IC, t,t) and y have the same order 67. Equating the orders of the principal terms x, yt and 9’ in (74), we obtain the equality Q = y + ,8 = 37. From this and (78) it follows that (Y = 3, p = 2 and y = 1. Th us, x = e3t, t = c2r, and the series for the function w(<, 7, e) = u(x, t, t) must begin with the term ~wl(J, T). The exact form of the inner expansion can be obtained from the condition for its matching with the outer expansion (76). For this we need to substitute the asymptotics (77) in the series (76) and pass to the variables t, 7. As a result we obtain the formal asymptotic series
k=l
j=o
Here each of the symbols wk,j (I, T) also denotes an asymptotic series as I[1 + 1~1 + co. From the matching principle it follows that the inner expansion must have the same form, that, is kl W=~t’Clu~t.wk,i(~,~), k=l
(79) j=o
V. The Boundary
Layer
201
where the functions wk,j (E, r) are defined for all <, r and are expanded in the asymptotic series tik,j(<, 3) as 7 + co. The equations for wk,j are obtained in the usual way after substituting the series (79) in equation (72). It turns out that explicit formulas can be written down for these solutions. We go into more detail concerning the principal term of the series (79), the function CW~,~(<,T). The equation for it is the Burgers equation 8%
awl,0 &?
and totic series tion mula:
the asymptotics as 7 series ‘LzIl,o(<,~). It is is the Whitney fold H”  TH + < = 0. w~,o(<, 7) = 2w[A(<,
+
0
w1,0*
a2w1 
*
0 =
+ CC is given by the not hard to see that the function H([,T), i.e., the The function ZU~,O(<, T) 7)1l, where A(<, T)
0,
(80)
abovedefined asympprincipal term of this solution of the equais defined by the foris the Pearcey function
J“, exp (;(z*  2~2~ + 421)) dz. It is not hard to see that the function ~ul,o defined in this way is in fact asymptotically equal to H(<, T) as 7 + CG. One can show that the asymptotic series of this function is equal to the abovedefined series Gl,o(<,~) as 7 4 oo. One can also find all the remaining functions wl,,j(e, T) so that the series (79) is a f.a.s. of equation (72) in a neighborhood of the origin and so that it will match with the series (76) in an intermediate domain for small t < 0 and large 3 < 0. It turns out that the domain of matching of these series is significantly wider: the matching is preserved in almost all of a neighborhood of zero in the X, t plane and, respectively, in almost all of a neighborhood of infinity in the E, 7 plane (see Fig. 7). More precisely, the matching is valid as IT/+ I<1 + co
Fig. 7
202
A. M. Il’in
everywhere for r < 0 and everywhere for r > 0, 111 > r1/2. Thus, there is no matching only in a narrow strip along the positive rhalfaxis. On the halfaxis for r > 0 matching is impossible, since in the small this halfaxis coincides with the curve I (s’(0) = 0), the functions uk(z,t) are discontinuous on the curve 1, and the wk,j are continuous functions. Along the whole curve I the outer expansion (76) also cannot be the correct asymptotic expansion of the solution ~(z,t,e), since the function u(x,~, t) is continuous for E > 0. For t > 0 the functions uk(z, t) are smooth on both sides of 1, the series (76) is a f.a.s. of (72) .m a neighborhood of 1 for t 2 ? > 0 and therefore along this curve we observe an exponential inner boundary layer. The correct scale is easy to choose, starting from equation (72). The new inner variables are < = [X  s(t)]tp4 and t > 0. We will look for the inner exnansion
v = 9 E41ct&(<, t)
(81)
k=O
in a neighborhood of the curve obtained in the usual way:
1. The
equations
for the coefficients
Uk are
(82)
d2uk x2 ~
+
$[b’(t)

(P’(%))vk]
=
Gk(<,t),
for
k
>
1,
(83)
where the righthand sides Gk are known functions of the preceding vi, i < k. Equations (82)(83) are essentially ordinary differential equations relative to <, since the time t only occurs parametrically in them. The functions ?&(<, t) must be found for all < E R1 and for 0 5 t 5 T. In order to match the series (76) and (81) in a neighborhood of 1 (that is, as z + s(t) f0 and C + fco; see Fig. 7), it is necessary to pose the correct boundary conditions for equations (82))(83) as c + &co. This is done by the method already indicated above. In the outer expansion (76) we need to replace z by s(t) + c4(. and to expand all the uk in Taylor series. Then
u = g E4kP;(C, t), k=O
where
the Pk are polynomials of degree k in < that are expressed explicitly by f 0, t). Therefore equations (82), (83) must be supplemented by the boundary conditions
$$(s(t)
uk(c, t)  P,‘((,
t) + 0 as < f 03.
(84)
The problem (82), (84) is. solvable in view of the Hugoniot condition; its solution tends exponentially fast to its limits at infinity, but this solution is not unique. Let Y(<, t) b e some fixed solution of the problem (82), (84). Then
V. The Boundary
Layer
203
Y(<+co(t), t) for any function co(t) is also a solution of the problem (82), (84). It turns out that the displacement cc(t) can be defined from equation (83) for Ic = 1. For the problem (83))(84) to b e solvable for Ic = 1 it is necessary and sufficient that &(ce(t)[Pz(t)  Pi(t)]) = go(t), where go(t) is a known function. Thus, the function va(<,t) = Y(C + cc(t),t) is defined up to one ar,bitrary constant. After the solution zlc(<,t) has been chosen, we can proceed to define the function vi(<, t). It is also not uniquely defined, up to the term ci(t)aY/a<, since aY/a< is a solution of the homogeneous equation (83) and tends exponentially to zero as ]C] + 00. The coefficient cl(t) is also chosen from the solvability condition for the problem (83))(84) for k = 2. It must satisfy the equality &(cl(t)[Pof(t)  P;(t)]) = gl(t), where gl(t) is a known function. Thus the solution vi(C, t) is also defined up to a constant. The rest of the process of constructing the coefficients of the series, (81) continues along the same line. After all the zli have been constructed for 0 < i < rE, the solution ~k+i(<, t) of the problem (83))(84) is defined up to an arbitrary constant. Thus the series (81) constructed in this way for any choice of the constants is a f.a.s. of equation (72) in a neighborhood of the curve I for t > % > 0. But as and the order of these singulart + 0 the functions vk(<, t) h ave singularities ities increases as k increases, which is an indicator of the bisingularity of the problem (72))(73). As t + 0 the series (81) cannot be the correct asymptotic expansion of the solution U(Z, t, E). In addition, the functions vk are so far not uniquely defined. It is not possible to determine them uniquely starting only from the problems (82))(84). All these difficulties are overcome by matching the series (81) and (79). In a neighborhood of the origin the series (79) is the correct asymptotic epansion of U(Z, t, E). It turns out that the series (79) and (81) can be matched and in the matching process the values of the constants occurring in the expressions for the wk((‘, t) are clarified. To do this, we need first of all to clarify the asymptotics as r + 03 of the previously constructed functions wk,l) ([, 7). In addition, it is necessary to substitute these asymptotic series for wk,j in the inner expansion (79), to pass from the variables 5, r to the variables <, t, and to obtain the formal series C,“=, 6’ )$i Id t . Gi,j (C, t), where the Gz,j are also formal series as t + 0. This is the asymptotics of the functions vk(<, t) as t + 0, from which they are already uniquely determined. Strictly speaking, we should also have looked for the inner expansion in the variables c, t in a more complicated form, including the terms t’ Id t. But it is possible to prove afterwards that the only nonzero coefficients are those where j = 0 and 1 is a multiple of four. Thus, the construction of the asymptotic expansions is complete. The series (79) gives the correct asymptotics of the solution ~(2, t, E) in the domain {z:, t : /x] < E”*, ItI < &}, the series (81) gives it in the domain {z., t : ]Z  s(t)1< P2 , c2pZ < t < T}, and the series (76) gives it in the remainder of R, where cri and ,& are arbitrary positive numbers such that al < 3, ~2 < 4, and ,& < 2. As was shown in $2.1, it is possible to construct a composite asymptotic
A. M. Il’in
204
Fig. 8 expansion, uniformly approximating more complicated form. We shall expansion. Let 19(z) = 2Y1(1 + sgn H(<, T) is understood as a solution continuous everywhere except for H(f0,~) = rJ’; for 7 > 0. Then qz,
t, E) = uo(z,
t) + vo(C, t) +
~(2, t, F) everywhere in Q. Here it has a only write down the principal term of this z) be the Heaviside function. The function of the equation H”  Hr + < = 0, which is the positive Thalfaxis (see Fig. 8), so that the function w,0(1,7)

uo(s(t)
+
o,tp(c)
 uo(s(t)  0,t)[l  Q(C)1  cH(J,7)  WWC) + J;,
(85)
where C = (z  s(t))~~, < = ICC~, 7 = tce2, uniformly approximates the solut,ion U(Z, t, E) of the problem (72)(73) everywhere in the strip R. Here the function uo(z, t) is defined by formula (75), wl,o(<, 7) is defined by formula (80) and VO(<, t) is th e solution of the problem (82), (84). As was explained above, this solution is defined up to a translation of the argument <. However in forrnula (85) this translation is completely concrete. It can be computed explicitly, but is not given here because it is cumbersome. The difference G(z, t, E)  ~(2, t, t) has order O(e4) for 1x1 + ItI > Const and has order O(e21 lncl) for Ix~~I + Itee < Const. In domains that are intermediate between these domains the error of the approximation also has order intermediate between ~‘1 In E/ and t4.
Comments The physics
on the Literature
mathematical problems considered in this paper and engineering. It is difficult to trace the whole
arose originally history of their
in ap
V. The Boundary
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205
pearance but, at least, by the 19th century in the works of Laplace, Maxwell and Kirchhoff concrete problems were studied that in fact contained boundary layers (see Nayfeh (1973) and Eckhaus (1973)). Later it became clear that all areas of natural science and engineering were rich in similar singular problems, but the birth of this theme and even the term “boundary layer” is customarily associated with Prandtl’s lecture at the Third International Congress of Mathematicians [Prandtl (1905)]. In this lecture he put forth the idea of considering a boundary layer in the flow around a body by a fluid with small viscosity and he wrote down the equation in the inner variables. It is necessary to point out that among many mathematicians and applied mathematicians the term “boundary layer” continues to be associated solely with this difficult and as yet not completely solved problem (see Schlichting (1960)). And mathematical boundary layer theory often means the study of the nonlinear equations for the boundary layer functions in the flow problem. There are many results in this area (see, for example the survey paper by ignored in the present paper. Oleinik [Oleinik (1968)]) w h’ ICh a,re completely After arising in fluid mechanics, the term “boundary layer” was actually extended to all similar singular problems, related by the commonness of their mathematical expression and the nature of the asymptotics of their solutions. This is the way the term is understood in the present paper. This also explains the terms “outer” and “inner” expansions: in Prandtl’s problem the flow of the fluid is first described outside of the body, outside of the boudary layer, and then it is described inside the boundary layer. A rather large amount of time passed before mathematicians arrived at a systematic study of singular problems with boundary layers. For partial differential equations this was done in the 1950s (see Levinson (1950), Oleinik and Pontryagin (1961)). A (1952)) Ladyzhenskaya (1957), and Mishchenko large role in attracting attention to similar problems was played by Friedrichs [Friedrichs (1955)]. (Singular problems for ordinary differential equations were considered in the present paper only as examples, preparatory to studying partial differential equations. In essence they were not considered here at all. One can learn more about this theme and the history of the problem from two monographs (Vasil’eva and Butuzov (1973) and Mishchenko and Rozov (1975)).) The most complete study of the mathematical questions for partial differential equations, encompassing a wide class of problems with an exponential boundary layer, was given by M. I. Vishik and L. A. Lyusternik (see their papers [Vishik and Lyusternik (1957) and (1960)] as well as the survey paper by Trenogin (1970)). In particular, these authors studied problems more general than those discussed in Examples 3 and 4. For a wide class of systems of differential equations they found conditions under which the asymptotics of a solution is the sum of an outer expansion and an exponential inner expansion. This situation was termed regular degeneration in their first paper [Vishik and Lyusternik (1957)]. Th e method of investigation presented in 31.2 above is often called the VishikLyusternik method.
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Corner boundary layers were introduced by V. F. Butuzov much later and studied by him [Butuzov (1973) and (1977)]. The problems which we called bisingular in 52 have been studied for a long time by applied mathematicians and researchers in mechanics. Properly speaking, this class also includes Prandtl’s flow problem, which we discussed above. The matching method also arose in mechanics. This method is often interpreted as a method for combining or joining asymptotic expansions. For more about the history of the question and various applications the reader is referred to several monographs [Cole (1968)) Nayfeh (1973)) Van Dyke (1964)]. The first rigorous mathematical results concerning the justification of the asymptotics constructed by this method appeared only in the 1970s. One must note the work of K. I. Babenko, who obtained and gave a justification for the asymptotics of the solution in the problem of the flow around a threedimensional body by a fluid for small Reynolds numbers [Babenko (1975); see also Proudman and Pearson (1957)]. In fact, the matching method was applied and rigorously justified by V. M. Babich and his students (see Babich’s paper III in this volume and the monographs by Babich and Buldyrev (1972) and by Babich and Kirpichnikova (1974)). F or certain elliptic boundary value problems the asymptotics were constructed and justified in various papers by the author and others [Il’in (1976), Il’in and Lelikova (1975), Sverdlovsk collection (1979)) Sverdlovsk collection (1980)]. The presentation given above is mainly based on this work. In fact this method is very close to the method of Van Dyke [Van Dyke (1964); see also Fraenkel (1969)]. In the works of V. G. Maz’ya, S. A. Nazarov and B. A. Plamenevskij one sees the development of a somewhat different approach to the study of bisingular problems. The solution is sought as a sum of outer and inner expansions. Here the coefficients of each of the expansions do not have increasing singularities. They belong to fixed classes of function spaces, which must be chosen. The righthand side of the equation and all the residuals arising in the righthand side are divided by a special method into suitable righthand sides of the equations for the outer and inner expansions. Above we made an attempt at demonstrating this method in $2.2 using the simplest example. For more details about this method and results see their work [Maz’ya,Nazarov and Plamenevskij (1981), (1984a), (198413); Nazarov (1981)]. In $3 we presented the result of a paper by Il’in and Lelikova [Il’in and Lelikova (1975)]. H owever, the proofs in that paper are too complicated. They can be significantly simplified, following work of Lelikova [Lelikova (1978)], where the analogous problem is solved for a parallelepiped. For a system of equations arising in magnetohydrodynamics similar asymptotics were studied by Kalyakin [Kalyakin (1982), (1983)]. The problem of $4 was presented in a paper by Il’in [Il’in (1976)], where most of the attention was occupied with the more difficult twodimensional case. The asymptotics of solutions of analogous problems for elliptic equations of higher order was obtained in the monograph by Maz’ya, Nazarov and Plamenevskij (1981). The diffraction of waves by thin prolate bodies is also
V. The Boundary
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related to problems of this type (for a survey and bibliography see the paper by Fedoryuk (1985)). The problem of 55 was studied in a paper by Il’in [Il’in (1985)], which also contains a bibliography relating to equation (72). Singular problems for equations with weak nonlinearities are often studied just as if they were linear. The interest is only in those nonlinear problems where new effects arise. From the works in which the asymptotics of the solution of a nonlinear bisingular problem are studied up to any power of the small parameter, we point out the paper by Kalyakin [Kalyakin (1984)]. The study of the asymptotics of solutions in a neighborhood of a discontinuity of the limit solution has been done for many problems arising in physics by V. P. Maslov and his students (see Maslov and Omel’yanov (1981), Dobrokhotov and Maslov (1982), and Maslov, Omel’yanov and Tsupin (1983)). In this case they did not study the influence of a small dissipation, as in 55, but the influence of a small dispersion, which is considerably more complicated. These works apply the method of twoscale expansions in combination with the methodology presented in $1. The method of twoscale expansions, in which a f.a.s. is sought directly as a function of the fast and slow variables (for example, as a function of x and [ in $2) has been completely left aside in the present paper. This is a very robust method, which apparently is irreplaceable in the study of rapidly oscillating solutions (as in Fedoryuk’s paper I in this volume). However, in boundary layer problems in the absence of rapid oscillations this method does not give any particular advantages in comparison with the method of matching asymptotic expansions. (However, the latter, with some tension concerning the composite expansion (38), can also be considered to be a variant of the method of twoscale expansions.) The asymptotics in a neighborhood of a single solitary wave, studied in the works of Maslov and others (see Maslov and Omel’yanov (1981), Dobrokhotov and Maslov (1982), and Maslov, Omel’yanov and Tsupin (1983)), can be carried out by any of these methods. But the study of these asymptotics in a neighborhood of a collision point of two waves requires an application of the method of twoscale expansions. We also note that the study of the asymptotics in a neighborhood of a point “where the wave breaks” represents a very complicated and as yet unstudied problem in problems with small dispersion. In this paper we have also not discussed another method, which is particularly useful in some nonlinear problems. It consists in expanding not only the desired function in a series in the small parameter, but also the independent variable. This method has the name of the stretched parameter method or the PLK (PoincarBLighthillKuo) method. One can find a survey of these methods and a bibliography in the book by Nayfeh (1973). From the papers in which various bisingular problems have been studied using the method of matching asymptotic expansions we also point out works by Novokshenov (1978), Sojbel’man (1984) and Shajgardanov (1985).
208
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References* Babenko, K. I. (1975): The perturbation theory of stationary flows of a viscous incompressible fluid at small Reynolds numbers. Preprint no. 79, Inst. Prikl. Mat. (Institut,e of Applied Mathematics), Moscow, 71 pp. (in Russian) Babich, V. M., and Buldyrev, V. S. (1972): Asymptotic Methods in ShortWave Diffraction Problems. Nauka, Moscow, 456 pp. English transl.: Springer Series on Wave Phenornena 4, Springer, Berlin Heidelberg New York 1991. Zbl. 255.35002 Babich, V. M., and Kirpichnikova, N. Ya. (1974): The Boundary Layer Method in Diffraction Problems. Izd. Leningrad. Gos. Univ. (Leningrad State University Press), Leningrad, 124 pp. Zbl. 317.35001. English transl.: Springer Series in Electronics and Photonics 3. Springer, Berlin Heidelberg New York 1979. Butuzov, V. F. (1973): Asymptotic properties of the solution of the equation p’du!?(x, y)~ = f(z, y) in a rectangular domain. Diff. Urav. 9, no. 9, 165441660. Zbl. 271.35016. English transl.: Differ. Equations 9, 12741279 (1975) Butuzov, V. F. (1977): Corner boundary layer in mixed singularly perturbed problems for hyperbolic equations. Mat. Sb. 104, no. 3, 460485. Zbl. 367.35007. English transl.: Math. USSR Sb. 33, 4033425 (1977) Cole, J. D. (1968): Perturbation Methods in Applied Mathematics. Blaisdell, Waltham, MA, 260 pp. Zbl. 162,126 Dobrokhotov, S. Yu., and Maslov, V. P. (1982): Multiphase Asymptotics of NonLinear Partial Differential Equations with a Small Parameter. Sov. Sci. Rev. Sect. C, Math. Phys. Rev. 3, 221~311, Harwood Acad. Publishers, New York. Zbl. 551.35072 Eckhaus, W. (1973): Matched Asymptotic Expansions and Singular Perturbations. NorthHolland, Amsterdam, 145 pp. Zbl. 255.34002 Fedoryuk, M. V. (1985): Scattering of a plane wave by a cylindrical surface with a long perturbation, Izv. Akad. Nauk SSSR, Ser. Mat. 49, no. 1, 160193. Zbl. 607.35068. English transl.: Math. USSR Izv. 26, 1533184 (1986) Fraenkel, L. E. (1969): On the method of matched asymptotic expansions. Parts IIII. Proc. Cambridge Philos. Sot. 65, 2099231,233251, and 2633284. Zbl. 187,241 Friedrichs, K. 0. (1955): Asymptotic phenomena in mathematical physics. Bull. Amer. Math. Sot. 61, no. 6, 4855504. Zbl. 68,164 Il’in, A. M. (1976): A boundary value problem for a secondorder elliptic equation in a domain with a narrow slit. I. The twodimensional case. Mat. Sb. 99, no. 4, 514537. Zbl. 333.35033 English transl.: Math. USSR Sh. 28, no. 4, 4599480 (1978) Il’in, A. M. (1977): A boundary value problem for a secondorder elliptic equation in a domain with a narrow slit. II. A domain with a small cavity. Mat. Sb. 103, no. 2, 265284. Zbl. 359.35017. English. transl.: Math. USSR Sb. 32, no. 2, 227244 (1977) Il’in, A. M. (1985): The Cauchy problem for a quasilinear parabolic equation with a small parameter. Dokl. Akad. Nauk SSSR 283, no. 3, 530~534. Zbl. 627.35044. English transl.: Sov. Math., Dokl. 32 (1985), no. 1, 133136 Il’in, A. M., and Lelikova, E. F. (1975): The method of matching asymptotic expansions for the equations EAU  a(~, y)u, = f(z,y) in a rectangle. Mat. Sb. 96, no.
* For the convenience of the reader, references to reviews in Zentralblatt fur Mathematik (Zbl.), compiled by means of the MATH database, and Jahrbuch iiber die Fortschritte der Mathematik (Jbuch) have, as far as possible, been included in this bibliography.
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4, 5688583. Zbl. 313.45004. English transl.: Math. USSR Sb. 25, no. 4, 533548 (1976) Kalyakin, L. A. (1982): Asymptotics of the solution of a system of two linear equations of MHD with a singular perturbation. I. The standard problem in an elliptic layer. Diff. Urav. 18, no. 10, 17241738. Zbl. 516.76029. English transl.: Differ. Equations 18, 123881249 (1983) Kalyakin, L. A. (1983): Asymptotics of the solution of a system of two linear equations of MHD with a singular perturbation. II. Complete asymptotic expansion. Diff. Urav. 19, no. 4, 628664. Zbl. 538.76119. English transl.: Differ. Equations 19, 461475 (1983) Kalyakin, L. A. (1984): Longwave asymptotics of the solution of a hyperbolic system of equations. Mat. Sb. 124, no. 5, 966120. Zbl. 566.35066. English transl.: Math. USSR, Sb. 52, no. 1, 91114 (1985) Ladyzhenskaya, 0. A. (1957): On equations with a small parameter in the highest derivatives in linear partial differential equations. Vestnik Leningrad. Gos. Univ. 7, no. 2, 104120 (in Russian) Zbl. 91,95 Lelikova, E. F. (1978): A method of matched asymptotic expansions for the equation &uau, = f in a parallelepiped, Dill’. Urav. 14, no. 9, 163881648. Zbl. 395.35005. English transl.: Differ. Equations 14, 11651172 (1978) Levinson, N. (1950): The first boundary value problem for cAu + A(z, y)u5 + B(5,y)uy + C(z,u)u = D(z,y) for small t. Ann. Math. 51, no. 2, 428445, Zbl.
36,68 Lomov, S. A. (1981): Introduction to the General Theory of Singular Perturbations. Nauka, Moscow, 398 pp. Zbl. 514.34049; English transl.: Am. Math. Sot., Providence (1992) Maslov, V. P., and Omel’yanov, G. A. (1981): Asymptotic solitonlike solutions of equations with small dispersion. Usp. Mat. Nauk 36, no. 3, 633126. Zbl. 463.35073. English transl.: Russ. Math. Surv. 36, no. 3, 733149 (1981) Maslov, V. P., Omel’yanov, G. A., and Tsupin, V. A. (1983): Asymptotics of certain differential, pseudodifferential equations and dynamical systems with small dispersion. Mat. Sb. 122, no. 3, 197219. Zbl. 567.35078. English transl.: Math. USSR Sb. 50, 191212 (1985) Maz’ya, V. G., Nazarov, S. A., and Plamenevskij, B. A. (1981): Asymptotics of Solutions of Elliptic Boundary Value Problems under Singular Perturbations of the Domain. Izdat. Tbilisi Univ. (Tbilisi University Press), Tbilisi, 206 pp. (in Russian), Zbl. 462.35001 Maz’ya, V. G., Nazarov, S. A., and Plamenevskij, B. A. (1984a): The DIrichlet problem in domains with thin bridges. Sib. Mat. Zh. 25, no. 2, 161179. Zbl. 553.35018. English transl.: Sib. Math. J. 25, 297313 (1984) Maz’ya, V. G., Nazarov, S. A., and Plamenevskij, B. A. (198413): Asymptotic expansions of eigenvalues of boundary value problems for the Laplace operator in domains with small apertures. Izv. Akad. Nauk SSSR, Ser. Mat. 48, no. 2, 347371. Zbl. 566.35031. English transl.: Math. USSR, Izv. 24, 321345 (1985) Mishchenko, E. F., and Pontryagin, L. S. (1961): 0 n a statistical problem of optimal control. Izv. Akad. Nauk SSSR, Ser. Mat. 25, no. 4, 477498 (in Russian). Zbl. 122,366 Mishchenko, E. F., and Rozov, N. Kh. (1975): Differential Equations With a Small Parameter and Relaxation Oscillations. Nauka, Moscow, 247 pp. English transl.: Plenum Press, New York London (1980). Zbl. 482.34004 Nayfeh, A. H. (1973): Perturbation Methods. Wiley, New York. Zbl. 265.35002 Nazarov, S. A. (1981): The VishikLyusternik method for elliptic boundary value problems in domains with conic points. I. The problem in a cone. Sib. Mat. Zh. 22, no. 4, 1422163. Zbl. 479.35032. English transl.: Sib. Math. J. 22, 5944611
A. M. Il’in
(1982). II. The problem in a bounded domain. Sib. Mat. Zh. 22, no. 5, 1322152. Zbl. 479.35033. English transl.: Sib. Math. J. 22, 753769 (1982) Novokshenov, V. Yu. (1978): A singular integral equation with a small parameter on a finite interval. Mat. Sb. 105, no. 4, 5433573. Zbl. 391.45002. English transl.: Math. USSR Sb. 34, no. 4, 4755502 (1978) Oleinik, 0. A. (1952): On equations of elliptic type with a small parameter in the highest derivative. Mat. Sb. 31, no. 1, 104117. Zbl. 49,76 (in Russian) Oleinik, 0. A. (1968): Mathematical problems of boundary layer theory. Usp. Mat. Nauk 23, no. 3, 365. Zbl. 184,319. English transl.: Russ. Math. Surv. 23, no. 3, l66 (1968) Prandtl, L. (1905): Uber Fliissigkeitsbewegung bei sehr kleiner Reibung. In: Verhandlunger des dritten Internationalen MathematikerKongresses, 1904. Teubner, Leipzig, pp. 4844491. Jbuch 36,800 Proudman, I., and Pearson, J. R. A. (1957): Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. J. Fluid Mech. 2, no. 3, 2377262, Zbl. 77,391 Schlichting, G. (1960): Boundary layer theory. Translated from Grenzschichttheorie, Karlsruhe (1958), McGrawHill, New York. Zbl. 81,196 Shajgardanov, Yu. Z. (1985): Asymptotics with respect to a parameter of a solution of a higherorder elliptic equation in a neighborhood of a curve of discontinuity of the limiting equation. Diff. Urav. 21, no. 4, 7066715. Zbl. 578.35019. English transl.: Differ. Equations 21, 482490 (1985) Sojbel’man, Ya. S. (1984): Asymptotics of the capacity of a condenser with plates of an arbitrary form. Sib. Mat. Zh. 25, no. 6, 167181, Zbl. 569.35020. English transl.: Sib. Math. J. 25, 966978 (1984) Sverdlovsk collection (1979): Application of the Method of Matching Asymptotic Expansions to Boundary Value Problems for Differential Equations. Ural Sci. Center Akad. Nauk SSSR, Sverdlovsk, 110 pp. (in Russian) Sverdlovsk collection (1980): Differential Equations with a Small Parameter. Ural Sci. Center Akad. Nauk SSSR, Sverdlovsk, 158 pp. (in Russian) Trenogin, V. A. (1970): Development and applications of the asymptotic method of Lyusternik and Vishik. Usp. Mat. Nauk 25, no. 4, 123156, Zbl. 198,145. English transl.: Russ. Math. Surv. 25, no. 4, 119156 (1970) Van Dyke, M. (1964): Perturbation Methods in Fluid Mechanics. Academic Press, New York. Zbl. 136,450 Vasil’eva, A. B., and Butuzov, V. F. (1973): Asymptotic Expansions of Solutions of Singularly Perturbed Equations. Nauka, Moscow, 272 pp. (in Russian) Zbl. 364.34028 Vishik, M. I., and Lyusternik, L. A. (1957): Regular degeneration and boundary layer for linear differential equations with a small parameter. Usp. Mat. Nauk 12, no. 5, 3122, Zbl. 87,286. English transl.: AMS Selected Translations, Ser. 2, vol. 20, 239364 (1962) Vishik, M. I., and Lyusternik, L. A. (1960): Solution of some perturbation problems in the case of matrices and selfadjoint and nonselfadjoint differential equations. Usp. Mat. Nauk 15, no. 3, 3380. Zbl. 96,87. English transl.: Russ. Math. Surv. 15, no. 3, l73 (1960)
VI. The Averaging Method for Partial Differential Equations (Homogenization) and Its Applications N. S. Bakhvalov,
G. P. Panasenko,
and A L. Shtaras
Contents Foreword
..213
.....................................................
Chapter 1. Problems of the Mechanics of Nonhomogeneous Described by Partial Differential Equations ............................... with Rapidly Oscillating Coefficients
Structures
............... $1. Media with Periodically Arranged Inhomogeneities 1.1. Stationary Temperature Field. ........................... 1.2. System of Equations of Elasticity Theory. ................. 1.3. Nonstationary Problems. ................................ 52. Strongly Nonhomogeneous Media. ............................. 2.1. Filamentary Structure: the Scale Effect ................... 2.2. Dispersive Structure .................................... Structures ............... 2.3. Other Strongly Nonhomogeneous Chapter 2. Asymptotic and Numerically Asymptotic Methods of Solving Problems of the Mechanics of Nonhomogeneous Structures $1. $2. 53. 54. 55.
Separation of the Fast and Slow Variables ...................... Averaged Equation of Infinite Order ........................... .................... Expansion with Respect to Two Parameters The Boundary Layer Method in Averaging Problems. ............ Description of Processes in Periodic Media by Means of Functions Depending on the Fast Variables ..........
213 213 .215 .217 ,218 .219 .219 ,221 .221
.. .222 .223 ,224 .226 .228 .230
212
N. S. Bakhvalov,
G. P. Panasenko,
Chapter 3. Numerically Asymptotic for Weakly Nonlinear Problems .. References
Methods .. . ..
..,...................................................238
and A. L. Shtaras
. . .. . .
. . .. . .. .
,231
VI. The Averaging
Method
for Partial
Differential
Equations
213
Foreword In this paper we consider numerically asymptotic methods for solving partial differential equations with rapidly oscillating periodic coefficients and their application in the solution of a number of problems of mechanics. The paper consists of three chapters. In the first chapter we construct the simplest mathematical models of fields and processes in nonhomogeneous periodic media and formulate the main results of the asymptotic analysis of such models, study the singularities of nonhomogeneous structures in the case when the physical properties of the components are differentiated into one or several orders. In the second chapter we present asymptotic and numerically asymptotic methods for solving problems of the mechanics of nonhomogeneous structures. The application of these methods to model problems is accompanied by brief generalizing summaries. The third chapter is devoted to the asymptotic analysis of weakly nonlinear problems. In what follows we use the following system of references to formulas inside a given section and to sections inside a given chapter. In referring to a formula from another section we give the number of that formula preceded by a period and the number of the section. We refer to sections from other chapters in the same way. For example: formula (1.2) means formula (2) of $1 of the present chapter, formula (2.1.3) means formula (3) of $1 of Chapter 2, and 52.1 means $1 of Chapter 2.
Chapter 1 Problems of the Mechanics of Nonhomogeneous Structures Described by Partial Differential Equations with Rapidly Oscillating Coefficients $1. Media
with
Periodically
Arranged
Inhomogeneities
A medium with a periodic structure is understood to be a medium consisting of a periodically repeated element (a cell). Such media are associated with composite materials, consisting of two or more homogeneous substances; the volumes of these substances are alternate periodically with period much larger than the characteristic size of the molecules, but at the same time much smaller than the characteristic spatial dimension of the problem. Further the characteristic spatial dimension of the problem is taken to be one; then the
214
N. S. Bakhvalov,
G. P. Panasenko,
and A. L. Shtaras
side (edge) of a cell of periodicity can be considered as a dimensionless small parameter E. We shall give examples of geometric models of the simplest periodic structures: layered, filamentary, and dispersive. Here the term “geometric model” means a partition of the space R” (s = 1,2 or 3) into sets that fill up each of the components of the composite material. In the examples we consider the twocomponent case, and the indicated partition of R” is defined by describing one of the sets. This set (denoted BE) corresponds to the part of the space occupied by the component, called the filling or the reinforcing material, while the component that “fills” the complement is called the matrix. a) A layered structure is a union of the layers C’, = {X E RS 1 x1 E (jt, (j + @E)}, 0 < 13 < 1, that is, B, = U,“_, Cj. b) Filamentary structure. Let ,/31, . . . , PM be bounded twodimensional domains, By, . . . , B& cylinders, Bj” = {I E R3 1 (52, &) E &, (1 E R}; BT is the cylinder obtained from By by some orthogonal transformation which maps the
line with directional cosines ($,$,~i), kz,&  kg) E Bjy}, ICI, kz and k3 are intesets (j, ICI, kz, k3) either the closures of the do not have common points, and that a fi
nite number of distinct cylinders Byklkzk3 intersect the unit cube. The union of the cylinders BTklk2k3 over all integers k1, ka, k3 is denoted Bj, and the union of the Bj over j from 1 to A4 is denoted by B. B is a lperiodic set, which is the union of disjoint cylinders oriented in < M directions. As a geometric model of a system of filaments we take the set B, = {x E R3 1 X/E E B}, where E is a small positive parameter. c) Dispersive structure. Let Go be a strictly interior subdomain of the unit cube Q = (0, I)“, Gkl...k, = {I E R” I (
(1) where K, and KM are the thermal the matrix, respectively.
conductivity
coefficients
of the filling
and
VI. The Averaging
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215
In the examples considered above, the set B = (5 E R” 1 t< E B,} does not depend on E and is periodic with period 1, Kc(x) = K(z/E), where K(t) is lperiodic with respect to
1.1. Stationary an example of the form eauation
in which boundary
Temperature Field. Above we already considered as of a physical characteristic the thermal conductivity coefficient (1). As a function of the point x the temperature satisfies the
there
is a summation
dB, of B, the following [u] = 0,
from 1 to s over the repeated indices. compatibility conditions hold:
[K(z/~)du/dn]
= 0,
On the
(3)
a jump of the function on the boundary of B, (in this case to be smooth). This problem corresponds to a generalized formulation as an integral identity. This remark is also related to the analogous problems formulated below. Suppose the sample fills a domain G, independent of 6, on whose boundary the temperature p(x) is given, also independent of E: where
[.I denotes
dB, is assumed
u(x)
= y(x),
x E dG.
(4
216
N. S. Bakhvalov,
G. P. Panasenko,
and A. L. Shtaras
temperature field in a sample The problem (2))(4) d escribes a stationary of a composite material. A more general statement of the problem, which allows for the possible anisotropy of thermal conductivity properties of the filling and the matrix, for the complexity of the geometry of the components and for the presence of thermal sources distributed in the composite, has the form:
=f(x)
for x E G,
where the ai, are bounded piecewise smooth respect to
functions, conditions
lperiodic with aij([) = a+(<)
with a positive constant K that is independent of q, <, and c. For s > 1 and values of the small parameter E realizable in practice the diin practice, since rect numerical solution of the problem (5), (6) IS . unrealizable it requires too dense a grid in order to “catch” the characteristic variability of the coefficients (the number of grid points is considerably greater than E“). In this connection it becomes necessary to study the problem (5), (6) asymptotically as E + 0. Problems of this type were posed and solved in 1973374 by SanchezPalencia (see SanchezPalencia (1980)), De Giorgi and Spagnolo (1973) and Bakhvalov (see Bakhvalov and Panasenko (1984)). For the problem (5), (6) it follows from their work that the principal approximation to a solution as F + 0 can be determined according to the following algorithm: 1) solve s (r = 1, . . , s is an index parameter) cell problems relative to the functions NT(<) of period 1
2) define the quantities 3) solve the averaged cients
iL,j = & aik(<)&(iVj problem: the elliptic
+ &)dE; equation with
= f(x), and with
boundary
x 6 G,
constant
coeffi
(8)
condition
Suppose dG E C”, f(x) and (p(x) E C”(G), a~([) do not depend on t. Then the solution ve of the problem (8), (9) is the principal term of the asymptotics of U(X) as E f 0; in fact, the following estimate holds: In$gc 12~ 7101 = O(E).
(10)
VI. The Averaging
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217
The problem (8), (9) d oes not contain rapidly oscillating coefficients and thus it can be solved numerically on a grid with a coarse stepsize. We see from the algorithm for constructing the averaged equation (8) that in order to determine its coefficients &j we must solve auxiliary cell problems of the type (7). A solution of (7) is understood to be a function N, that belongs to the completion r/i/,’ of the set of lperiodic C” functions in the W;(Q)norm and satisfies the following integral identity for all ‘p E I@;:
sQ
aiig(IVr
+&)$dC 3
2
= 0.
The problems (7) are solved numerically; in this connection the method of constructing an approximate solution to the problem (5), (6) is termed numerically asymptotic. To determine the socalled effective coeficients i&j packages of standard programs have been created. The estimate (10) proves that if the coefficients a%j(l) d o not depend on t, then the hypothesis of equivalent homogeneity is true. 1.2. System of Equations of Elasticity Theory. The same arguments as at the beginning of 51 show that a physicalmechanical characteristic such as the modulus of elasticity tensor a:;, defining the connection between the stresses g” and the deformations es from Hooke’s law (which is linear (gf = abler)), in media with a periodic structure, is an eperiodic function of the form a,“,“(%/~), where ufj (E) is lperiodic with respect to [i, . . . , ES. In the case of a twocomponent composite the tensor a,“j”(<) is piecewise constant and takes the values afja for 5 E B and aijlclrn for < $ B. The linear stationary system of equations of elasticity theory in the displacements has the form x E
where the Aij (5) are piecewisesmooth, tions with elements u;;(l), satisfying
(11)
G,
bounded, lperiodic the conditions
s x s matrix
for all < E R”, for any symmetric matrix ll$ll, K > 0 is a constant; U(X) f(x) are sdimensional vector functions, f(z) E C(G), dG E C”, and a bounded domain in R”. Suppose that the displacements u are equal to the boundary of G: 4G = 0. The statement corresponding
of the problem integral identity.
(ll),
(12)
is understood
func
and G is 0 on
02)
in the sense of the
218
N. S. Bakhvalov,
G. P. Panasenko,
and A. L. Shtaras
If the at;(<) do not depend on E, then, as in $1.1, the hypothesis of equivalent homogeneity holds, that is, the solution of the problem (II), (12) is close to the solution of some problem of elasticity theory with constant coefficients, describing the stressdeformed state of a homogeneous material:
coefficients & of the In fact, IJu‘u~IIL~(GJ = O(&). Th e constant s x smatrix by the aboveindicated algorithm, with averaged problem (13) are determined the following changes: the functions Nj are sdimensional square matrices, and the sum N, + & occurring in formula (7) is replaced by N, + JTE, where E is the identity matrix. The problems (7) are solved numerically using a package of programs for the computation of the effective elasticity tensor aij. 1.3. equation
Nonstationary
Problems.
du d 4x/~) dt  dzi with
boundary
The solution
du aij (x/e) dzj
where C(G
= f(x,t),
heat
x E G, t E (&to)
condition 4m
and initial
>
of the nonstationary
= 0
condition
4t,o = 0, c(t) is a lperiodic piecewise smooth function, is close to the solution of the averaged x [O,tol),
= f(x,t),
c(t) > K. > 0, f problem
E
x E G, t E (&to),
I&=0,wOlt,O =0,2=sc(E)@; Q
the estimate 11~  weII~~(~~(e,~~)) = O(d) holds. The solution of the nonstationary system of equations
with
boundary
and initial
theory
condition
conditions UltzO = 0,
where c(72
of elasticity
wdtIt=o
= 0,
p(J) is a lperiodic piecewisesmooth function, x [0,trJ]), is c 1ose to the solution of the averaged
p(E) > po > 0, f problem
E
VI. The Averaging
Method
for Partial
Differential
Equations
x E G,
t E (&to),
219
In a number of problems of the mechanics of composite materials it is necessary to compute the stresses
where u1 is the lth component of the vector U. One can prove that
II&
 2
+ gy I,=z,~~llL2 3
= mm
from which we obtain the following formula for the computation of stresses 0,” = af;(x/r)e; where $ = 0.5(dv&/dxj + dvA/dxl) the elements of the matrices &(t)
+ O(A),
is the average deformation, and &f;(t) are
= A&&(E)
I
+ t&.
Averaged problems can also be obtained for nonlinear problems of heat conduction, elasticity theory, and also for a number of nonlocal problems.
52. Strongly
Nonhomogeneous
Media
The results of the previous section are related to the case when only one small parameter E is present in the problem. Actually other dimensionless parameters are already implicit in the model, and these may be small or large. Composite materials are as a rule constructed according to the principle of constrasting the physical properties of the components. In the mathematical realm this means that, besides the small parameter E that characterizes the relationship of the period of the structure to the characteristic size of the domain G, the problem actually includes a parameter w that characterizes the relationship of the physical constants of the components. In the model examples (1.2)( 1.4) we shall assume that w is a large parameter, K(t) = w for E E B, K(E) = 1 for 6 q! B. 2.1. Filamentary Structure: the Scale Effect. It turns out that the introduction of a second parameter in the problem (1.2)(1.4) has an
N. S.Bakhvalov, G. P. Panasenko,and A. L. Shtaras
220
essential influence on the qualitative behavior of the solution. This, in the case of a filamentary structure B, the hypothesis of equivalent homogeneity does not always hold (but only when c2w is a small parameter). For e2w < 1 in a number of cases explicit formulas have been obtained for the eflective coeficients of the equation defining the principal term of the asymptotics of the solution as E + 0, w + 00. This circumstance removes the necessity of solving the cell problems (1.7) (or parts of them) numerically. For the difference of solutions of the problem (1.2)(1,4) and the new averaged problem an estimate is obtained in which in addition to E there are small parameters wl and c2w. We proceed to more precise statements. We write M l&j
=
c
l9,y’$,
(1)
q=l
where 8, = mes(B,Y n Q) is the volume fraction of the cylinders of the qth direction, and $, yz, and 73”are their directional cosines. Lemma 1. For the matrix III?ijII to be positive definite and suficient that there be three noncoplanar vectors among (Y:,Y;, $1, q = 1,. . . , M.
it is necessary the M vectors
In the sequel we shall assumethat this condition holds. Theorem 1. Suppose that the cylinders B: have a C” boundary and suppose there exist a,0 > 0 such that w = O(P), t2w = O(8). Let there exist a solution to the problem
& (i?$$
=O,
XEG,
VOOI~~ = ‘~1~~ and
voo
E C”(c).
Then
max IuUeeI =O(e+w1+e2w). ZEG Theorem 1 supplies a justification for the hypothesis of equivalent homowith egective coeficients determined from the explicit formulas (l), in the case e2w << I.
geneity
C”,
Theorem 2. There are a set B, a domain G, and a function ‘p, all of class such that for any function v(x) that does not depend on E and w,
max Iu  211f+ 0 as 6 + 0, w + 00, and e2w + 00 za
For the system of equations of elasticity theory (1.11) with coefficients of order w in B, explicit formulas are obtained in the book by Bakhvalov and Panasenko (1984) for effective coefficients analogous to (1). An expansion of the solution u in the cases e2w > 1 and e2w = const was constructed by Panasenko (1990).
VI. The Averaging Method for Partial Differential
Equations
221
2.2. Dispersive Structure. We again consider the problem (1.2))(1.4), when B, is a dispersive structure with a smooth boundary. In this caseit turns out that the abovementioned effect of having no averaged description of the problem as E’W + co (scale eflect) does not occur. The principal terms of the asymptotics of ue(z) is discovered according to the following algorithm: 1) solve three cell problems relative to the lperiodic functions AT,
[ E R3\B,
= 0
m,+&=O,
l~dB,
where r = 1,2,3 is an index parameter. 2) solve three cell problems relative to N:(c): AN;
= 0,
dN1 dn
‘=&(~r+Ev),
< E Go,
~E~Go,r
= 1,2,3;
3) determine the quantities
gj=s
(Nj’ + &)nids
+ (1  mes GO)&,
8GO
where (nl, n2, n3) is an exterior normal vector to dGc; 4) solve the averaged problem = 0,
x E G;
vo&G =
Theorem O(@)
‘PIBG’
3. Suppose there exigt a,P > 0 such that 6 = O(wPa), and there exists a solution of the problem (2) belonging to C”(G).
maxzcE Iu  ~001= O(E + w‘)
wl
=
Then
as E+ 0 and w + co.
2.3. Other Strongly Nonhomogeneous Structures. Bakhvalov and Panasenko (1984) constructed the full asymptotic expansions of solutions of the systems of equations of elasticity theory with rapidly oscillating strongly varying coefficients defined in the whole space. Equations and systems of elliptic and hyperbolic types were considered. Another large class of strongly nonhomogeneous structures consists of the socalled frame structures, which contain two small parameters. The averaged description of these structures was obtained by Bakhvalov and Panasenko (1984)) and by Panasenko (1992). Perforated media (media with periodically located holes) have been studied by Cioranescu and Saint Jean Paulin (1979)) Bakhvalov and Panasenko (1984)) Berdichevskij (1983)) and SanchezPalencia (1980). The question of the computation of effective characteristics of nonhomogeneous media with a periodic structure was also raised in the classical work of
222
N. S. Bakhvalov,
G. P. Panasenko,
and A. L. Shtaras
Poisson, Maxwell, Rayleigh, Voigt and Reiss, and thus it has an old history. In §§l2 we considered an approach to the study of processesin nonhomogeneous media, based on the asymptotic analysis of partial differential equations that describe these processeson a microlevel. This approach is presented in detail in the monographs by Bakhvalov and Panasenko (1984), Bensoussan, Lions and Papanicolaou (1978), Berdichevskij (1983), Kozlov, Oleinik, and Zhikov (1992), Oleinik, Yosifian, and Shamaev (1992)) and SanchezPalencia (1980). Another approach, based on the analytic solution of twodimensional problems of elasticity theory by the methods of the function theory of one complex variable, is developed in books by Grigolyuk and Fil’shtinskij (1970) and Van Fo Fy (1971). We also note that questions, related closely to the averaging of processesin composites, of the asymptotic study of equations in domains with a finegrained boundary were studied by Marchenko and Khruslov (1974). Nonperiodic nonhomogeneous media were considered by Bensoussan, Lions and Papanicolaou (1978), and by Zhikov, Kozlov, Oleinik and Kha (1979).
Chapter 2 Asymptotic and Numerically Asymptotic Methods of Solving Problems of the Mechanics of Nonhomogeneous Structures There are usually two requirements concerning the asymptotic methods with respect to a small parameter E that are applied in differential (or more general functional) equations: 1) Each one of the sequence of recursive problems obtained in the asymptotic solution must be simpler than the original problem. For example, a nonlinear problem must be replaced by a linear one, a differential problem by an algebraic one, a problem not integrable in quadratures by one that is integrable, and so on. 2) Each of these problems can be written so that the parameter E does not appear explicitly in it. Both of the requirements reflect a fully developed historical tradition of solving problems with a small parameter. These traditions arose during the years when there was no powerful computational technique, and they corresponded to the classical representation of the solvabilty of the problems. A problem was considered to be solved if explicit analytic formulas had been obtained for a solution, and in turn, such formulas are obtained more simply when the problem contains fewer undefined parameters. The appearance of computers and the corresponding mathematical software, based on effective numerical methods, permitted researchers to con
VI. The Averaging
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Equations
223
sider as elementary many mathematical problems that previously could not be solved in any practical way. This new class of methods, combining asymptotic procedures with the numerical solution of elementary problems, are usually called numerically asymptotic. Below we give some examples of such methods. To simplify the presentation these methods are illustrated using examples of applying them to linear partial differential equations, although they have also been successfully used for the solution of nonlinear problems and operator equations (see Bakhvalov and Panasenko (1984), Berdichevskij (1983), and SanchezPalencia (1980))
51. Separation We present
the method
of the Fast and Slow Variables using
the example
L’u z &
of the elliptic
 f(x)
= 0,
equation
x E R”,
where the akl(<) are the lperiodic functions from 5 1.1 and f(x) is a sufficiently smooth function with zero mean (f)z,~ = 0 that is Tperiodic in ~1,. . ,x,. Here and later on (.)y,a denotes the integral over the sdimensional cube y E (0, CX)~. We look for a solution of the equation that is Tperiodic in ~1,. . . , 5, and has zero mean ((u),,~ = 0). An asymptotic solution is sought as a series
d’=)  C&(x,
J)IE=s,c’
i=O
where u~(z, 6) are lperiodic functions in 61, . . . , &. We substitute (2) in (l), apply the chain rule, group the terms according to like powers of E and equate the sum to zero. We have a sequence of equations in the ui:
where the Ti depend on Ui1 and ui2. The condition for the existence solution in the class of functions that are lperiodic in Jr is the equality
of a
Pi)<,1
(4)
A solution
of equation
= 0.
(3) is determined ui(z,
E)
=
Wi(27
up to an arbitrary t)
+
function
vi(x):
%(x)7
where (wi)c,i = 0. The validity of condition (4) is ensured by means of a suitable choice of the functions vJ for j < i. The equalities (3), (4) are equivalent to the recursive chain of equations
L<<wi = Ti(x, E),
= .fiCx),
224
N. S. Bakhvalov,
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and A. L. Shtaras
where the Ti depend on we,. . , wi1, ~0,. . . , vi1 and the fi(x) depend on fe = f. The constant coefficients &l are determined wo )...) Wi,tlO )...) vii; from the algorithm of 51.1. All the Wi, zli are determined successively from (5), and, hence, so are the ui. Thus, the problem (1) containing the small parameter E is asymptotically (that is, up to terms of order any power of E) reduced to the sequence of problems (5) which do not contain the small parameter. These problems are solved by numerical methods. Using a priori estimates for the original problem, one can show that the difference of the exact solution of the original problem from the Nth partial sum of the series has order O(fN) in the norm W,‘( (0, T)“) and, in particular, 21 = ue + O(E) in the Lz( (0, T)“) norm, where vc is a solution of the averaged equation
Eva = f(x), and us is Tperiodic
in xi,.
(VO)z,T= 0,
(6)
, x,.
Remark 1. The equation (1) IS . considered in the whole space R”. In the case of the boundary value problem (1.1.4), (1.1.5) the series ~(““1 satisfies the boundary condition only with error O(F) and thus ceases to be a formal asymptotic expansion of the solution; only estimates for the zeroth and first approximations (of the type (1.1.10)) h ave been obtained. For boundary value problems in a halfspace or a layer the asymptotic expansion of a solution is constructed in $4 below. The method is intended for the solution of equations with rapidly oscillating coefficients (Bakhvalov and Panasenko (1984)) SanchezPalencia (1980)). As in the multiscale method, an asymptotic solution u(a) is sought as a series in powers of 6’ with coefficients that depend on the fast (< = X/E) and slow (z, t) variables. Substituting the series u(03) in the equation, grouping the terms in like powers of E and equating the resulting coefficients to zero gives a recursive chain of equations relative to ul (cell equations). These equations are solvable, but not uniquely, up to some functions ~1, and only for righthand sides that satisfy solvability conditions. The solvability conditions also form a recursive chain of equations relative to the functions Vi, “parallel” to the chain for the ~1. Thus, the construction of the asymptotic solution u(03) is reduced to the solution of two chains of problems that do not depend on a parameter.
$2. Averaged
Equation
of Infinite
Order
We demonstrate how the method works using the example of the problem from $1. A formal asymptotic solution of equation (1.1) is sought in the form
VI. The Averaging
Method
for Partial
Differential
Equations
225
where i = (ii,. . , il); ij E (1,. . , s} is a multiindex, (ij = I is the number of components of the multiindex, called its length, Div = d’w/dxi, . . . dxi,, the Ni([) are periodic functions in
 2 cl’ l=O
c H,(E)D2~lc=z,t [iI=
 f(x),
(2)
where Hi(F)
= &Ni,...i,
+ &
(ai,
(Wiz...il)
Hi, (I) = LEENil + &Q/ci, (
hi, starting from the requirement that the coefficients Hi(<) in the t and the derivatives Diu in (2) be constant: Hi([) E hi = const 0 if [iI 5 1. Ordering the Ni so that the relation Nj 4 N, holds (i is “longer” than j), we will find a recursive chain of equations the Ni: < E RS, LEti”& + T,(t) = hi, (3)
where
(4 We want to find solutions functions that belong to I&i. and sufficient that hi = ./(Q
of the equations For there to exist dN,...i, sja&
(3) in the class of lperiodic solutions of (3) it is necessary
+ ai,izN,...i,
Thus, the quantities hi and Ti depend on Nj for Ijl < Iii; this gives us the possibility of defining Ti(J) and I 1, se q uentially by induction on Iii, and then, solving equation (3), the possibility of finding N,(J) E I&‘;. Then, as all the Ni (and parallelly the h,) have been found, we have:
We define
V(X)
as the asymptotic
solution hiDiu 121=1
of the relation  f(x)
 0,
(5)
226
N. S. Bakhvalov,
G. P. Panasenko,
and A. L. Shtaras
called the “averaged equation of infinite order”. It turns out that hi = 0 for (iI = 0,l and hiliz = iLiliz, so that the principal term of the averaged equation (5) coincides with (1.6). We look for an asymptotic solution of (5) in the series form w 
(6)
C&j(X),
j=o where the vj(x) do not depend on e. If we substitute this series in (5), we can obtain a recursive chain of “moment equations”, from which the vj(x) ^ are determined: Lvj = fj(x), the fj depend on the functions va, . . . , vjuj1 and their derivatives, and fe(x) = f(x). The method is applied to a wide class of problems with rapidly oscillating coefficients, and, in particular, to the problems of Chapter 1 above (see Bakhvalov and Panasenko (1984)), although Remark 1 from $1 remains valid. Thus, the method is useful for the solution of equations with rapidly oscillating coefficients. An unknown function is introduced (in the general case this is a vectorfunction) that depends regularly on a small parameter, and the goal is set of obtaining an asymptotic equation for this function that does not contain the fast variables 6 = x/e, a socalled averaged equation of infinite order. We look for an asymptotic solution u (ml of the original problem in the form of a power series in E with coefficients that are lperiodic with respect to < and depend on the fast variables < and the derivatives of this function v, where the order of these derivatives is higher as the corresponding power of t increases. We substitute this series in the original equation and group the terms with like powers of t (using the chain rule); we impose the condition that the coefficients of the series that is obtained as a result of the substitution do not depend on the fast variables. This condition gives a recursive chain of equations (cell equations), from which we can successively determine the coefficients of the series ~(~1.
53. Expansion As an example
with
of a problem
with
Respect
to Two Parameters
two parameters
we consider
the equation
(1) with the compatibility 31.2, taking the value assumed to be smooth;
conditions (1.1.3), where K,(t) is the function from on the set B, of the dispersive structure (dB, is see s1.1); f( x ) .is a Tperiodic function with respect
w
to x1, x2, x3 from Cm(R3), JOT JOT JeT f(x)dx = 0; we look for a solution in the class of Tperiodic functions. We give an algorithm for constructing an asymptotic expansion of a solution of the problem (l), (1.1.3), (1.1.4) as E + 0, w + co, E = O(w“), wl = O(@), where o, ,0 are positive numbers.
VI. The Averaging As in §2, an asymptotic
NC<) the N&
Method solution
Differential
is sought
in the form
Coo wy,(t)

are lperiodic
for Partial
functions
4x)
(2.1),
227
where
for < E B,
“+lNj,(J) ,
CEOW
Equations
(2)
for 6 +! B,
of
(3)
 7; ii: E~WTv.jr(x), j=o T=0
Use are Tperiodic functions in zr,x2,xs that belong to Cm(R3) and do not depend on E, w. At the first stage of constructing an expansion we substitute the formal series (2.1) in (1) and, just as in 52, we will obtain a recursive chain of cell equations (2.3). At the second stage these problems are solved asymptotically: the series (2) are substituted in equation (2.3) and the compatibility conditions. Here it is proved by induction that the righthand sides Ti and the constants hi have the expansions
k=O
k=O
After grouping the terms with the same powers of w we set them zero and find a recursive chain of problems for the N,T,: 1) the N$ are sought as lperiodic solutions of the problems AN:,
f Ti,k = hi,k, N$
2) the Ntk
are lperiodic AN:,
with
boundary
t E dB;
of the equation = ht>k,
I E B,
(4)
condition
tlN,l, d
+ Z,k
to
( E R3\B,
= N&l,
solutions
equal
dn
8N,z, =
dn
+
nil
(N!...il,k

Nty...il,k)
)
t
’
dB.
(5)
After constructing the functions N:k and NZtk we will find that the compatibility conditions (1.1.3) are satisfied asymptotically exactly (that is, up to arbitrary powers of E and wl), and relation (2.2) becomes an averaged equation of infinite order of the form
228
N. S. Bakhvalov,
G. P. Panasenko,
 f + Fw~+’
and A. L. Shtaras
c
k=2
hi,kD%
Iii=2
hi,kDiV k=l
N 0,
(61
lil=k
whose asymptotic “solution” at the third stage is sought in the form (3). Substituting (3) in (6) after reducing similar terms gives a chain of elliptic equations for v,k of the form
i”t&k
= fmk(x),
3: E R3,
J
tl,kdX
=
0
(O>TP
with Lo
the =
conditions
&Gfj&),
that x fmk(
)
the v,k are Tperiodic in x1,x2, and x3, where are smooth Tperiodic functions with zero mean on
combinations of the derivatives of u,,kl for m~+k~<m+k,m~~m,k~~k. In case B, is a filamentary structure, the solution is also constructed in two stages; however, the procedure of constructing the asymptotics of the solutions of the cell problems is more complicated: it is necessary to consider solvability conditions for problems of the type (4), (5) in each of the cylinders which leads to an increase of the order in w of the solutions Ni, and, B;ilkakg) hence, of the constants hi. The method is intended for the solution of equations with rapidly oscillating coefficients which depend additionally on one of several small (large) parameters. In the first stage one applies the method of separation of the fast and slow variables or the method of the averaged equation of infinite order under the assumption that the additional parameters are finite. In the second stage the cell equations (problems), depending only on the additional parameters, are solved asymptotically. Here one can use different asymptotic methods; the peculiarity in applying them is the requirement that they be compatible with the original expansion in E (the period of the structure). After substituting the asymptotic solutions of the cell problems into the series u(O”) we will obtain an averaged equation of infinite order containing both the basic and the additional parameters. In the third stage the asymptotic solutions of the averaged equation are sought in the form of a regular series in the basic and the additional parameters, as well as combinations of them. Recently another type of equations with many parameters was considered by Bakhvalov and Eglit (1994, 1998). (0,T)3,
foe
=
$4. The
f,
fmk
are linear
Boundary
Layer
Method
in Averaging
Problems
We will demonstrate how this method works using the example (l.l), defined in the halfspace {zr > 0) with boundary conditions
uIzl=o = dx’),
of equation
(1)
VI. The Averaging
Method
for Partial
Differential
Equations
229
where 2’ = (~2,. . . ,x,), If(~)1 5 crecZZ1, f and g are Tperiodic in 22,. . . , Z, (where TN l), and f,g E C”. The coefficients of (1.1) satisfy the conditions from $1 and cl, c2 > 0. As in 52, we look for asymptotics in the form (2.1), but we replace the requirement that the Ni (I) be lperiodic in &, . . , & by the requirement that N&t) be a sum N:(l) + N,!(t), where the N:(t) are lperiodic functions of , Es, equal to the functions Ni(<) from $2, and the N:(E) are “boundary El,... layer components”, lperiodic in 12,. . . , Es, tending to zero exponentially fast as [r + +oc together with their first derivatives. After substituting this series (2.1) in the operator L’ and the boundary condition (1) we require that the coefficients of the resulting expansion be independent of the fast variables. Then, starting with the equations (2.3), we will obtain the conditions Ni(O, E’) = hi = const for Ni. Since the N,” were defined in 33, we have a chain of equations for the Nt : LCCN; with
boundary
+ T;(t)
= 0,
5 E R; = (0, +cm) x R”1
(2)
condition N;(O,<‘)
= N,o(O,J')+
h,l,
& = 0.
(3)
Here the N,’ are ordered as in $2, the T,” are defined by the equalities (2.4) with the Ni replaced by N/ , and the constant hi is chosen (uniquely) from the condition for the existence of a lperiodic solution in 12, . . . , Is, tending exponentially fast to zero as [i + +co, for the problem (2), (3). After setting L’u(O”)  f and u(~) Ill=c  g asymptotically equal to zero we obtain the averaged equation of injhite order (2.5) and an averaged boundary condition of the form 2.c' z=o
c h;(Dia)lz,=, lil=Z
g(d)
 0.
(4)
In addition an asymptotic solution for this problem appears as a regular series chain (2.6). After substituting (2.6) into (2.5), (4) we will obtain a recurrent of problems for the uj. The method is intended to solve contact problems of several periodic media and boundary values for equations with rapidly oscillating coefficients, defined in a layer or a halfspace. As in the method of the averaged equation of infinite order, we introduce a function v, depending regularly on a parameter, and the asymptotic solution ~(~1 appears as a power series in E with coefficients depending on the fast variables < and the derivatives of the functions U. These coefficients are equal to the sum of a periodic component in 6 and a boundarylayer component (the latter tends to zero as we go farther away from the boundary). After substituting ~(~1 in the equation and the boundary conditions we define the periodic and boundarylayer components from the requirements that the coefficients of the series obtained as a result of this substitution be independent from the fast variables. As a result we will
N. S. Bakhvalov,
230
G. P. Panasenko,
and A. L. Shtaras
obtain asymptotic equations and a boundary condition (averaged problem) for the unknown function v. Another variant of the method proposes a search for a solution as the series from 51, where the functions ZL~are also sums of two components: a periodic one and a “boundarylayer” component. In addition, after substitution we will obtain a recurrent chain of problems for the components 211 and the conditions for solving them in the corresponding classes give a “parallel chain” of boundary value problems for vl analogous to the one obtained in $1. In conclusion we discuss the main ideas of the justification of the methods and the proof of theorems of the type of those in Chapter 1. The formalism for constructing an asymptotic solution is such that, when a partial sum of sufficiently large length of the series u cco) is substituted in the equation, we can obtain a remainder of arbitrary order of magnitude in the parameter. Using an a priori estimate for the original problem, we can show that an exact solution is close to a partial sum of u (O”). In the proof of the results of Chapter 1 additional difficulties arise, which are related to the necessity of taking the remainder into account in the boundary conditions. These difficulties are overcome either by using a maximum principle or by the technique of “cuttingoff” the boundary layer of the functions (see Bakhvalov and Panasenko (1984)).
35. Description of Processes by Means of Functions Depending
in Periodic Media on the Fast Variables
The special characteristic of the methods considered above is obtaining an averaged equation that does not contain the fast variables. However, in a number of cases such an approach can turn out to be rather complicated. Thus, for some nonlinear problems the construction of averaged equations and their solution require unjustifiably large outlays of computational work. It may turn out to be appropriate to describe such processes using functions depending on both the fast and the slow variables, with a characteristic size of variation with respect to both groups of variables that does not tend to ( see Bakhvalov zero as t + 0 (the multiscale method) and Eglit (1983)). We consider the simplest onedimensional model for problems of plasticity in a onedimensional periodic medium: the displacement vector u and the first column g of the stress tensor are connected by the equation
the first column c are connected
of the tensor of relative by the equation
extensions
e = du/dz
and the vector
VI. The Averaging
Method
for Partial
where R(E) and G([, ., ., .) are periodic totic solution in the form
functions
21  ug(z, t) + tu1(5, CT ql(2,
Differential
t, z/t)
Equations
231
of I. We look for an asymp+ ... (3)
t) + EUl(Z, t, 2/E) + ‘. . )
where ui(z, t, I) and cri(z, t, I) are lperiodic functions of <. After suitable transformations the substitution of (3) into (l), (2) leads to a system of equations relative to the unknown functions ~0, 00 and ei (2, t, <) = &Q/d<:
eo = 2,
(el)
Starting with functions uo and cre that do not depend contains an unknown function ei that depends on <.
= 0. on E, the system
(4)
Chapter 3 Numerically Asymptotic Methods for Weakly Nonlinear Problems As was noted in Chapter 2, the development of numerical methods has brought about new possibilities in the development and application of asymptotic methods and, in particular, methods for the construction of uniformly suitable asymptotic expansions for the desired solutions. Below we present one of the methods, the averaging method, in a form in which requirement 1) of Chapter 2 does not always hold. The concepts used below were applied earlier in special cases  see the books of Bogolyubov and and Moseenkov (1976)) Mitropol’skij (1974)) Lomov (1981)) and Mitropol’skij and the article by Shtaras (1977). W e note that in the best known variant of the averaging method, the KrylovBogolyubovMitropol’skij (KBM) method (see Bogolyubov and Mitropol’skij (1974)), both of those requirements do not always hold. For example, the system of ordinary differential equations du/dt corresponds
to the averaged
= tF(u,
t)
(1)
system duo/d7
where 7 = et and (F(u)) = limT,, (2) is a system of integrodifferential
= (F(vo)),
(2)
T’ JoF F(u,t)dt. In the general case equations and in this respect it is more
232
N. S. Bakhvalov,
G. P. Panasenko,
and A. L. Shtaras
complicated than system (1). On the other hand, a solution of (2) is a function ‘uc(r), the principal approximation of an asymptotic solution, and for the next approximation VI(T) we obtain the system h/d7
= (F(w))
+ ~G(v)),
(3)
where G(u, t) is some function which is found from the function F(u, t). System (3), just like the analogous systems for succeeding approximations, explicitly contains E. Numerous examples show that the violation of requirements 1),2) does not obstruct the successful application of the KBM method. One can say the same thing about many other asymptotic methods. We consider a generalization of (l), a system of evolution equations
where ‘u. = (~1,. . . ,u,) T is the desired solution, depending possibly on the space variables (~1,. . , z,) E Z, L is a linear is a nonlinear operator. Suppose that the initial condition u(0)
on time operator
t and and f
= ZLO
(5)
holds for u(t) (the dependence on x will be omitted for brevity). An asymptotic solution of the problem (4), (5), uniformly suitable for t = O(el), is constructed from the solutions of the homogeneous equation
and the inhomogeneous
equation
2
+ Lw = F(t).
We assume that the following conditions hold. Cl. There exists a linear operator M = M(t, F(t) equation (7) has the special solution
s) such that
for any function
t
w=
I
MF(s)ds. 0
C2. Any solution of equation (6) is bounded as t f co. C3. If v(t) is an arbitrary solution of (6) and F(t) = f[v(t)], function MF(s) is bounded as t, s + 00. C4. For the functions MF defined in C3 the following identity
then
the
holds:
t
J to
MF(s)ds = (t  to)(F)0 + G(t),
(8)
VI. The Averaging where
the function
Method
for Partial
Differential
Equations
233
as t + 00, and
G is bounded
ta+T (F)o =J$mTl J
(9)
MF(s)ds,
to
and the limit in (9) does not depend on to. It is also assumed that this limit is uniform in to and the other variables on which MF(s) depends, and equality (9) can be differentiated with respect to these variables, exchanging of the differentiation and the passage to the limit. Suppose also that the equality L(tv) = tLv holds for arbitrary functions v(t) satisfying equation (6). These conditions and also the assumption that the operator f and the solution u are sufficiently smooth allow us to apply the method of twoscale expansions and to construct an asymptotic solution in the form u  vo(t, r) + c
r”[vrc(t,
T) + u&(t,
T)],
7 = et,
(10)
k>l
where the functions IC. For ve we obtain
uk satisfy equation (6) with the averaged problem dv0lar
The
function
wi
respect
to the variables
?J(O,O) = uo.
= (.f[~ol)o,
t and
(11)
Wl =JtM(f[vO] 
is represented
in the form
th,/c3r)ds.
0
It follows from (8), (9) and (11) that the function For succeeding terms of the asymptotic solution problems dtlk/dr
and the explicit
=
expressions Wkfl
=
(A’Uk
t J
M(Avk
+
gk)O,

wi is bounded as t + co. we will obtain the averaged
vk(o,
&Jk/d?
0)
+
=
0,
gk)ds.
(12)
(13)
0
In these equalities we have denoted by A the Frechet derivative of the operator f at the point ~0 and defined the functions gk (t, T) by the previously found terms of the asymptotic solution (10). Formulas (12) and (13) are meaningful if the operator f and the terms of the asymptotic solution are sufficiently smooth and if conditions analogous to C3 and C4 hold at each stage of the asymptotic integration. It follows from (11) and (12) that in order to construct an asymptotic solution it is necessary to solve a Cauchy problem for systems of integrodifferential equations, and the operator M, which may have a very complicated analytic form, occurs in the definition. Moreover, the solutions of the averaged problems must be looked for among functions satisfying equation (6). This shows that in the general case the practical construction of an asymptotic
234
N. S. Bakhvalov,
G. P. Panasenko,
and A. L. Shtaras
solution (10) is not trivial. However, this complexity cannot be assumed to be an insufficiency of the method; this complexity is the “cost” for the accuracy of the results. One can make an even stronger assertion: the principal term of the asymptotic solution must satisfy equations (6) and (ll), otherwise there will have to be secular terms in the asymptotic solution. In the simplest cases of (lo)(13) we obtain formulas for the known asymptotic methods. For example, if L z 0 and f = f(u, t,e) is a nonlinear vectorfunction, depending periodically on t, then the function ~0 is equal to the principal approximation obtained in the KBM method (see Bogolyubov and Mitropol’skij (1974)). Th e same thing is also true in the case when L = diag[iXi, . . , ix,], where the Xj, j = 1,. . . , n, are real numbers. If L = diag[/\i(d),
cf = fo(ct) + df(d)u,
. . . , X,(ct)],
where H is a square n x n matrix, then the formulas of the regularization method (see Lomov (1981)) can be derived from (lo)( 13). If L = P,(d/dq,. . . , a/axs) of degree T with (where p,(Jl,. , b) is. a polynomial constant coefficients), and the desired solution is lperiodic in 2, then the operator M can be constructed using Fourier series expansions and thus one can generalize the results of the book by Mitropol’skij and Moseenkov (1976). Thus we have obtained a satisfactory description of the weakly nonlinear interaction of strongly dispersive waves. But if the dispersion is small in the system under consideration, which happens, for example, for m = 1 and L = diag[Xi,. . . , X,]d/dz, then the use of Fourier series is associated with significant difficulties. Such problems can be considered using the averaging method along characteristics (Shtaras (1977)), which is also a special case of the method (10))(13). W e consider the simplest example, the Cauchy problem for the system of equations of gasdynamics, written in the Riemann invariants,
where X(c) = 2 + Xi< +. . . . In the case of smooth solutions the averaged problem for the principal term of the asymptotic solution, the functions us,1 (y, r), wc,z(z,r),y=z+t,z=st,hastheform dUOl/dT
+
h(VOl

(~02)0)~~01/&/
=
0,
VOl(Y,O)
du02/d7
+
Xl

(7Jol)o)dvo2/a~
=
0,
2102(z,
=
To(Y,
O),
=
so(z,
0).
(15) (7Jo2
0)
Using the divergence of the equations (15), we can show that (~ci)a = the system (15) into (To(YY, O))o, (vo2)o = bo(z, 0)) 0, and thus we can divide two independent Cauchy problems. This allows us to simplify the solution of the nonlinear problem (15) in an essential way. In the method being described, despite its abstractness, the main concept is the classical concept of a secular term: condition C4 allows us via averaging to extract the secular terms from the solutions of the inhomogeneous
VI. The Averaging
Method
for Partial
Differential
Equations
235
equations (7), and the method of twoscale expansions allows us to construct asymptotic solutions that do not contain secular terms. The problems (4), (5), for which conditions ClC4 are already assumed to hold, are merely the simplest examples of problems in which secular terms arise. We consider a class of problems in which the appearance of strong (in comparison with those problems considered above) secular terms is possible. Consider a Cauchy problem of the form a2u/at2
+ Lu = Ef[U],
(16) (17)
where the same assumptions are made on u, L and f as were made above. The class of problems under consideration is described using the following conditions: C5. uo E TO, ~1 E ri, where TO and rr are sets of initial data such that . the solution of the Cauchy problem a2v/at2
+ Lv = 0,
v(0)
= 60,
g(o)
= ii1
(18)
is bounded as t + co, UO E Fe, Ur E rr. C6. There exists a linear operator M = M(t, s) such that for any function F(t) the equation dw2/dt2 + Lw = F(t) has the special solution w = s,” MF(s)ds. C7. If v is any bounded solution of the problem (18) and F(s) = f[v(s)], s ), w h ere Ml and M2 are nonlinear operators then MF(s) = sMrF(s) +M#( such that the functions MlF(s) and MzF(s) are bounded as t, s + co. Moreover, MlF(s) e 0 if {F} = 0, where {.} is some mean such that {F} = F for F independent of t. C8. If {F} = 0, then condition C4 holds for such an F. We look for an asymptotic solution of the problem (16), (17) in the form U
+avo(t,rl) + vo(T,rl) + c Pk[Vk(h7) + Kc(7,77)+ Wk(GTT, 7) + Wk(4T>VII, k>l
for r =
pt,
7 =
p’t,
and p = &,
(19)
where the functions vk, Vj, and WI; satisfy equation (18) and WI = 0. Substituting (19) in (16) leads to a chain of equations of which the first has the form a2w, d2W1 p2vo a2v, p+Lw2=2~~
at2
atar at&j aT2+ f[vo+ vol.
In order to construct a special solution of equation (20) it is natural condition C6. It is essential that the first two terms in the righthand (20) can be integrated “explicitly”. For W2 we obtain
(20)
to use side of
T
w, = t(dw1/dr
+ dV()/d~) +
M(f[vo s0
+ Vo]  d2Vo/Ch2)ds.
(21)
N. S. Bakhvalov,
236
Condition Eliminating
C7 implies that such quadratic
G. P. Panasenko,
In order obtain
to eliminate
the linear
dWl/dT
in (21) can have order t2 as t + co. we obtain the first averaged equation
the integral secularities, d2Va/dr2
and A. L. Shtaras
= {f[vo secularities
+ vi)]}.
(22)
in (21) we use condition
= (f[zJo + Vo]  d2Vo/dT2)
C8 and
 dv,/dq.
An asymptotic solution can further be constructed in a different way. To begin with we assume that the following conditions hold: C9.All the solutions of equation (22) are bounded as r ) co. ClO. If ~0 is any bounded solution of equation (18) and VO is any solution of equation (22), then the function Fl (t, 7,~) is also bounded as 7 + 00 and the following equality holds:
7 J
Fl(t,s,q)ds
= (T  ~o)(Fl)l + Gl(t,7,v),
70
where
the function
Gr is bounded
as r + 00 and
and the properties of the integral (23) are analogous to the properties integral (9). Now the function 201 will be bounded as 7 + 00 if dvc/dq = ((f[~e
of the + VO]
a”vo/aT2)) 1. If conditions C9 and Cl0 do not hold, then the function ZIO must be defined to be independent of T. We may also assume that the remaining terms of the asymptotic solution also do not depend on n. This simplifies the asymptotic integration procedure, but in general it allows us to obtain a uniformly suitable asymptotic solution only for t = O(pl) = O(E‘.~). For succeeding terms of the asymptotic solution we obtain analogous averaged equations if at each step some additional conditions of the type of C7, C8 and Cl0 hold. We obtain
d2Vj/d2
= (4~
ih/av
+ Vc) + h/c),
= ((A(Q
+ h) + hk  d2h/d?2))1,
where the functions hk(t, 7,~) are determined by the previously found terms of the asymptotic solution. For ‘U&+1 and wk+2 we have the explicit formulas Wkfl
=
J
k+k
+
v(%
17))
+
auk/aq)

wk+2
=
t(dWk+l/dT
 d2vk/tk2 and the functions
t J
(a2vk/aT2)(%
0
+
M(A(%(s,
v)
+
hk(t,
7)
s, 7))
+
vk)
+ hk(s, ‘r, Q))ds, ’
wk+2
are bounded
as t + 00 in view of (24).

dvk/dq)ds,
(24)
VI. The Averaging After
we substitute
Method
for Partial
(19) in the initial Vk(O,O)
+
Differential
conditions
h(O,O)
Equations
we will
+w%1(0)
=
6OkU0,
+%2(o)
=
bOk%,
237
obtain
(25) %(0,0)+%
(O,O)
and wk2 = dwkl/dt + 8W&1,1/dT + d‘dk2J/d~ + ince the function wki is defined earlier than the functions vk and vk, it follows from (25) that vk(O,O) can be chosen so that ?&(O, 0) E ri. Since (dwk/dt)(O,O,O) = 0, the function ‘J&?(O) is also defined earlier than the function T/ki. Therefore for Ic 2 1 one can choose so that (d?&/at)(O,O) E ri. (av,lla~)(o,o) The simplest example of problems of the form (16)) (17) can be obtained for function f depends periodically L = 0 and f = f(U, t, c), w h ere the nonlinear on t. However, problems with partial derivatives are more familiar, for exampe the equations for the vibrations of strings, beams, rods, etc. In their book Mitropol’skij and Moseenkov (1976) consider a number of such problems, in which no quadratic secular terms arise, which allows us to circumvent the difficulties with the intermediate parameter p = J; and the variable r = pt. As an example we take the model problem of the KleinGordon equation where
wkl
= wk + wk
~?&2/&,+~v,,/&,.
s
d2u/dt2  d2U/dX2 The
principal
v02b,rl)
we will
and
= Ef(U).
approximation for u(t, x) consists of three terms: where y = z + t and .z = x  t. For these the averaged system
vO(7,rl),
obtain
vei(y, v), functions
11
d2Vo/dr2
=
JJ J J
f(vo1(yy,77)+~02(%rl)
0
d2vol/a~ay
+
Vo(7,rl))4f+
+
Vo(7,77))
0
= 0.5(
l~f(vol(Y,~~
+
Vo2(%77)
0
 d2Vo/d72]dz)
(26)
1)
1
6%()2/d+
= 0.5( 
82&r2,czy)
[f(vol(Y,
77) + Vo2(?
71) +
vo(T,
7))
1.
Here it is assumed that the initial data for u(t, x) are lperiodic in x and that all the solutions of equation (26) are bounded as r + co. It is seen from this example that it is only possible to eliminate the variable r and the parameter ,LLin the case when (26) has solutions that do not depend on 7. This imposes very rigid restrictions on the function f(u).
N. S. Bakhvalov,
G. P. Panasenko,
and
A. L. Shtaras
References* Bakhvalov, N. S., and Panasenko, G. P. (1984): Averaging of Processes in Periodic Media. Nauka, Moscow, Zbl. 607.73009. [English translation: Kluwer, 19891 Bakhvalov, N. S., and Eglit, M. E. (1983): P recesses in a periodic medium which are not describable by averaged characteristics. Dokl. Akad. Nauk SSSR 268, 836840, Zbl. 569.35067. [English translation: Sov. Phys., Dokl. 28, no. 2, 12551271 Bakhvalov, N. S., and Eglit, M. E. (1994): A n estimate of the error of averaging the dynamics of small perturbations of very inhomogeneous mixtures. Zh. Vychisl. Mat. Mat. Fiz. 34, no. 3, 3955414. [English translation: Comp. Math. Math. Phys. 34, no. 3, 3333491 Bakhvalov, N. S., and Eglit, M. E. (1998): Effective moduli of composites reinforced by systems of plates and bars. Zh. Vychisl. Mat. Mat. Fiz. 38, no. 5, 813834. [English translation: Comp. Math. Math. Phys. 38, no. 5, 7838041 Bensoussan, A., Lions, J.L., and Papanicolaou, G. (1978): Asymptotic Analysis for Periodic Structures. NorthHolland Publishing Co., Amsterdam, Zbl. 404.35001 Berdichevskij, V. L. (1983): Variational Principles in Mechanics of a Solid Medium. Nauka, Moscow (in Russian), Zbl. 526.73027 Bogolyubov, N. N., and Mitropol’skij, Yu. A. (1974): Asymptotic Methods in the Theory of Nonlinear Vibrations. Nauka, Moscow (in Russian), Zbl. 303.34043 (see also 2nd edn. Nauka, Zbl. 83,81) Christensen, R. M. (1979): Mechanics of Composite Materials. Wiley, New York Chichester, Zbl. 537.73053 Cioranescu, D., and Saint Jean Paulin, J. (1979): Homogenisation in open sets with holes. J. Math. Anal. Appl. 71, no. 2, 590607 De Giorgi, E., and Spagnolo, S. (1973): Sulla convergenza degli integrali dell’energia per operatori ellittici de1 second0 ordine. Boll. Un. Mat. Ital. no. 8, 391411, Zbl 274.35002 Grigolyuk, E. I., and Fil’shtinskij, L. A. (1970): Perforated Plates and Shells. Nauka, Moscow (in Russian), Zbl. 224.73079 Kozlov, S. M., Oleinik, 0. A., and Zhikov, V. V. (1992): Homogenization of Partial Differential Operators and Integral Functionals. Springer, Berlin Heidelberg New York Lomov, S. A. (1981): Introduction to the General Theory of Singular Perturbations. Nauka, Moscow, Zbl. 514.34049 [English translation: Transl. Math. Monogr. 112 (AMS, Providence 1992)] Marchenko, V. A., and Khruslov, E. Ya. (1974): Boundary Value Problems in Domains with a FineGrained Boundary. Naukova Dumka, Kiev (in Russian) Mitropol’skij, Yu. A., and Moseenkov, B. I. (1976): Asymptotic Solution of Partial Differential Equations. Vishcha Shkola, Kiev (in Russian) Oleinik, 0. A., Yosifian, G. A., and Shamaev, A. S. (1992): Mathematical Problems in Elasticity and Homogenization, NorthHolland, Amsterdam Panasenko, G. P. (1990): Multicomponent homogenization of processes in strongly nonhomogeneous structures. Mat. Sbornik 181, no. 1, 134142 (in Russian). [English translation: Math. USSR Sbornik (1991) 69, no. 1, 14331531
* For the convenience of the reader, references to reviews in Zentralblatt fur Mathematik (Zbl.), compiled by means of the MATH database, and Jahrbuch iiber die Fortschritte der Mathematik (Jbuch) have, as far as possible, been included in this bibliography.
VI.
The
Averaging
Method
for
Partial
Differential
Equations
239
Panasenko, G. P. (1992): Asymptotic solutions of the system of elasticity theory for rod and frame structures. Mat. Sbornik 183, no. 1, 899113 (in Russian). [English translation: Russian Acad. Sci. Sb. Math. (1993) 75, no. 1, 8551101 Pobedrya, B. E. (1984): Mechanics of Composite Materials. Izdat. Mosk. Univ., Moscow (in Russian), Zbl. 555.73069 SanchezPalencia, E. (1980): NonHomogeneous Media and Vibration Theory. Springer, Berlin Heidelberg New York, Zbl. 432.70002 Shtaras, A. L. (1977): Asymptotic integration of weakly linear partial differential equations. Dokl. Akad. Nauk SSSR 237, no. 3, 5255528, Zbl. 395.35062 [English translation: Sov. Math., Dokl. 18, 146221466 (1978)] Van Fo Fy, G. A. (1971): Theory of Reinforced Materials. Nauka, Moscow (in Russian) Zhikov, V. V., Kozlov, S. M., Oleinik, 0. A., and Kha Tien Ngoan (1979): Averaging and Gconvergence of differential operators. Usp. Mat. Nauk 34, no. 5, 655133, Zbl. 421.35076 [English translation: Russ. Math. Surv. 34, 655147 (1979)]
Author Agranovich, M. S. 86 Airy, G. B. 32,108,111,112,126 Akhiezer, N. I. 138,168 Anosov, D. V. 146 Arnol’d, V. I. 17, 18,29, 32,49, 50, 137, 139, 140, 142, 144, 147, 148, 150, 159,160,163,164,167,168 Arsen’ev, A. A. 85,86 Babenko, K. I. 206,208 Babich, V. M. 23, 58,8486,92,95,99, 101, 102, 105, 107, 108, 110, 114, 117, 123,127130,135,160,165,166,168, 206,208 Bakhvalov, N. S. 215,216,220224, 226,228,230,238 Bardos, C. 85,86 Bensoussan, A. 222,238 Berdichevskij, V. L. 221&223,238 Berezin, F. A. 138,139,168 Berry, M. V. 159,168,169 Birkhoff. G. D. 49,164 Birman, M. Sh. 138140,146,169 Blekher, P. M. 60,86 Bogaevskii, V. N. 51 Bogolyubov, N. N. 231,234,238 Bohr, N. 40,159 Borovikov, V. A. 129,130 Brillouin, L. 69 Buldyrev, V. S. 85,86,92,95,99,102, 107,108,110,114,117,127~130,135, 160,165,166,168,206,208 Bunimovich, L. A. 167,169 Burgers, J. M. 201 Buslaev, V. S. 85,86,92,102,127,130 Butuzov, V. F. 205,206,208,210 Bykov, V. P. 160,169
Index Dobrokhotov, Douady, R. Duistermaat,
S. Yu. 48,50,207,208 149,169 J. J. 136, 169
Eckhaus, W. 205,208 Eglit, M. E. 228, 230, 238 Eidus, D. M. 84,86,87 Einstein, A. 159,169 Epstein, P. 159 Faddeev, L. D. 167,171 Fedoryuk, M. V. 35,37,38,4951,58, 70,71, 73,76,88,95, 123, 129, 131, 137,152,163,170,207,208 Feynman, R. 12 Fil’shtinskij, L. A. 222,238 Filippov (ZPhilippov), V. B. 102, 128, 129.131 Fock (=Fok), V. A. 108,125,126,131 Fomin, S. V. 145,146,169 Fourier, J. 20,61 Fraenkel, L. E. 206,208 Fredholm, I. 60 Friedrichs. K. 0. 138,139,205,208 Glazman, I. M. 138,168 Grigolyuk, E. I. 222, 238 Grigor’ev, N. S. 58,84,86 Guillemin, V. 35, 50,85,87, 169 Guillot, J. C. 85,86 GuseinZade, S. M. 29,50 Gushchin, A. K. 85,87
Cauchy, A. L. 7,14,33,47,129 Christensen, R. M. 215,238 Cioranescu, D. 221,238 Cole, J. D. 50,206, 208 Colin de Verdi&e, Y. 136,152,159, 169 Courant, R. 96,104,130
Hadamard, J. 104,129,131 Halpern, B. 145,169 Hamilton, W. 5,6,19 Hankel, H. 59,103 Heading, J. 51,135, 169 Hejhal, D. A. 167,169 Helmholtz, H. 3,4,21 Hooke, R. 217 Hormander, L. 58,87 Hugoniot, C. 199 Huygens, C. 56
De Giorgi, E. 216,238 Debye, P. 335,49 Dirac, P. A. M. 3,103 Dirichlet, P. G. L. 112, 113, 136
Ikawa, M. 85,87 Il’in, A. M. 206,208 Ince, E. L. 44 Ivrij, V. I. 58,85,87
136, 150,
242 Iwasaki, Jacobi,
Author N.
85,87
C. G. J.
5,119
Kalinin, V. L. 167, 171 Kalyakin, L. A. 206,207,209 Kaufman, A. N. 168,170 Keller, J. B. 114, 131, 159, 169 Kevorkian, J. 50 Kha Tien Ngoan 222,239 Khruslov, E. Ya. 222,238 Kinber, B. E. 129,130 Kirchhoff, G. 205 Kirpichnikova, N. Ya. 127,129131, 206,208 Kolmogorov. A. 137,149 Kornfel’d, I. P. 145,146,169 Kozlov, S. M. 222,238,239 Kozlov, V. V. 137,160,163,167,168 Krakhnov, A. D. 159,165,167,169 Kramers, H. 69 Kravtsov, Yu. A. 129,131 Kreiss, H.O. 87 Kronecker, L. 137,140,161 Krylov, N. M. 231 Kucherenko, V. V. 23,40,50,85,87 Kuo, Y. H. 207 Kuzmak, G. E. 50 Ladyzhenskaya, 0. A. 85,87,205,209 Landau, L. D. 135,146,159,170 Laplace, P. S. 135,205 Lax, P. 84,85,87 Lazutkin, V. F. 85,87,113,131,135, 141,149,158,159,163166,170 Lelikova, E. F. 206,208,209 Leray, J. 29,40,49,50 Levinson, N. 205,209 Lichtenberg, A. G. 146,170 Lieberman, M. A. 146,170 Lifshits, E. M. 135, 146, 159, 170 Lighthill, M. J. 207 Lion, G. 50,51 Lions, J.L. 57,87,222,238 Liouville, J. 146, 147 Lomov, S. A. 209,231,234,238 Lopatinskij, Ya. B. 57 Ludwig, D. 58,84,88,101,110,127, 129,131 Lukin, D. S. 32,37,51 Lyapunov, A. M. 47 Lyusternik, L. A. 205,210 Maclaurin,
C.
111
Index Magenes, E. 57,87 Majda, A. 58,85,87,88 Marchenko, V. A. 222,238 Markus, L. 147,148,170 Maslov, V. P. 3,5,23, 25,26,35,36, 38,40,4851,58, 70, 71,73, 76,88, 123,129,131,137,141,152,159,163, 170,207209 Mathieu, E. 44 Matsumura, M. 84,88 Maxwell, J. C. 3,205, 222 Maz’ya, V. G. 206,209 McDonald, S. V. 168,170 Melrose, R. 58,85,88 Menzala, G. 85,88 Meyer, K. R. 147,148,170 Mikhajlov, V. P. 84,85,87,88 Mishchenko, E. F. 205,209 Mishchenko, A. S. 23,49Q51,70&73, 76,88,152,153,163,170 Mitropol’skij, Yu. A. 231,234,237, 238 Mizohata, S. 84,88 Mochizuki, K. 84,85,88 Mogilevskij, D. Sh. 112 Molotkov, I. A. 95,114,128%131 Morawetz, C. S. 58,84,85,87,88,101 Moseenkov, B. I. 231,234,237,238 Moser, J. 137 Murata, M. 85,88 Muravei, L. A. 84,88 Nayfeh, A. A. 50,51,205,206,209 Nazaikinskij, V. E. 51 Nazarov, S. A. 206,209 Nejshtadt, A. I. 137,160,163,167, 168 Neumann, C. 113 Neumann, F. E. 57 Newton, I. 14 Nomofilov, V. E. 131 Novokshenov, V. Yu. 207,210 Oleinik, 0. A. 205,210,222,238, 239 Omel’yanov, G. A. 207,209 Orlov, Yu. I. 129, 131 Osher, S. 58,87 Osipov, Yu. V. 85,87 Osmolovskij, V. G. 160, 170 Palkin, E. L. 32,37,51 Panasenko, G. P. 215,216,220,221, 223,224,226,238
Author Pankratova, T. F. 92,123,130,141, 159,170 Papanicolaou, G. 222,238 Pearcey, T. 201 Pearson, J. R. A. 206,210 Petkov, V. 85,88 Phillips, R. 84,85,87 Plamenevskij, B. A. 206,209 Planck, M. 135 Pobedrya, B. E. 239 Poeschel, J. 149,170 Poincark, H. 94,207 Poisson, S. D. 222 Pontryagin, L. S. 205,209 Popov, G. S. 85,88,89 Popov, M. M. 92,123,128,131 Povzner, A. Ya. 51 Prandtl, L. 205,206,210 Protas, Yu. N. 85,89 Proudman, I. 206,210 Radon, J. 105 Ralston, J. 8589 Rauch, J. 85,89 Rayleigh, J. W. 115, 116,222 Reed, M. 138,170 Reiss, E. 222 Riccati, J. F. 118 Rozov, N. Kh. 205,209 Rubinow, S. 159,169 Saint Jean Paulin, J. 221,238 Sakamoto, R. 89 Samarskij, A. A. 59,89 SanchezPalencia, E. 89, 216, 221224, 239 Schlichting, G. 205, 210 Schonbek, T. 85,88 Schrodinger, E. 3,4,135,159 Shajgardanov, Yu. Z. 207,210 Shamaev, A. S. 222,238 Shapiro, Z. Ya. 57 Shatalov, V. E. 23,4951,70,71,73, 76,88,152,153,163,170 Shnirel’man, A. I. 167, 170 Shtaras, A. L. 231,234,239 Shubin, M. A. 85,87,88,138,139, 168
Index
243
Simon, B. 138,170 Sinai, Ya. G. 145,146,167,169,171 Sjostrand, J. 58,88 Smirnov, V. I. 96,131 Smyshlyaev, V. P. 92,128,130 Sojbel’man, Ya. S. 207,210 Solomyak, M. Z. 1388140,146,169 Sommerfeld, A. 4,40,49,129,159 Spagnolo, S. 216,238 Sternberg, S. 35,49,50,150,169 Sternin, B. Yu. 23,49951,70,71,73, 76,88,152,153,163,170 Stoneley, R. 115,116 Strauss, W. 84,85,88 Svanidze, N. V. 163,167,171 Taylor, M. 58,85,88,89 Terman, D. Ya. 165,170 Tikhonov, A. N. 59,89 Trenogin, V. A. 205,210 Tsupin, V. A. 207,209 Ulin, V. V. 101,130 Ursell, F. 101 Vainberg, B. R. 3,36,40,51,59,63, 64,6668,70,71,73,76,85,86,89, 150 Vainshtein, L. A. 160,171 Van Dyke, M. 206,210 Van Fo Fy, G. A. 222,239 Varchenko, A. N. 29,50 Vasil’eva, A. B. 205,210 Venkov, A. B. 167,171 Vergne, M. 50,51 Vishik, M. I. 86,205,210 Voight, W. 222 Wentzel, G. 69 Whitham, G. B. 50,51,95,120,131 Whitney, H. 201 Wilcox, C. 85,89 Yosifian,
G. A.
Zachmanoglou, Zayaev, A. B. Zhikov, V. V.
222,238 E. 84,89 102,131 222,239
Subject amplitude, scattering 69 ansatz 93 asymptotics, longwave (lowfrequency) 64 asymptotics, semiclassical (also called “quasiclassical”) 69 ssymptotics, shortwave (highfrequency) 60 asymptotics, WKB 69 atlas, canonical 17 basis, symplectic beam, Gaussian bicharacteristic bracket, Poisson
18 118 6 19
caustic 8,9, 106, 116 chart 142 chart, canonical 17 chart, symplectic 144 coefficient, diffraction 111 coefficients, effective 217,220 complete vector field 143 condition, averaged boundary 229 condition, Dirichlet boundary 136 condition, ellipticity 56 condition, Hugoniot 199 condition, impedance boundary 114 condition, nondegeneracy, of the Kolmogorov set relative to projection 157 condition, nontrapping 58,70 condition, radiation 59 conditions, Maslov quantization 39 conditions, semiclassical quantization 151 conditions, Sommerfeld 58 coordinates, focal 152 coordinates, ray 8, 10,97,98 cycle, singular 23,33,36 degeneration, discrepancy 151 dynamics effect, scale eigenfunctions eigenvalue
regular of quantum 143 219,221 135, 136 136
205 conditions
Index eigenvalue of the Schrodinger operator 135 eiaenvector 135,139 eikonal 8,95 ’ energy, local 56 equation for oscillations of a pendulum 46 equation, averaged 215 equation, averaged, of infinite order 224,226,227,229 equation, cell 50 equation, Dirac 50 equation, dispersion 99 equation, Duffing’s 46 equation, eikonal 5,8,95 equation, HamiltonJacobi 5,6,43,48, 49,119 equation, Heisenberg 48 equation, Helmholtz 4,40,49,59 equation, homological 156 equation, Jacobi 119 equation, KleinGordon 68 equation, Mathieu’s 44 equation, Schrijdinger 4,40,135 equation, wave 4,36,135 equations, bicharacteristic 6 equations, transport 5,9,31,71,95 evolute 9 expansion, asymptotic 93 expansion, asymptotic, composite 185 expansion, asymptotic, formal 94 expansion, asymptotic, inner (inner expansion) 176 expansion, asymptotic, outer (outer expansion) 176 expansion, Debye 4 expansion, ray 95 extension, Friedrichs 138 family of quasimodes 140 family, finitely meromorphic Fredholm 60 field, shortrange 68 field, stationary temperature 215,216 filling 214 flow 143 flow, phase 19 focus 9 formula, commutation 24,25 formula, composition 11
246
Subject
formula, formula, frequencies function, function, function, function, function, function, function, functions, functions, functions, 32 gallery, germ, group,
Fock’s 126 Liouville’s 10 147 Airy 32, 108, 111, 126 boundary layer 174 generating 16,71 Hamiltonian 142 Pearcey 201 phase 8 spectral 76,82 boundary layer 174 elliptic 46 wavecatastrophe special
whispering complex symplectic
Index
method of matching asymptotic expansions 182,185 method of stationary phase 26,125 method of summation of Gaussian beams 123 method, stretched parameter 207 method, VishikLyusternik 205 method, WKB 40 methods, numericallyasymptotic 217,223 model, geometric 214 momenta dual to coordinates 142 multiplicity of an eigenvalue 139 multiplicity, total, of a spectrum approximated by quasimodes 141 multiplier 120
110 23 17
Hamiltonian 3,56,142 hypothesis of equivalent 215,218,220 index, boundary index, intersection index, Maslov (or 25,27, 28,30, 71, index, Morse 30 integral, Pearcey integral of a classical 147 isotropic 15
norm
operator, elliptic 56 operator, Fredholm 60 operator, hyperbolic 56 operator, LaplaceBeltrami operator, Maslov canonical 137,154,155 operator, precanonical 23 operator, pseudodifferential operator, Schrodinger 135 orthogonality relation 42
homogeneity
151 28 “KellerMaslov”) 150, 153 32 dynamical
138
136 25,30,70,
20
system
Xpseudodifferential operator 20 Xsymbol 4, 20 Xtransform, Fourier 2022 layer, boundary 174 layer, boundary, corner 181 layer, boundary, exponential 178 layer, boundary, parabolic 191 layer, boundary, power 182 manifold, Lagrangian 15,70, 147 manifold, Riemannian 136 map, canonical 18 map, Poincare 163 map, symplectic 17 mapping, reflection 143 material, reinforcing 214 matrix, Green 36,57 matrix, symplectic 17 method of comparing asymptotic expansions (compatibility of asymptotic expansions) 182
prepresentation 21,22 parametrix 36,60,76 parametrix of a Cauchy problem 36, 76 parametrix of a nonstationary problem 62 parametrix of an elliptic problem 76 perturbations, singular, of a boundary 194 plane, coordinate 17 plane, isotropic 17 plane, Lagrangian 17 point, focal 8 point, nonsingular 71 point, ordinary 23 point, regular, of an operator 138 point, singular 23 point, stationary 26 points, turning 40,43 potential 33 principle, Huygens 56 problem, averaged 216,217,221,230, 233 problem, bisingular 182
Subject
problem, problem, oscillating problem, problem, problem, problem, problem, problem, singular product, product, quasimode
Cauchy Lagrangian 7 Cauchy, with rapidly initial data 33 cell 216, 221, 228 eigenvalue 37 elliptic 57 elliptic, with parameter 57 scattering 40,69 singular (problem with a perturbation) 182 scalar 138 skewscalar 16 113,120,139
radius of curvature, ray 6,69,96 ray tube 7 resolvent 58,139 rule, BohrSommerfeld 40 rule, quantization
effective
107
quantization 39
set, caustic 71 set, Kolmogorov 149 set, Liouville 147 signature 153 singularities, Lagrangian 32 solution, formal asymptotic 4,41,69, 93,175,224 solution, fundamental, of a Cauchy problem 36 space, phase 6,141 space, phase, of a classical dynamical system 141 space, Sobolev 57 spectrum 138 spectrum, discrete 138 spectrum, operator 139 spreading, geometric 98 structure, dispersive 214,221,226 structure, filamentary 214,220,228 structure, layered 214
Index
247
structure, structure, summation support surface symbol system, 144 system, 147 system, Cantor system, system,
symplectic 16, 144 symplectic, canonical 142 of Gaussian beams 123 139 of section 163 of a differential operator 3 classical dynamical 141, 143, completely
integrable
dynamical
completely integrable, set 148,149 Hamiltonian 658 strictly hyperbolic
term, secular 234 terminator 126 theorem, Liouville’s 18 torus, Kolmogorov 149 torus, Liouville 147 trajectory 6 trajectory, closed, of general type 164 trajectory, elliptic 163 trajectory, phase 6 transformation, canonical tube, bicharacteristic 7 tube, trajectory 7,16,24 variables, variables, vibrations 110
actionangle dual 3, 142 of whispering
deep shadow 102 Fock 126 of caustic shadow penumbral 101
a
36
elliptic
18
148 gallery
wave front 8,9,99 wave, scattered 69 waves, grazing 114 wellposedness 57 zone, zone, zone zone,
on
109
type