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May 7, 2010 - 1 Discussion and plan. We start our analysis in subsection 2 ...... Springer-Verlag, SMM, 1998, xv+731. [Ivr2] V. Ivrii. Microlocal Analysis andย ...
arXiv:1005.0936v1 [math.AP] 6 May 2010

Operators with Periodic Hamiltonian Flows in Domains with the Boundary Victor Ivrii May 7, 2010 Abstract We consider operators in the domains with the boundaries and derive sharp spectral asymptotics (containing non-Weyl correction) in the case when Hamiltonian flow is periodic. Even if operator is scalar but not second order (or even secondorder but there is an inner boundary with both refraction and reflection present) Hamiltonian flow is branching on the boundary and the notion of periodicity becomes more complicated.

0

Introduction

In section 6.2 of [Ivr2]1) we considered the case when there is no boundary and Hamiltonian flow is periodic and derive very sharp spectral asymptotics. Now we want to achieve similar results when there is a boundary. We do not expect asymptotics to be that sharp, but better than O(h๐Ÿฃโˆ’d ) and have non-Weyl correction term. However the presence of the boundary, more precisely, branching of the rays on the boundary brings the new possibilities. 1)

This article is a rather small part of the huge project to write a book and is just section 8.3 consisting entirely of newly researched results. section 6.2 roughly corresponds to Section 4.7 of its predecessor V. Ivrii [Ivr1]. External references by default are to [Ivr2].

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Chapter 1. Discussion and plan

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Even if operator is scalar but not second order (or even second-order but there is an inner boundary with both refraction and reflection present) Hamiltonian flow is branching on the boundary and the notion of periodicity becomes more complicated. For simplicity we consider only Schrยจodinger operators and derive sharp spectral asymptotics (containing non-Weyl correction) in the case when Hamiltonian flow is periodic.

1

Discussion and plan

We start our analysis in subsection 2 from the case when there is no branching and the Hamiltonian flow (with reflections) is periodic. Then if period is T๐Ÿข , ๐ญT๐Ÿข = I we have the same equality (1.1)

e ih

โˆ’๐Ÿฃ T

๐ŸขA

Q = e iT๐Ÿข L Q

as before where ๐—Œ๐—Ž๐—‰๐—‰ q is disjoint from the boundary and only transversal to the boundary trajectories originate at ๐—Œ๐—Ž๐—‰๐—‰ q and L is h-pseudo-differential operator. Then (๏ธ€ )๏ธ€ t (1.2) Ftโ†’hโˆ’๐Ÿฃ ๐œ ๐œ’( ฬ„ โˆ’ n)๐˜ Q๐Ÿฃx u tQ๐Ÿคy = (๐Ÿค๐œ‹h)๐Ÿฃโˆ’d J(n) + O(h๐Ÿฃโˆ’d+๐›ฟ ) T๐Ÿข with โˆซ๏ธ (1.3)

e inโ„“ q๐Ÿฃ q๐Ÿค dยต๐œ

J(n) = ๐จ๐œ

as ๐œ’ฬ„ is supported in (โˆ’ ๐Ÿค๐Ÿฅ , ๐Ÿค๐Ÿฅ ) and equals ๐Ÿฃ in (โˆ’ ๐Ÿฃ๐Ÿฅ , ๐Ÿฃ๐Ÿฅ ) and n โˆˆ โ„ค, |n| โ‰ค hโˆ’๐›ฟ . Here as before ๐จ๐œ = {a(x, ๐œ‰) = ๐œ } and ยต๐œ = dxd๐œ‰ : da is the measure there, โ„“ is the principal symbol of L. If โ„“ is โ€œvariable enoughโ€ on ๐จ๐œ so the right-hand expression is decaying as n โ†’ โˆž we can improve remainder estimate O(h๐Ÿฃโˆ’d ) (to what degree depends on the rate of the decay). In section 6.2 essentially โ„“ was defined by integral of the subprincipal symbol along periodic trajectories. Now however โ„“ can pick up extra terms at the moment of reflection: for example, for Schrยจodinger operator increment of โ„“ is ๐Ÿข and ๐œ‹ for Neumann and Dirichlet boundary condition respectively; however more general boundary condition brings increment which depends on the point (x โ€ฒ , ๐œ‰ โ€ฒ ) โˆˆ T * Y of reflection. Then in section 3 we consider a more complicated case when two operators are intertwined through boundary condition and all Hamiltonian trajectories

Chapter 2. Simple Hamiltonian flow

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of one of them are periodic but of another is not and, moreover, if the generic Hamiltonian billiard is periodic, it is in fact Hamiltonian billiard of the first operator. Then (1.1) fails but (1.2)โ€“(1.3) is replaced by (1.2), (1.4) โˆซ๏ธ โ€ฒ (1.4) J(n) = e inโ„“โˆ’|n|โ„“ q๐Ÿฃ q๐Ÿค dยต๐Ÿฃ.๐œ ๐จ๐Ÿฃ,๐œ

where ๐จ๐Ÿฃ,๐œ and ยต๐Ÿฃ,๐œ correspond to the first operator and โ„“โ€ฒ โ‰ฅ ๐Ÿข depends on the portion of energy which was whisked away along trajectory of the second operator at reflection point. Then if (๏ธ€ )๏ธ€ (1.5) ยต๐Ÿฃ,๐œ {(x, ๐œ‰) โˆˆ ๐จ๐Ÿฃ,๐œ , โ„“โ€ฒ = ๐Ÿข, โˆ‡โ„“ = ๐Ÿข} = ๐Ÿข we conclude that J(n) = o(๐Ÿฃ) as n โ†’ โˆž and we will be able to improve remainder estimate O(h๐Ÿฃโˆ’d ) to o(h๐Ÿฃโˆ’d ) and again there will be a non-Weyl correction term but it will depend also on โ„“ and โ„“โ€ฒ . Finally, in subsection 3 we consider even more complicated when two operators are intertwined through boundary condition and Hamiltonian trajectories of both of them are periodic and moreover after a while all trajectories issued from some point at T * Y assemble there. Then we can construct a matrix symbol and its eigenvalues play a role of symbol โ„“.

2 2.1

Simple Hamiltonian flow Inner asymptotics

Let ๐’ฐ be an open connected subset in T * X disjoint from ๐œ•X . Consider Hamiltonian billiards issued from ๐’ฐ, let ๐ซt denote generalized Hamiltonian flow. Assume that (2.1)

|โˆ‡x,๐œ‰ a(x, ๐œ‰)| โ‰ฅ ๐œ–

(2.2)

|๐œ‘(x)| + |{a, ๐œ‘}(x, ๐œ‰)| โ‰ฅ ๐œ–

(2.3)

๐ซt (x, ๐œ‰) = (x, ๐œ‰)

โˆ€t โˆ€(x, ๐œ‰) โˆˆ ๐ซt (๐’ฐ), โˆ€t โˆ€(x, ๐œ‰) โˆˆ ๐ซt (๐’ฐ),

with t = t(x, ๐œ‰) > ๐Ÿข โˆ€(x, ๐œ‰) โˆˆ ๐’ฐ,

ฬ„ ), ๐œ‘ > ๐Ÿข in X and ๐œ‘ = ๐Ÿข on ๐œ•X and where ๐œ‘ โˆˆ C โˆž (X (2.4) Along ๐ซt (๐’ฐ) reflections are simple without branching, i.e. ๐œ„โˆ’๐Ÿฃ ๐œ„(y , ๐œ‚) โˆฉ {a(y , ๐œ‚) = ๐œ } consists of two (disjoint) points (y , ๐œ‚ ยฑ ) such that ยฑ{a, ๐œ‘} > ๐Ÿข as (y , ๐œ‚) โˆˆ ๐ซt (๐’ฐ), y โˆˆ ๐œ•X and ๐œ = a(y , ๐œ‚).

Chapter 2. Simple Hamiltonian flow

4

Let us recall that ๐œ„ : T * X |๐œ•X โ†’ T * ๐œ•X is a natural map. Then exactly as in the case without boundary (2.5) One can select t(x, ๐œ‰) = T (a(x, ๐œ‰)) in ๐’ฐ with T โˆˆ C โˆž (so period depends on energy level only2) and (2.6) As (x, ๐œ‰) โˆˆ ๐’ฐ and ๐Ÿข < t < T (x, ๐œ‰) ๐ซt (x, ๐œ‰) meats ๐œ•X exactly (N โˆ’ ๐Ÿฃ) times with N = ๐–ผ๐—ˆ๐—‡๐—Œ๐—; so as N โ‰ฅ ๐Ÿฃ billiard consists of N segments with the ends on the boundary. We can consider instead of A with domain (2.7)

D(A) = {u โˆˆ H m (X ), รฐBu = ๐Ÿข}

operator f (A) with โˆซ๏ธ (2.8)

f (๐œ ) =

T โˆ’๐Ÿฃ (๐œ ) d๐œ

โˆ’๐Ÿฃ

and introduce propagator e ih tf (A) so that the Hamiltonian flow ๐ญt of f (A) is periodic with period ๐Ÿฃ. However now structure of f (A) near ๐œ•X is a bit murky; but it is not really serious obstacle since we can consider problem (๏ธ€ )๏ธ€ โˆ’๐Ÿฃ (2.9) fฬŒ(hDt ) โˆ’ A e ih tf (A) = ๐Ÿข, (2.10)

รฐBe ih

โˆ’๐Ÿฃ tf (A)

=๐Ÿข

where fฬŒ is an inverse function. Therefore its Schwartz kernel U(x, y , t) satisfies problem (2.9)โ€“(2.10) with respect to x and transposed problem with respect to y . โˆ’๐Ÿฃ Note that due to assumptions (2.1)โ€“(2.2) e ih tf (A) Q is a Fourier integral operator with the symplectomorphism ๐ญt provided Q is h-pseudo-differential operator with the symbol supported in ๐’ฐ. Further, due to (2.3), (2.8) ๐ญ๐Ÿฃ = I โˆ’๐Ÿฃ and therefore e ih f (A) Q is h-pseudo-differential operator. So, we arrive to But there could be subperiodic trajectories with periods nโˆ’๐Ÿฃ T (a(x, ๐œ‰)), n = ๐Ÿค, ๐Ÿฅ, ... . The structure of subperiodic trajectories described in subsection 6.2.5. 2)

Chapter 2. Simple Hamiltonian flow

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Proposition 2.1. Let A be a Schrยจodinger operator with the scalar principal symbol a(x, ๐œ‰) satisfying (2.1)โ€“(2.3) and (2.8). Then (2.11)

e ih

โˆ’๐Ÿฃ f (A)โˆ’i๐œ…

โ‰ก e ih

โˆ’๐Ÿฃ ๐œ€L

in ๐’ฐ

with ๐œ€ = h and h-pseudo-differential operator L; here ๐œ… is Maslovโ€™s constant and the principal symbol โ„“ of L is defined by โˆ๏ธ โ€ฒ (2.12) e iโ„“ = e iโ„“k e iโ„“k ๐Ÿฃโ‰คkโ‰คN

where โ„“k corresponds to k-th segment (see theorem 6.2.3) and โ„“โ€ฒk corresponds to k-th reflection. Example 2.2. Consider Schrยจodinger operator. Assume that at the reflection point a(x, ๐œ‰) = ๐œ‰๐Ÿฃ๐Ÿค + a(x โ€ฒ , ๐œ‰ โ€ฒ ), b = ๐œ‰๐Ÿฃ + i๐›ฝ(๐œ‰ โ€ฒ ). Then (2.13)

(๏ธ€ )๏ธ€โˆ’๐Ÿฃ (๏ธ€ )๏ธ€ ๐Ÿฃ ๐Ÿฃ โ€ฒ e iโ„“k = โˆ’ (๐œ โˆ’ aโ€ฒ ) ๐Ÿค โˆ’ i๐›ฝ (๐œ โˆ’ aโ€ฒ ) ๐Ÿค + ฤฑ๐›ฝ

with the right-hand expression calculated in the corresponding reflection point. Taking ๐›ฝ = ๐Ÿข or ๐›ฝ = โˆž (formally) we get โ„“โ€ฒk = ๐Ÿข and โ„“โ€ฒk = ๐œ‹ for Neumann and Dirichlet boundary conditions respectively. Then we can use all the local results of section 6.2, assuming that (2.11) holds with hl โ‰ค ๐œ€ โ‰ค h and derive local spectral asymptotics with the same precision (up to O(h๐Ÿคโˆ’d )) as there. Corollary 2.3. Let conditions of proposition 2.1 be fulfilled. Then asymptotics with the remainder o(h๐Ÿฃโˆ’d ) holds provided (๏ธ€ )๏ธ€ (2.14) ยต๐œ {(x, ๐œ‰) โˆˆ ๐จ๐œ , โˆ‡๐จ๐œ โ„“(x, ๐œ‰) = ๐Ÿข} = ๐Ÿข where โˆ‡๐จ๐œ means a gradient along ๐จ๐œ ; (ii) Asymptotics with the remainder O(h๐Ÿฃโˆ’d+๐›ฟ ) with small exponent ๐›ฟ > ๐Ÿข holds provided (๏ธ€ )๏ธ€ โ€ฒ (2.15) ยต๐œ {(x, ๐œ‰) โˆˆ ๐จ๐œ , |โˆ‡โ€ฒ โ„“(x, ๐œ‰)| โ‰ค ๐œ€} = o(๐œ€๐›ฟ ) with the small exponent ๐›ฟ โ€ฒ > ๐Ÿข;

Chapter 2. Simple Hamiltonian flow

6

(iii) Furthermore asymptotics with the remainder o(h๐Ÿฃโˆ’d ) holds as (2.1)โ€“(2.2) are replaced by their non-uniform versions (2.2)โ€ฒ

|๐œ‘(x)| + |{a, ๐œ‘}(x, ๐œ‰)| > ๐Ÿข

โˆ€t โˆ€(x, ๐œ‰) โˆˆ ๐ซt (๐’ฐ โˆฉ ๐จ๐œ โˆ– ๐ ๐œ ),

where ยต๐œ (๐ ๐œ ) = ๐Ÿข. We leave to the reader to formulate statement similar to (iii) but with the remainder estimate O(h๐Ÿฃโˆ’d+๐›ฟ ). Usually our purpose is the asymptotics with Q๐Ÿฃx = ๐œ“๐Ÿฃ (x), Q๐Ÿคy = ๐œ“๐Ÿค (y ) where ๐œ“j are smooth functions but not vanishing near the ๐œ•X . Corollary 2.4. Asymptotics with the remainder estimate o(h๐Ÿฃโˆ’d ) holds provided conditions (2.1), (2.2)โ€ฒ , (2.3) and (2.14) are fulfilled in ๐’ฐ = T * X , ฬ„ ) compactly supported. Qj = ๐œ“j โˆˆ C โˆž (X Proof. Due to corollary 2.3(iii) contribution of the zone {x๐Ÿฃ > ๐œ€} to the remainder is o๐œ€ (h๐Ÿฃโˆ’d ) while contribution of zone {x๐Ÿฃ โ‰ค ๐œ€} to the remainder does not exceed O(๐œ€h๐Ÿฃโˆ’d ) as ๐œ€ โ‰ฅ h. These arguments could be improved to ๐œ€ = h๐›ฟ and the remainder estimate could be improved to O(h๐Ÿฃโˆ’d+๐›ฟ ). We leave the precise statements and arguments to the reader. Our goal is to derive more sharp remainder estimate.

2.2

Asymptotics near boundary

Let us consider X = {x๐Ÿฃ > ๐Ÿข} and ๐’ฐ open subset in T * โ„dโˆ’๐Ÿฃ ร— [๐Ÿข, ๐œ€]x๐Ÿฃ ร— โ„๐œ xโ€ฒ where ๐œ€ > ๐Ÿข is a very small constant. We are interested in (๏ธ€ โˆ’๐Ÿฃ )๏ธ€ (2.16) Ftโ†’hโˆ’๐Ÿฃ ๐œ ๐œ’ฬ„T (t) ๐–ณ๐—‹ e ih tf (A) Q with Q = Q(x, hDxโ€ฒ , hDt ) with symbol supported in ๐’ฐ; let us rewrite it as (๏ธ€ โˆ’๐Ÿฃ )๏ธ€ โˆ’๐Ÿฃ โˆ’๐Ÿฃ Ftโ†’hโˆ’๐Ÿฃ ๐œ ๐œ’ฬ„T (t) ๐–ณ๐—‹(e ih (tโˆ’ฬ„t )f (A) Q โ€ฒ e ih ฬ„t f (A) ) = Ftโ†’hโˆ’๐Ÿฃ ๐œ ๐œ’ฬ„T (t) ๐–ณ๐—‹ e ih tf (A) Q โ€ฒ with Q โ€ฒ = e ih

โˆ’๐Ÿฃ ฬ„ t f (A)

Qe โˆ’ih

โˆ’๐Ÿฃ ฬ„ t f (A)

.

Chapter 2. Simple Hamiltonian flow

7

Note that if (2.1), (2.2) are fulfilled, ๐œ€ > ๐Ÿข and ฬ„t โ‰ฅ C ๐œ€ are small enough then Q โ€ฒ = Q โ€ฒ (x, hDx , hDt ) is h-pseudo-differential operator with the symbol supported in {๐œ–โ€ฒฬ„t โ‰ค x๐Ÿฃ โ‰ค C โ€ฒฬ„t } and we can rewrite (๏ธ€ )๏ธ€ Q โ€ฒ = Q โ€ฒโ€ฒ (x, hDx ) + Q โ€ฒโ€ฒโ€ฒ (x, hDx , hDt ) hDt โˆ’ f (A) ; therefore we can rewrite (2.16) in the same form but with Q replaced by Q โ€ฒโ€ฒ . Now we can apply all the arguments of the previous subsection and of 6.2 and derive asymptotics with the remainder estimate as sharp as O(h๐Ÿคโˆ’d ) provided conditions (2.1)โ€“(2.3) are fulfilled in the full measure as well as corresponding conditions of section 6.2 to โ„“. Here as usual we lift the points of (y โ€ฒ , ๐œ‚ โ€ฒ , ๐œ ) โˆˆ T * ๐œ•X to (y โ€ฒ , ๐œ‚ ยฑ ) โˆˆ ๐จ๐œ . So, we need to analyze only near tangent zone {x๐Ÿฃ โ‰ค ๐œ€, |{a, x๐Ÿฃ }| โ‰ค ๐œ€}. Unfortunately the theory of the manifolds with the boundaries and periodic Hamiltonian (or even geodesic) flow with reflections is completely undeveloped. Let us assume that (2.1), (2.2)โ€ฒ and (2.3) are fulfilled for all (x, ๐œ‰) โˆˆ T * X : ๐œ๐Ÿฃ โ‰ค a(x, ๐œ‰) โ‰ค ๐œ๐Ÿค with ๐œ๐Ÿฃ < ๐œ๐Ÿค . Then trajectories tangent to ๐œ•X cannot penetrate into X and therefore x๐Ÿฃ = ๐Ÿข, ๐œ๐Ÿฃ โ‰ค aโ€ฒ (x, ๐œ‰ โ€ฒ ) โ‰ค ๐œ๐Ÿค is incompatible with axโ€ฒ ๐Ÿฃ (x, ๐œ‰) < ๐Ÿข. Further, x๐Ÿฃ = ๐Ÿข, ๐œ๐Ÿฃ โ‰ค aโ€ฒ (x, ๐œ‰ โ€ฒ ) โ‰ค ๐œ๐Ÿค is incompatible with axโ€ฒ ๐Ÿฃ (x, ๐œ‰) > ๐Ÿข as well; really otherwise almost tangent to boundary billiards make very small jumps which contradicts to periodicity after N reflections with fixed N. Therefore ๐œ•X is bicharacteristically flat (2.17)

x๐Ÿฃ = ๐Ÿข, ๐œ๐Ÿฃ โ‰ค aโ€ฒ (x, ๐œ‰ โ€ฒ ) โ‰ค ๐œ๐Ÿค =โ‡’ axโ€ฒ ๐Ÿฃ (x, ๐œ‰) = ๐Ÿข.

Further, (2.3) implies that (2.18) All solutions of equation (2.19)

z โ€ฒโ€ฒ + b(๐ญt (x โ€ฒ , ๐œ‰ โ€ฒ ))z = ๐Ÿข

๐Ÿฃ with b(x โ€ฒ , ๐œ‰) = axโ€ฒ ๐Ÿฃ x๐Ÿฃ (๐Ÿข, x โ€ฒ , ๐œ‰ โ€ฒ ) ๐Ÿค

are T๐Ÿข -periodic for any (x โ€ฒ , ๐œ‰ โ€ฒ ) โˆˆ T * ๐œ•X , ๐œ๐Ÿฃ โ‰ค aโ€ฒ (x, ๐œ‰ โ€ฒ ) โ‰ค ๐œ๐Ÿค . Therefore (2.3) implies that (2.20) ๐œŒ(x, ๐œ‰) โˆ˜ ๐ญt โ‰ ๐œŒ with ๐œŒ = x๐Ÿฃ + |{a, x๐Ÿฃ }| as ๐œ๐Ÿฃ โ‰ค a(x, ๐œ‰) โ‰ค ๐œ๐Ÿค .

Chapter 2. Simple Hamiltonian flow

8

Consider zone ๐’ฐ๐œ€ = {(x, ๐œ‰) : ๐œŒ(x, ๐œ‰) โ‰ ๐œ€}. Blowing up (x๐Ÿฃ , ๐œ‰๐Ÿฃ ) โ†’ ๐œ€โˆ’๐Ÿฃ (x๐Ÿฃ , ๐œ‰๐Ÿฃ ) we can prove easily that after this (๏ธ€ )๏ธ€ (2.21) ๐ญt (x, ๐œ‰) = ๐ญโ€ฒt,๐œ€ (x, ๐œ‰), ๐ญโ€ฒโ€ฒ(x โ€ฒ ,๐œ‰โ€ฒ ,t),๐œ€ (x๐Ÿฃ , ๐œ‰๐Ÿฃ ) , with โˆ‘๏ธ (2.22) ๐œ€n ๐ญโ€ฒt,(n) (x, ๐œ‰), ๐ญโ€ฒt,๐œ€ (x, ๐œ‰) โˆผ ๐ญโ€ฒt,๐Ÿข (x โ€ฒ , ๐œ‰ โ€ฒ ) + nโ‰ฅ๐Ÿค

(2.23)

๐ญโ€ฒโ€ฒ(x โ€ฒ ,๐œ‰โ€ฒ ,t),๐œ€ (x๐Ÿฃ , ๐œ‰๐Ÿฃ ) โˆผ

โˆ‘๏ธ

๐œ€n ๐ญโ€ฒโ€ฒ(x โ€ฒ ,๐œ‰โ€ฒ ,t),(n) (x๐Ÿฃ , ๐œ‰๐Ÿฃ )

nโ‰ฅ๐Ÿฃ

where ๐ญโ€ฒt,๐Ÿข is the Hamiltonian flow on T * ๐œ•X and ๐ญโ€ฒโ€ฒt,(๐Ÿฃ) is the linearized with respect to (x๐Ÿฃ , ๐œ‰๐Ÿฃ ) billiard flow (described by (2.19)). One can prove easily that ๐ญโ€ฒt,(n) and ๐ญโ€ฒโ€ฒ(x โ€ฒ ,๐œ‰โ€ฒ ,t),(n) are uniformly smooth. ๐Ÿฃ

Then in blown-up coordinates (2.11) holds provided ๐œ€ โ‰ฅ h ๐Ÿค โˆ’๐›ฟ because โ„ = ๐œ€โˆ’๐Ÿค h and then we can apply all the previous arguments and derive asymptotics; the remainder estimate is as sharp as O(h๐Ÿคโˆ’d ) provided (in not blown-up coordinates) (2.24)

|โˆ‡(x,๐œ‰) โ„“(x, ๐œ‰)| โ‰ฅ ๐œ–๐Ÿข

as x๐Ÿฃ โ‰ ๐œŒ(x, ๐œ‰) โ‰ค ๐œ–๐Ÿฃ ;

x| โ‰ ๐œŒ(x, ๐œ‰) means exactly that x๐Ÿฃ โ‰ฅ ๐œ–|๐œ‰๐Ÿฃ |. Therefore Proposition 2.5. Let A be a Schrยจ odinger operator with the principal symbol a(x, ๐œ‰) satisfying (2.1)โ€“(2.3) and (2.17)โ€“(2.18). Let the subprincipal symbol and the boundary condition be such that (2.24) is fulfilled. All these conditions are supposed to be fulfilled in {(x, ๐œ‰) : ๐œŒ(x, ๐œ‰) โ‰ค ๐œ–๐Ÿข }. ๐Ÿฃ Then the contribution of zone {h ๐Ÿค โˆ’๐›ฟ โ‰ค ๐œŒ(x, ๐œ‰) โ‰ค ๐œ–๐Ÿค } to the remainder is O(h๐Ÿคโˆ’d ). Since the contribution of zone {๐œŒ(x, ๐œ‰) โ‰ค ๐œ€} to the remainder does not ๐Ÿฃ exceed C ๐œ€๐Ÿค h๐Ÿฃโˆ’d and we can take ๐œ€ = h ๐Ÿค โˆ’๐›ฟ we conclude that the contribution of zone {๐œŒ(x, ๐œ‰) โ‰ค ๐œ–๐Ÿค } to the remainder is O(h๐Ÿคโˆ’dโˆ’๐›ฟ ). One can also estimate this way the contribution of subperiodic trajectories. Thus we arrive (details are left to the reader) to Theorem 2.6. Let A be a Schrยจodinger operator with the principal symbol a(x, ๐œ‰) satisfying (2.1)โ€“(2.3) and (2.17)โ€“(2.18). Let the subprincipal symbol and the boundary condition be such that (2.24) is fulfilled. All these conditions are supposed to be fulfilled in {(x, ๐œ‰) : ๐œŒ(x, ๐œ‰) โ‰ค ๐œ–๐Ÿข }.

Chapter 2. Simple Hamiltonian flow

9

Then contribution of zone {๐œŒ(x, ๐œ‰) โ‰ค ๐œ–๐Ÿค } to the remainder is O(h๐Ÿคโˆ’dโˆ’๐›ฟ ) while its contribution to the principal part of the asymptotics is given by the standard two term Weyl formula plus a correction term constructed according to section 5.2 for symbol โ„“. Problem 2.7. Recover O(h๐Ÿคโˆ’d ) remainder estimate. This should not be extremely hard especially if there are no subperiodic trajectories but definitely worth of publishing. Example 2.8. Consider Laplacian h๐Ÿค ๐™ on the standard hemisphere ๐•Šd+ := {|x| = ๐Ÿฃ, x๐Ÿฃ > ๐Ÿข} in โ„d+๐Ÿฃ or harmonic oscillator h๐Ÿค ๐™ + |x|๐Ÿค on the standard half-space โ„d+ = {x๐Ÿฃ > ๐Ÿข}. (i) Consider boundary operator of example 2.2. Then N = ๐Ÿค and โ„“(z, ๐œ ) = โ„“โ€ฒ (z, ๐œ ) + โ„“โ€ฒ (๐œš(z), ๐œ ) where z = (x โ€ฒ , ๐œ‰ โ€ฒ ), ๐œš(z) is the antipodal point and โ„“โ€ฒ is defined by (2.13). Then one can express conditions to โ„“ as conditions to ๐›ฝ; we leave exact statements to the reader; (ii) Consider Dirichlet or Neumann boundary conditions. Then as perturbation is hx๐Ÿฃ one can calculate easily that โ„“ = ๐œ…๐œŒ + O(๐œŒ๐Ÿค ) where ๐œŒ is the incidence angle of trajectory and ๐œ… > ๐Ÿข. Then condition (2.24) is fulfilled.

2.3

Discussion

As we mentioned almost nothing is known about manifolds with all billiards closed; we are discussing geodesic billiards but one can ask the same questions about Hamiltonian billiards or Hamiltonian billiards associated with the Schrยจodinger operator; in the two last cases phase space S * X is replaced by the portion of the phase space {๐œ๐Ÿฃ โ‰ค a(x, ๐œ‰) โ‰ค ๐œ๐Ÿค }. We assume that the phase space is connected: Problem 2.9. (i) So far our only examples are a standard hemisphere ๐•Šd+ and a standard half-space โ„d+ (see figure below); are any other there really different examples? (ii) In particular, are there manifolds with N ฬธ= ๐Ÿค (i.e. with N = ๐Ÿฃ, ๐Ÿฅ, ๐Ÿฆ, ... )? We can consider N copies X๐Ÿฃ , ... , XN of the standard quarter-sphere ๐•Š๐Ÿค โˆฉ {๐Ÿข โ‰ค ๐œƒ โ‰ค ๐œ‹๐Ÿค } โˆฉ {๐Ÿข โ‰ค ๐œ‘ โ‰ค ๐œ‹}, then glue {Xk , ๐œ‘ = ๐œ‹} with {Xk+๐Ÿฃ , ๐œ‘ = ๐Ÿข} for k = ๐Ÿฃ, ... , N, where XN+๐Ÿฃ := XN . Then for resulting manifold we have N reflections but for N ฬธ= ๐Ÿค there will be singularity at North Pole.

Chapter 3. Branching Hamiltonian flow with โ€œscatteringโ€

z

10

ฯ(z)

(a) Hemisphere, projected to equatorial plane

(b) half-plane

Figure 1: Different billiards (actually, their x-projections) are shown by different colors, they reflect from the boundary at two mutually antipodal points.

(iii) In particular, are there manifolds with ๐œ•X which is not connected? (iv) In particular, are there manifolds with subperiodic billiards and (or) with subperiodic boundary trajectories?

3 Branching Hamiltonian flow with โ€œscatteringโ€ Let us consider two manifolds X๐Ÿฃ and X๐Ÿค glued along the connected component Y of the boundary. We consider Schrยจodinger operators Aj on Xj and these operators are intertwined through the boundary conditions. We assume that Hamiltonian flows have simple (no branching) reflections on each of them, so branching comes from โ€œreflection-refractionโ€. Further we are interested in the case when Hamiltonian flow on X๐Ÿฃ is periodic while on X๐Ÿค majority of trajectories are not periodic.

Chapter 3. Branching Hamiltonian flow with โ€œscatteringโ€

3.1

11

Analysis in X๐Ÿค

Since we do not know any other examples of manifolds with periodic Hamiltonian flows with reflections but hemisphere or half-space and N = ๐Ÿค then we assume that (3.1) For z โˆˆ T * Y ๐ซ๐Ÿฃ,๐œ (z) = ๐œš(z) 3) , ๐œš is an antipodal map on T * Y . We also assume that ๐ซ๐Ÿฃ,๐œ and ๐ซ๐Ÿค,๐œ commute (3.2)

๐ซ๐Ÿฃ,๐œ โˆ˜ ๐ซ๐Ÿค,๐œ = ๐ซ๐Ÿค,๐œ โˆ˜ ๐ซ๐Ÿฃ,๐œ .

Then for points of ๐จ๐Ÿค,๐œ periodicity properties of the branching Hamiltonian flow with reflections ๐ญt coincide with those of ๐ญ๐Ÿค,t . Now we need to impose condition to the boundary operator except that {A, B} is self-adjoint. Namely, consider a point z = (x โ€ฒ , ๐œ‰ โ€ฒ ) โˆˆ T * Y , and consider an auxiliary problem (3.3) (3.4)

aj (x โ€ฒ , D๐Ÿฃ , ๐œ‰ โ€ฒ )uj = ๐Ÿข

j = ๐Ÿฃ, ๐Ÿค.

รฐb(x โ€ฒ , D๐Ÿฃ , ๐œ‰ โ€ฒ )u = ๐Ÿข

(where u = (u๐Ÿฃ , u๐Ÿค )) and consider solutions which are combinations of exponents e i๐œ†j x๐Ÿฃ where for each j either ๐–จ๐—† ๐œ†j > ๐Ÿข or ๐–จ๐—† ๐œ†j > ๐Ÿข and {aj , x๐Ÿฃ }(x โ€ฒ , ๐œ†j , ๐œ‰ โ€ฒ ) > ๐Ÿข. Denote by ๐  the set of points for which such non-trivial solutions exist. We assume that (3.5)

๐—†๐–พ๐—ŒT * Y ๐  = ๐Ÿข.

Then we arrive to the following Proposition 3.1. Let in the described setup conditions (3.1), (3.2) and (3.5) be fulfilled. Let Q๐Ÿฃ , Q๐Ÿค be h-pseudo-differential operators with the symbols supported in T * X๐Ÿค and (3.6)

ยต๐Ÿค,๐œ (๐ฅ๐Ÿค,๐œ โˆฉ ๐—Œ๐—Ž๐—‰๐—‰ q๐Ÿฃ โˆฉ ๐—Œ๐—Ž๐—‰๐—‰ q๐Ÿค ) = ๐Ÿข

where ๐ฅ๐Ÿค,๐œ is the set of point (y , ๐œ‚) โˆˆ ๐จ๐Ÿค,๐œ periodic with respect to ๐ญ๐Ÿค,t . Then for ๐˜(Q๐Ÿฃx e(., ., ๐œ ) tQ๐Ÿฃy ) the standard Weyl two-term asymptotics holds with the remainder estimate o(h๐Ÿฃโˆ’d ). We define ๐ซj,๐œ in the following way: we lift z โˆˆ T * Y to z โˆˆ TjX |Y โˆฉ ๐จj,๐œ with {aj , x๐Ÿฃ }(z) > ๐Ÿข, launch trajectory forward until it hits the boundary at (y โ€ฒ , ๐œ‚) โˆˆ T * Xj |Y and then project to T * Y . 3)

Chapter 3. Branching Hamiltonian flow with โ€œscatteringโ€

12

Remark 3.2. (i) Note that (3.7)

ยตj,๐œ (๐ j,๐œ ) = ๐Ÿข

where ๐ j,๐œ = ๐ โ€ฒj,๐œ โˆช ๐ โ€ฒโ€ฒj,๐œ , ๐ โ€ฒj,๐œ is the set of dead-end points of billiard (not generalized billiard) flow and ๐ โ€ฒโ€ฒj,๐œ is the set of points (y , ๐œ‚) โˆˆ ๐จj,๐œ such that ๐œ„j ๐ญj,t (y , ๐œ‚) โˆˆ ๐ฃ๐Ÿฅโˆ’j,๐œ where ๐ฃk,๐œ = ๐œ„k {(x, ๐œ‰) โˆˆ ๐จk,๐œ , {ak , x๐Ÿฃ } = ๐Ÿข. (ii) Note that the grows properties of the branching Hamiltonian flow with reflections ๐ญt coincide with those of ๐ญ๐Ÿค,t as long as we ensure that along billiards we do not approach to glancing rays on X๐Ÿฃ : (3.8) {a๐Ÿฃ , x๐Ÿฃ }(x, ๐œ‰) โ‰ฅ h๐›ฟ๐Ÿฃ

โˆ€(x, ๐œ‰) โˆˆ ๐œ„โˆ’๐Ÿฃ ๐Ÿฃ ๐œ„๐Ÿค ๐ญ๐Ÿค,t (y , ๐œ‚) โˆฉ ๐จ๐Ÿฃ,๐œ

โˆ€(y , ๐œ‚) โˆˆ ๐—Œ๐—Ž๐—‰๐—‰ q๐Ÿค โˆ€t : ยฑt โˆˆ [๐Ÿข, T ] ๐ญ๐Ÿค,t (y , ๐œ‚) โˆˆ T * X๐Ÿค |Y where T = T (h). Then in assumptions of section 7.4 one can recover similar results: namely that two-term Weyl asymptotics holds with the remainder estimate O(T (h)โˆ’๐Ÿฃ h๐Ÿฃโˆ’d ) which is usually O(h๐Ÿฃโˆ’d | ๐—…๐—ˆ๐—€ h|โˆ’๐Ÿฃ ) or O(h๐Ÿฃโˆ’d+๐›ฟ ) (depending on the growth and non-periodicity conditions to ๐ญ๐Ÿค,t . Exact statement and proof are left to the reader (which is a relatively easy problem). (iii) In (ii) we need also to assume a more sharp version of (3.5): namely if we consider approximate solutions to (3.3)โ€“(3.4) with precision ๐œ€ we get set ๐ ๐œ ,๐œ€ and we need to impose condition ๐—†๐–พ๐—Œ ๐ ๐œ ,๐œ€ = o(๐œ€๐›ฟ ) or ๐—†๐–พ๐—Œ ๐ ๐œ ,๐œ€ = o(| ๐—…๐—ˆ๐—€ ๐œ€|โˆ’๐Ÿฃ ) as ๐œ€ โ†’ +๐Ÿข. (iv) In subsubsection โ€œ7.4.3โ€ of subsection 7.4.3 we did not consider points z โˆˆ T * Y where some of operators aj are elliptic on ๐œ„โˆ’๐Ÿฃ z rather than hyperbolic. If we want to consider such points we need to assume (3.5) or more sharp version of it described in (iii). (v) (3.8) is guaranteed if either a๐Ÿค is cylindrically symmetric on T * X๐Ÿค or if there no internal reflection for billiards from X๐Ÿค at all: (3.9) |{a๐Ÿฃ , x๐Ÿฃ }(x, ๐œ‰)| โ‰ฅ ๐œ–|{a๐Ÿค , x๐Ÿฃ }(x, ๐œ‚)|l โˆ€(x, ๐œ‰) โˆˆ T * X๐Ÿฃ |Y โˆฉ ๐จ๐Ÿฃ,๐œ โˆ€(x, ๐œ‚) โˆˆ T * X๐Ÿค |Y โˆฉ ๐จ๐Ÿค,๐œ : ๐œ„๐Ÿฃ (x, ๐œ‰) = ๐œ„๐Ÿค (x, ๐œ‚).

Chapter 3. Branching Hamiltonian flow with โ€œscatteringโ€

3.2

13

Analysis in X๐Ÿฃ

However if operators Q๐Ÿฃ and Q๐Ÿค have symbols supported in X๐Ÿฃ situation is very different: trajectories of a๐Ÿฃ are periodic but the typical closed branching billiard is the one which does not contain segments of Hamiltonian trajectories of a๐Ÿค . More precisely, due to conditions (3.10)

ยต๐Ÿค,๐œ (๐ฅ๐Ÿค,๐œ ) = ๐Ÿข

and (3.7) we conclude that (3.11)

ยต๐Ÿฃ,๐œ (๐ฅโ€ฒ๐Ÿฃ,๐œ ) = ๐Ÿข

where ๐ฅโ€ฒ๐Ÿฃ,๐œ is the set of points z โˆˆ ๐จ๐Ÿฃ,๐œ such that there exists a billiard trajectory of ๐ญt starting and ending in z and which is not entirely billiard of ๐ญ๐Ÿฃ,t . Then for arbitrarily large T > ๐Ÿข and arbitrarily small ๐œ€ > ๐Ÿข there exists a set ๐’ฐ = ๐’ฐT ,๐œ€ โŠ‚ T * X such that (3.12)

ยต๐Ÿฃ,๐œ (๐จ๐Ÿฃ,๐œ โˆ– ๐’ฐ) โ‰ค ๐œ€,

(3.13)

๐–ฝ๐—‚๐—Œ๐—(๐’ฐ, ๐ฅโ€ฒ๐Ÿฃ,๐œ ,T โˆช ๐ ๐Ÿฃ,๐œ ,T ) โ‰ฅ ๐›พ

(3.14)

๐–ฝ๐—‚๐—Œ๐—(Y , ฯ€x ๐’ฐ) โ‰ฅ ๐›พ

with ๐›พ = ๐›พ(T , ๐œ€) > ๐Ÿข were ๐ฅโ€ฒ๐Ÿฃ,๐œ ,T s the set of points z โˆˆ ๐จ๐Ÿฃ,๐œ such that there exists a billiard trajectory of ๐ญt of the length not exceeding T , starting and ending in z and which is not entirely billiard of ๐ญ๐Ÿฃ,t and ๐ ๐Ÿฃ,๐œ ,T is the set of points z โˆˆ ๐จ๐Ÿฃ,๐œ such that billiard trajectory of a๐Ÿฃ hits the boundary at point of ๐ ๐Ÿค,T ,๐œ . As usual, these sets increase as T increases, their unions (with respect to T ) coincide with ๐ฅโ€ฒ๐Ÿฃ,๐œ and ๐ ๐Ÿฃ,๐œ respectively and ๐ ๐Ÿฃ,๐œ ,T and ๐ ๐Ÿฃ,๐œ ,T โˆฉ ๐ฅโ€ฒ๐Ÿฃ,๐œ ,T are closed sets. Therefore there exists operator Q = QT ,๐œ€ such that ๐—Œ๐—Ž๐—‰๐—‰ q โŠ‚ ๐’ฐ and )๏ธ€ (๏ธ€ (3.15) |๐˜ Q๐Ÿฃx e(., ., ๐œ ) t((I โˆ’ Q)Q๐Ÿค )y โˆ’ ๐œ…โ€ฒ๐Ÿข hโˆ’d โˆ’ ๐œ…โ€ฒ๐Ÿฃ h๐Ÿฃโˆ’d | โ‰ค C๐Ÿข ๐œ€h๐Ÿฃโˆ’d with ๐œ…โ€ฒj = ๐œ…โ€ฒj,Q๐Ÿฃ ,Q๐Ÿค ,Q (๐œ ). This estimate holds for any (๏ธ€ )๏ธ€ fixed ๐œ satisfying (3.10). Therefore we need to consider ๐˜ Q๐Ÿฃx e(., ., ๐œ ) tQ๐Ÿคy with ๐—Œ๐—Ž๐—‰๐—‰ q๐Ÿค โŠ‚ ๐’ฐ. We can assume without any loss of the generality that (3.13), (3.14) are fulfilled for all ๐œ . Then for ๐œ– โ‰ค |t| โ‰ค T (๏ธ€ )๏ธ€ (๏ธ€ )๏ธ€ (3.16) ๐˜ Q๐Ÿฃx U(., ., t) tQ๐Ÿคy โ‰ก ๐˜ Q๐Ÿฃx Z (., ., t) tQ๐Ÿคy

Chapter 3. Branching Hamiltonian flow with โ€œscatteringโ€

14

where as ยฑt > ๐Ÿข Z (t) is Schwartz kernel of โ€œapproximate propagation with scattering semigroupโ€ ๐—ญ(t) constructed in the following way (for a sake of simplicity in notations we consider t > ๐Ÿข: (i) We represent Q๐Ÿค = Q๐Ÿค,๐Ÿฃ + ยท ยท ยท + Q๐Ÿค,N where symbols Q๐Ÿค,๐œˆ have small supports; (ii) For each ๐œˆ we select ๐Ÿข = t๐Ÿข,๐œˆ < ยท ยท ยท < tM,๐œˆ = t in such a way that for any z โˆˆ ๐—Œ๐—Ž๐—‰๐—‰ q๐Ÿค,๐œˆ ๐ญt โ€ฒ (z) is disjoint from Y as t โ€ฒ = tj,๐œˆ and ๐ญt โ€ฒ (z) hits Y no more than once as tj,๐œˆ โ‰ค t โ€ฒ โ‰ค tj+๐Ÿฃ,๐œˆ ; (iii) We set ๐—ญ(t)Q๐Ÿค,๐œˆ =

(๏ธ

โˆ๏ธ

๐œ“e ih

โˆ’๐Ÿฃ (t j+๐Ÿฃ,๐œˆ โˆ’tj,๐œˆ )f (A)

)๏ธ

ยท Q๐Ÿค,๐œˆ

๐Ÿขโ‰คjโ‰คMโˆ’๐Ÿฃ

where ๐œ“ โˆˆ C โˆž is supported in {x โˆˆ X๐Ÿฃ , ๐–ฝ๐—‚๐—Œ๐—(x, Y ) โ‰ฅ ๐›พ โ€ฒ } and equals ๐Ÿฃ in {x โˆˆ X๐Ÿฃ , ๐–ฝ๐—‚๐—Œ๐—(x, Y ) โ‰ฅ ๐Ÿค๐›พ โ€ฒ }. Then (3.17) and (3.18)

๐—ญ(t๐Ÿฃ + t๐Ÿค )Q๐Ÿค โ‰ก ๐—ญ(t๐Ÿค )๐—ญ(t๐Ÿฃ )Q๐Ÿค

as ยฑ t๐Ÿฃ > ๐Ÿข, ยฑt๐Ÿค > ๐Ÿข

๐—ญ(t)* โ‰ก ๐—ญ(โˆ’t)

and due to periodicity of ๐ญ๐Ÿฃ,t with period T๐Ÿข ๐—ญ(ยฑT๐Ÿข ) are h-pseudo-differential operators. Remark 3.3. In contrast to what we had before they are not necessarily unitary because reflection coefficients are not unitary anymore. More precisely, (i) If ๐ญ๐Ÿฃ,t hits T * X |Y at points (x โ€ฒ , ๐œ‰) with (x โ€ฒ , ๐œ‰ โ€ฒ ) โˆˆ / ๐œ„๐Ÿค ๐จ๐Ÿค,๐œ reflection coefficient ๐œ˜๐Ÿฃ๐Ÿฃ (x โ€ฒ , ๐œ‰ โ€ฒ , ๐œ ) still satisfies |๐œ˜๐Ÿฃ๐Ÿฃ (x โ€ฒ , ๐œ‰ โ€ฒ , ๐œ )| = ๐Ÿฃ. (ii) However if ๐ญ๐Ÿฃ,t hits T * X |Y at points (x โ€ฒ ,(๏ธ€๐œ‰) with (x โ€ฒ ,)๏ธ€๐œ‰ โ€ฒ ) โˆˆ ๐œ„๐Ÿค ๐จ๐Ÿค,๐œ then reflection-refraction matrix ๐œ˜(x โ€ฒ , ๐œ‰ โ€ฒ , ๐œ ) = ๐œ˜jk (x โ€ฒ , ๐œ‰ โ€ฒ , ๐œ ) j,k=๐Ÿฃ,๐Ÿค is unitary but reflection coefficient ๐œ˜๐Ÿฃ๐Ÿฃ (x โ€ฒ , ๐œ‰ โ€ฒ , ๐œ ) satisfies only |๐œ˜๐Ÿฃ๐Ÿฃ (x โ€ฒ , ๐œ‰ โ€ฒ , ๐œ )| โ‰ค ๐Ÿฃ and the equality means exactly that ๐œ˜๐Ÿฃ๐Ÿค (x โ€ฒ , ๐œ‰ โ€ฒ , ๐œ ) = ๐œ˜๐Ÿค๐Ÿฃ (x โ€ฒ , ๐œ‰ โ€ฒ , ๐œ ) = ๐Ÿข (then |๐œ˜๐Ÿค๐Ÿค (x โ€ฒ , ๐œ‰ โ€ฒ , ๐œ )| = ๐Ÿฃ as well). So the principal symbol of ๐—ญ(T๐Ÿข ) is the product of e iโ„“๐Ÿข (x,๐œ‰) where โ„“๐Ÿข (x, ๐œ‰) is given by the usual formula for manifolds without boundary (but with

Chapter 3. Branching Hamiltonian flow with โ€œscatteringโ€

15

trajectories replaced by billiards) and also of reflection coefficients coefficients in the point of reflection. If the reflection coefficients do not vanish we can rewrite the answer in the form ๐—ญ(T๐Ÿข ) โ‰ก e i๐œ€h

(3.19)

โˆ’๐Ÿฃ L

๐—ญ(โˆ’๐œ€hโˆ’๐Ÿฃ T๐Ÿข ) โ‰ก e โˆ’iL

,

*

with h-pseudo-differential operator L, L + L* โ‰ค ๐Ÿข and ๐œ€ = h (but in calculations below we are not assuming this). We will use this formula even if the coefficients vanish: in this case we simply formally allow ๐–จ๐—† โ„“ = +โˆž. Then as ๐œ’ฬ„ โˆˆ C๐Ÿขโˆž [๐œ– โˆ’ ๐Ÿฃ, ๐Ÿฃ โˆ’ ๐œ–] (๏ธ€ )๏ธ€ (3.20) Ftโ†’hโˆ’๐Ÿฃ ๐œ ๐œ’ฬ„T๐Ÿข (t โˆ’ nT๐Ÿข )๐˜ Q๐Ÿฃx Z (., ., t) tQ๐Ÿคy = ฬ‚๏ธ€ฬ„ h๐Ÿฃโˆ’d J(h, ๐œ€, n, ๐œ ) ยท ๐œ’

(๏ธ€ )๏ธ€ (๏ธ€ ๐œ )๏ธ€ + O h๐Ÿคโˆ’d , h

with (3.21)

๐Ÿฃโˆ’d

โˆซ๏ธ

J(h, ๐œ€, n, ๐œ ) = (๐Ÿค๐œ‹)

e ih

โˆ’๐Ÿฃ ((๐œ€ ๐–ฑ๐–พ โ„“โˆ’๐œ )n+๐œ€ ๐–จ๐—† โ„“|n|)

dยต๐Ÿฃ,๐œ .

๐จ๐Ÿฃ,๐œ

We can see easily that (3.22) J(h, ๐œ€, n, ๐œ ) = o(๐Ÿฃ) as n โ†’ โˆž provided (๏ธ€{๏ธ€ }๏ธ€)๏ธ€ (3.23) ยต๐œ (x, ๐œ‰) โˆˆ ๐จ๐œ : ๐–จ๐—† โ„“(x, ๐œ‰) = โˆ‡ ๐–ฑ๐–พ โ„“(x, ๐œ‰) = ๐Ÿข = ๐Ÿข. On the other hand, note that (3.24)

โˆ‘๏ธ nโˆˆโ„คโˆ–๐Ÿข

e ih

โˆ’๐Ÿฃ (n๐›ผโˆ’|n|๐›ฝ)

=

๐Ÿคe โˆ’๐›ฝ ๐–ผ๐—ˆ๐—Œ ๐›ผ โˆ’ ๐Ÿคe โˆ’๐Ÿค๐›ฝ ๐Ÿฃ โˆ’ ๐Ÿคe โˆ’๐›ฝ ๐–ผ๐—ˆ๐—Œ ๐›ผ + e โˆ’๐Ÿค๐›ฝ

as ๐›ฝ > ๐Ÿข

and formally we can extend this for ๐›ฝ = ๐Ÿข as well. Then after summation with respect to n for ๐›ฝ = ๐Ÿข we get exactly ๐œ•๐›ผ ๐ช(๐›ผ) with ๐ช defined by (6.2.63) and for ๐›ฝ > ๐Ÿข we get that the expression (3.24) (๏ธ€ )๏ธ€โˆ’๐Ÿฃ is equal to ๐œ•๐›ฝ ๐ช *๐›ผ (๐Ÿค๐œ‹)โˆ’๐Ÿฃ G (๐›ผ, ๐›ฝ) where G (๐›ผ, ๐›ฝ) = ๐Ÿค๐›ฝ ๐›ผ๐Ÿค + ๐›ฝ ๐Ÿค . Note that

Chapter 3. Branching Hamiltonian flow with โ€œscatteringโ€

16

(3.25) Function ๐ช(๐›ผ, ๐›ฝ) := ๐ช *๐›ผ G (๐›ผ, ๐›ฝ)

(3.26)

is harmonic function as ๐›ฝ > ๐Ÿข, coincides with ๐ช(๐›ผ) as ๐›ฝ = ๐Ÿข, is o(๐Ÿฃ) as ๐›ฝ โ†’ +โˆž and is periodic with respect to ๐›ผ. Taking in account results of this subsection we arrive to Theorem 3.4. Let in the described setup conditions (3.1), (3.2), (3.5), (3.10) and (3.23) be fulfilled. Then asymptotics (3.27) ๐˜e(., ., ๐œ ) = ๐œ…๐Ÿข (๐œ )hโˆ’d + โˆซ๏ธ (๏ธ )๏ธ (๏ธ€ )๏ธ€ )๏ธ€ ๐œ…๐Ÿฃ (๐œ ) + ๐ช hโˆ’๐Ÿฃ (๐œ€ ๐–ฑ๐–พ โ„“ โˆ’ ๐œ ), โˆ’hโˆ’๐Ÿฃ ๐œ€ ๐–จ๐—† โ„“ dยต๐œ dยต๐œ + o(h๐Ÿฃโˆ’d ) ๐จ๐œ

holds. Problem 3.5. Calculate all coefficients ๐œ˜jk for two Schrยจodinger operators A๐Ÿฃ and A๐Ÿค . We will solve this problem in the very special case: Example 3.6. Assume that (at the given point z โˆˆ T * Y ) aj = cj๐Ÿค |๐œ‰|๐Ÿค and the boundary condition is (3.28) (3.29)

u๐Ÿค = ๐›ผu๐Ÿฃ , D๐Ÿฃ u๐Ÿค = โˆ’๐›ฝD๐Ÿฃ u๐Ÿฃ

(recall that x๐Ÿฃ > ๐Ÿข in both X๐Ÿฃ , X๐Ÿค ; in more standard notations of x๐Ÿฃ < ๐Ÿข in X๐Ÿค one needs to skip โ€œโˆ’โ€). Then {A, B} to be self-adjoint requires ๐›ผ๐›ฝ โ€  = c๐Ÿฃ๐Ÿค c๐Ÿคโˆ’๐Ÿค . We do not consider gliding points since their measure is ๐Ÿข. (i) Consider point ๐œ‰ โ€ฒ with c๐Ÿฃ๐Ÿค |๐œ‰ โ€ฒ |๐Ÿค < ๐œ < c๐Ÿค๐Ÿค |๐œ‰ โ€ฒ |๐Ÿค . One can prove easily that (3.30) with (3.31)

๐œ˜๐Ÿฃ๐Ÿฃ (๐œ‰ โ€ฒ , ๐œ ) = e ๐Ÿคi๐œ™ (๏ธ (๏ธ€ )๏ธ€ ๐Ÿฃ (๏ธ€ )๏ธ€โˆ’ ๐Ÿฃ )๏ธ ๐œ™(๐œ‰ โ€ฒ , ๐œ ) = ๐–บ๐—‹๐–ผ๐—๐–บ๐—‡ |๐›ฝ|โˆ’๐Ÿค c๐Ÿคโˆ’๐Ÿค c๐Ÿฃ๐Ÿค c๐Ÿค๐Ÿค |๐œ‰ โ€ฒ |๐Ÿค โˆ’ ๐œ ๐Ÿค ๐œ โˆ’ c๐Ÿฃ๐Ÿค |๐œ‰ โ€ฒ | ๐Ÿค

Chapter 4. Two periodic flows

17

and therefore assumption (3.23) is fulfilled provided c๐Ÿฃ ฬธ= c๐Ÿค , cj , ๐›ผ, ๐›ฝ are symmetric (i.e. cj ๐œš = cj ) and the subprincipal symbol is ๐Ÿข. If c๐Ÿฃ > c๐Ÿค this case does not appear. (ii) Consider point ๐œ‰ โ€ฒ with cj๐Ÿค |๐œ‰ โ€ฒ |๐Ÿค < ๐œ for both j = ๐Ÿฃ, ๐Ÿค. One can prove easily that (๏ธ€ )๏ธ€(๏ธ€ )๏ธ€โˆ’๐Ÿฃ (3.32) ๐œ˜๐Ÿฃ๐Ÿฃ (๐œ‰ โ€ฒ , ๐œ ) = ๐œ” โˆ’ ๐Ÿฃ ๐œ” + ๐Ÿฃ with (๏ธ€ )๏ธ€ ๐Ÿฃ (๏ธ€ )๏ธ€โˆ’ ๐Ÿฃ (3.33) ๐œ”(๐œ‰ โ€ฒ , ๐œ ) = |๐›ฝ|โˆ’๐Ÿค c๐Ÿคโˆ’๐Ÿค c๐Ÿฃ๐Ÿค ๐œ โˆ’ c๐Ÿค๐Ÿค |๐œ‰ โ€ฒ |๐Ÿค ๐Ÿค ๐œ โˆ’ c๐Ÿฃ๐Ÿค |๐œ‰ โ€ฒ | ๐Ÿค and |๐œ˜๐Ÿฃ๐Ÿฃ | < ๐Ÿฃ so assumption (3.23) is fulfilled. The following problems seem to be rather straightforward but not easy and worth publishing: Problem 3.7. Under stronger conditions to ๐ญ๐Ÿค,t and โ„“ (see (2.15)) prove remainder estimate O(h๐Ÿฃโˆ’d+๐›ฟ ). Problem 3.8. Prove the same results as Hamiltonian billiards of a๐Ÿฃ are assumed to be periodic only on one energy level ๐œ rather than on neighboring levels.

4

Two periodic flows

Now we consider the case of both flows ๐ญ๐Ÿฃt and ๐ญ๐Ÿคt generated by f (a๐Ÿฃ ) and f (a๐Ÿค ) being periodic and satisfying (3.1) as in the following figures:

4.1

Examples and discussion

Let us start from an example: Example 4.1. Consider two Schrยจodinger operators in โ„d+ = โ„+ ร— โ„dโˆ’๐Ÿฃ with the principal symbols (4.1)

)๏ธ€ ๐Ÿฃ (๏ธ€ aj (x, ๐œ‰) = ๐œ”j๐Ÿค |๐œ‰|๐Ÿค + |x|๐Ÿค | โˆ’ Ej ๐Ÿค

intertwined through boundary conditions. Then Tj = ๐Ÿค๐œ‹๐œ”jโˆ’๐Ÿค .

Chapter 4. Two periodic flows

(a) Hemisphere, projected to equatorial plane

18

(b) half-plane

Figure 2: x-projection of billiard trajectory on two half-planes, glued together: original (red), reflected (blue), refracted (magenta) We are interested in the number ๐–ญโˆ’ h (๐Ÿข) of the negative eigenvalues of operator AB . In order to consider the eigenvalue counting function ๐–ญโˆ’ h (๐œ†) for arbitrary spectral parameter ๐œ† one needs to redefine (4.2)

Ej := Ej + ๐Ÿค๐œ”j๐Ÿค ๐œ†.

But ๐–ญโˆ’ (๐Ÿข) is also the maximal dimension of the negative subspace of ๐Ÿฃ ๐Ÿฃ the operator A which does not change if we replace A by J โˆ’ ๐Ÿค AJ โˆ’ ๐Ÿค with the positive self-adjoint operator J and the latter equals to ๐–ญโˆ’ h (๐Ÿข) for the ๐Ÿค latter operator. Picking up J = J(x) equal ๐œ”j in Xj we arrive to the case of ๐œ”j = ๐Ÿฃ and Tj = ๐Ÿค๐œ‹. Therefore in this example one can make periods equal T๐Ÿฃ = T๐Ÿค . The dependence of ๐œ”j will come through Ej redefined by (4.2). There are however many deficiencies of the described approach; most important is that it does not work with the following example (unless some unnatural conditions to ๐œ”j , Ej are imposed): Example 4.2. Consider two operators on hemisphere ๐•Šd+ with the principal symbols (๏ธ€ )๏ธ€ (4.3) aj (x, ๐œ‰) = ๐œ”j๐Ÿค |๐œ‰|๐Ÿค โˆ’ Ej

Chapter 4. Two periodic flows

19

where |๐œ‰| is calculated in the standard Riemannian metrics. ๐Ÿฃ

The main obstacle here is that multiplying operator by J โˆ’ ๐Ÿค from both sides and calculating f (A) do not play well together.

4.2

Reduction to the boundary

First of all notice that the contribution to the remainder of the domain {z, ๐–ฝ๐—‚๐—Œ๐—(z, T * Y ) โ‰ค ๐œ€๐Ÿข } does not exceed C๐Ÿข ๐œ–๐Ÿค๐Ÿข h๐Ÿฃโˆ’d and therefore one needs to consider the contribution of {z, ๐–ฝ๐—‚๐—Œ๐—(z, T * Y ) โ‰ฅ ๐œ€๐Ÿข } with arbitrarily small but fixed ๐œ€๐Ÿข . In this zone one can consider U(x, y , t) as the solution of the Cauchy problem with respect to x๐Ÿฃ (or y๐Ÿฃ ) with data at {x๐Ÿฃ = ๐Ÿข} (or {y๐Ÿฃ = ๐Ÿข}). Rewriting therefore U = Rx U tRy where R is an operator resolving this Cauchy problem is given by an oscillatory integral we can rewrite (๏ธ€ )๏ธ€ (๏ธ€ )๏ธ€ (4.4) ๐˜ U(., ., t) tQy โ‰ก ๐˜โ€ฒ ๐’ฌ(x โ€ฒ , hDxโ€ฒ , hDt , h)V with h-pseudo-differential operator R and V = รฐx,mโˆ’๐Ÿฃ รฐy ,mโˆ’๐Ÿฃ U the Cauchy data for U (assuming that m is an order of A). One can see easily that as Q is an operator with the symbol supported in T * Xj , the principal symbol of ๐’ฌ at (x โ€ฒ , ๐œ‰, ๐œ ) is defined as an averaging of Q along ๐ญjt (zj ) with t โˆˆ [๐Ÿข, tj (zj )] where zj = ๐œ„โˆ’๐Ÿฃ (x โ€ฒ , ๐œ‰ โ€ฒ ) โˆฉ ๐จj๐œ and tj (zj ) is the time of the next hit of the boundary. โˆ’๐Ÿฃ Then instead of continuous family of Fourier integral operators e ih tA we can consider a discrete family โ„ฑ n defined by the following way: consider vj = รฐmโˆ’๐Ÿฃ uj = (vjโˆ’ , vj+ ) with vjโˆ“ corresponding to incoming and outgoing solutions. We have two types of trajectories: those in ๐จj๐œ which hit T * X |Y at points elliptic for (a๐Ÿฅโˆ’j โˆ’ ๐œ ) where j = ๐Ÿฃ, ๐Ÿค and those which hit T * X |Y at points hyperbolic for (ak โˆ’ ๐œ ) for both k = ๐Ÿฃ, ๐Ÿค. The analysis of the former is of no different from what we have seen in the previous subsection: we just consider ๐ญjt and construct the (real-valued) symbol โ„“ as long as we assume that (x โ€ฒ , ๐œ‰) โˆˆ / ๐  where ๐  is defined in the paragraph preceding (3.5). The analysis of the latter is more interesting. Therefore as before we have a matrix of reflection-refraction (๐œ˜jk )j,k=๐Ÿฃ,๐Ÿค which is a symbol defined on ๐’ฐ ร— (๐œ โˆ’ ๐œ€โ€ฒ , ๐œ + ๐œ€โ€ฒ ) where ๐’ฐ is a a zone described above. This matrix is

Chapter 4. Two periodic flows

20

unitary in the norm โ€–V โ€– =

(4.5)

(๏ธโˆ‘๏ธ(๏ธ€

|Vj+ |๐Ÿค

+

|Vjโˆ’ |๐Ÿค

)๏ธ€

|{aj , x๐Ÿฃ }|x๐Ÿฃ =๐Ÿข

)๏ธ ๐Ÿค๐Ÿฃ

.

j

We also have diagonal unitary matrices M๐œˆโ€ฒ = ๐–ฝ๐—‚๐–บ๐—€(e iโ„“๐Ÿฃ๐œˆ , e iโ„“๐Ÿค๐œˆ ) with โ„“j๐œˆ calculated along ๐œˆ-th leg of the closed trajectory of ๐ญjt with t = ๐Ÿฃ๐Ÿค T๐Ÿข and ๐œˆ = ๐Ÿฃ, ๐Ÿค. Consider closed branching billiard originated from point z โˆˆ T * Y ; first there are two trajectories (in X๐Ÿฃ and X๐Ÿค ) both hitting T * X |Y at ๐œš(z) and then there are two trajectories (in X๐Ÿฃ and X๐Ÿค ) returning to z. Let us recall that ๐œš(z) is an antipodal point. Therefore branching billiard is characterized by an unitary (in (4.5)-norm) matrix M(z, ๐œ ) = ๐œ˜(z, ๐œ ) M โ€ฒ (z, ๐œ ) ๐œ˜(๐œš(z), ๐œ ) M โ€ฒ (๐œš(z), ๐œ ).

(4.6)

4.3

Analysis of the evolution

One can prove easily that modulo OT (h๐Ÿคโˆ’d ) (๏ธ€ )๏ธ€ (4.7) Ftโ†’hโˆ’๐Ÿฃ ๐œ ๐œ’T (t)๐˜โ€ฒ ๐’ฌ(x โ€ฒ , hDxโ€ฒ , hDt , h)V โ‰ก โˆซ๏ธ โˆซ๏ธ โˆ‘๏ธ ๐Ÿฃโˆ’d (๐Ÿค๐œ‹h) ๐œ’T (T๐Ÿฃ + ยท ยท ยท + Tn )

โˆ‘๏ธ

Nฬธ=๐Ÿฃ j=(j๐Ÿฃ ,...,jN )โˆˆ{๐Ÿฃ,๐Ÿค}N

Mj๐Ÿฃ j๐Ÿค Mj๐Ÿค j๐Ÿฅ ยท ยท ยท MjNโˆ’๐Ÿฃ jN e โˆ’ih

โˆ’๐Ÿฃ ๐œ (T

j๐Ÿฃ +...TjN )

๐’ฌjN j๐Ÿฃ dx โ€ฒ d๐œ‰ โ€ฒ

where ๐œ’ โˆˆ C๐Ÿขโˆž is supported in [ ๐Ÿฃ๐Ÿค , ๐Ÿฃ] and symbols Mjk and ๐’ฌ and Tj are calculated at (x โ€ฒ , ๐œ‰ โ€ฒ , ๐œ ) where Tj = Tj (๐œ ) since all trajectories are periodic on energy levels close to ๐œ . One can rewrite terms with equal N as ๐Ÿฃโˆ’d

(4.8) (๐Ÿค๐œ‹h)

โˆซ๏ธ ๐œ’(๐œ†)e ฬ‚

iT โˆ’๐Ÿฃ ๐œ†(Tj๐Ÿฃ +ยทยทยท+TjN )

โˆซ๏ธ โˆซ๏ธ

โˆ‘๏ธ j=(j๐Ÿฃ ,...,jN )โˆˆ{๐Ÿฃ,๐Ÿค}N

Mj๐Ÿฃ j๐Ÿค Mj๐Ÿค j๐Ÿฅ ยท ยท ยท MjNโˆ’๐Ÿฃ jN e โˆ’ih

โˆ’๐Ÿฃ ๐œ (T

j๐Ÿฃ +...TjN )

๐’ฌjN j๐Ÿฃ dx โ€ฒ d๐œ‰ โ€ฒ d๐œ†

Chapter 4. Two periodic flows

21

which in turn equals (4.9)

(๐Ÿค๐œ‹h)

๐Ÿฃโˆ’d

โˆซ๏ธ

โˆซ๏ธ โˆซ๏ธ ๐œ’(๐œ†) ฬ‚

(๏ธ€ )๏ธ€ ๐—๐—‹ S(๐œ†, ๐œ , x โ€ฒ , ๐œ‰ โ€ฒ )N ๐’ฌ dx โ€ฒ d๐œ‰ โ€ฒ d๐œ†

where (4.10)

S(๐œ†T

โˆ’๐Ÿฃ

โ€ฒ

(๏ธ‚

โ€ฒ

, ๐œ, x , ๐œ‰ ) =

โˆ’๐Ÿฃ

โˆ’๐Ÿฃ

โˆ’๐Ÿฃ

โˆ’๐Ÿฃ

e i(๐œ†T +๐œ h )T๐Ÿฃ m๐Ÿฃ๐Ÿฃ e i(๐œ†T +๐œ h )T๐Ÿฃ m๐Ÿฃ๐Ÿค โˆ’๐Ÿฃ โˆ’๐Ÿฃ โˆ’๐Ÿฃ โˆ’๐Ÿฃ e i(๐œ†T +๐œ h )T๐Ÿค m๐Ÿค๐Ÿฃ e i(๐œ†T +๐œ h )T๐Ÿค m๐Ÿค๐Ÿค

)๏ธ‚

and M = (mjk ). Consider eigenvalues of this matrix; as T โ†’ โˆž they tend to eigenvalues of the same matrix with ๐œ‡ = ๐Ÿข. If (๏ธ€ )๏ธ€ (4.11) ๐—†๐–พ๐—Œ{(x โ€ฒ , ๐œ‰ โ€ฒ ) : ๐œŒ โˆˆ ๐–ฒ๐—‰๐–พ๐–ผ S(๐Ÿข, ๐œ , x โ€ฒ , ๐œ‰ โ€ฒ ) } = o(๐Ÿฃ) โˆ€๐œ โˆˆ [ฬ„ ๐œ โˆ’ ๐œ–hT โˆ’๐Ÿฃ , ๐œฬ„ + ๐œ–hT โˆ’๐Ÿฃ ]

as T โ†’ โˆž

then this term is o(h๐Ÿฃโˆ’d ) and therefore we estimated expression (4.7) by o(Th๐Ÿฃโˆ’d ) which in the end of the day returns remainder estimate o(h๐Ÿฃโˆ’d ) with the Tauberian main part. Note that instead of S we can consider matrix (๏ธƒ )๏ธƒ i๐œ hโˆ’๐Ÿฃ T * e m m ๐Ÿฃ๐Ÿฃ ๐Ÿฃ๐Ÿค โ€ฒ โ€ฒ ฬƒ ,x ,๐œ‰ ) = (4.12) S(๐œ โˆ’๐Ÿฃ * m๐Ÿค๐Ÿฃ e โˆ’i๐œ h T g๐Ÿค๐Ÿค ๐Ÿฃ

โˆ’๐Ÿฃ

with T * = ๐Ÿฃ๐Ÿค (T๐Ÿฃ โˆ’T๐Ÿค ) which is different from S(๐Ÿข, ., ., .) by factor e ๐Ÿค ih ๐œ (T๐Ÿฃ +T๐Ÿค ) . Consider correction to the main part; we are interested at ๐œ = ๐Ÿข; so we integrate (4.7) with ๐œ’ = ๐Ÿฃ from ๐œ = โˆ’โˆž to ๐œ = ๐Ÿข resulting in (after multiplication by hโˆ’๐Ÿฃ and modulo o(h๐Ÿฃโˆ’d )) โˆซ๏ธ โˆซ๏ธ โˆ‘๏ธ โˆ‘๏ธ ๐Ÿฃโˆ’d (4.13) (๐Ÿค๐œ‹h) ๐œ’T (T๐Ÿฃ + ยท ยท ยท + Tn ) Nฬธ=๐Ÿฃ jโˆˆ{๐Ÿฃ,๐Ÿค}N

Mj๐Ÿฃ j๐Ÿค Mj๐Ÿค j๐Ÿฅ ยท ยท ยท MjNโˆ’๐Ÿฃ jN (Tj๐Ÿฃ + ... TjN )โˆ’๐Ÿฃ ๐’ฌjN j๐Ÿฃ dx โ€ฒ d๐œ‰ โ€ฒ where we plug ๐œ = ๐Ÿข. This latter equals to (4.14)

๐Ÿฃโˆ’d

๐ฎ(๐œ ) = (๐Ÿค๐œ‹h)

โˆซ๏ธ โˆซ๏ธ

๐—๐—‹ ๐ช(L(x โ€ฒ , ๐œ‰ โ€ฒ , hโˆ’๐Ÿฃ ๐œ )) ๐’ฌ(x โ€ฒ , ๐œ‰.๐œ )dx โ€ฒ d๐œ‰ โ€ฒ

where L is defined by e iL = S and ๐ช is defined by (6.2.63).

Chapter 4. Two periodic flows

22

Theorem 4.3. Consider two Schrยจodinger operators A๐Ÿฃ and A๐Ÿค with all periodic trajectories on levels close to ๐œ , satisfying (3.1). Further, let (3.5) and (4.11) be fulfilled. Then asymptotics (๏ธ€ )๏ธ€ (๏ธ€ )๏ธ€ (4.15) ๐—๐—‹ ๐˜ e(., ., ๐œ ) = ๐œ˜๐Ÿข (๐œ )hโˆ’d + ๐œ˜๐Ÿฃ (๐œ ) + ๐ฎ(๐œ ) h๐Ÿฃโˆ’d + o(h๐Ÿฃโˆ’d ) holds. Example 4.4. Consider example 3.6 in the new conditions to Hamiltonian flows. We can use results of that example immediately to treat billiards with complete internal reflections (thus non-branching). To treat branching billiards we need to calculate eigenvalues of the matrix (๐œ˜jk ). It follows from calculations of example 3.6 that ๐œ˜๐Ÿค๐Ÿค = โˆ’๐œ˜๐Ÿฃ๐Ÿฃ ; thus we arrive to (๏ธ€ (4.16) ๐œ† = e ยฑi๐œ™ , ๐œ™ = ๐–บ๐—‹๐–ผ๐–ผ๐—ˆ๐—Œ ๐œ˜๐Ÿฃ๐Ÿฃ = ๐–บ๐—‹๐–ผ๐–ผ๐—ˆ๐—Œ (๐œ” โˆ’ ๐Ÿฃ)(๐œ” + ๐Ÿฃ)โˆ’๐Ÿฃ ; then obviously (4.11) is fulfilled and asymptotics (4.15) holds. Remark 4.5. On the contrary assume that (4.11) is not fulfilled. Then there will be eigenvalues of high multiplicity or clusters of eigenvalues located in o(h)-vicinities of solutions ๐œ of equation ๐–ฝ๐–พ๐—(S โˆ’ ๐Ÿฃ) = ๐Ÿข. One can rewrite this equation as )๏ธ€ (๏ธ€ )๏ธ€ (๏ธ€ ๐Ÿฃ ๐Ÿฃ (4.17) ๐–ผ๐—ˆ๐—Œ โˆ’๐œ™ + ๐œ hโˆ’๐Ÿฃ (T๐Ÿฃ + T๐Ÿค ) = ๐–ผ๐—ˆ๐—Œ ๐›ผ ยท ๐–ผ๐—ˆ๐—Œ โˆ’๐œ“ + ๐œ hโˆ’๐Ÿฃ (T๐Ÿฃ โˆ’ T๐Ÿค ) ๐Ÿค ๐Ÿค (๏ธ‚ i(๐œ‘+๐œ“) )๏ธ‚ e ๐–ผ๐—ˆ๐—Œ ๐›ผ e i(๐œ‘+๐œ’) ๐—Œ๐—‚๐—‡ ๐›ผ as M = which is the general form of the i(๐œ‘โˆ’๐œ’) โˆ’e ๐—Œ๐—‚๐—‡ ๐›ผ e i(๐œ‘โˆ’๐œ“) ๐–ผ๐—ˆ๐—Œ ๐›ผ unitary matrix. The following problem seems to be a difficult one and worth of publication: Problem 4.6. Prove the same results as flows ๐ญj,t are assumed to be periodic only on one energy level ๐œ rather than on neighboring levels. On the contrary, the following problem seems to be relatively easy: Problem 4.7. (i) Prove the same results as there are more than two manifolds Xj j = ๐Ÿฃ, ... , mโ€ฒ with the all billiards periodic. (ii) Extend these results to the case when there are also manifolds Xj j = mโ€ฒ + ๐Ÿฃ, ... , m with almost all billiards non-periodic.

Bibliography

23

Bibliography [Ivr1]

V. Ivrii. Microlocal Analysis and Precise Spectral Asymptotics, Springer-Verlag, SMM, 1998, xv+731.

[Ivr2]

V. Ivrii. Microlocal Analysis and Sharp Spectral Asymptotics, in progress: available online at http://www.math.toronto.edu/ivrii/futurebook.pdf Department of Mathematics, University of Toronto, 40, St.George Str., Toronto, Ontario M5S 2E4 Canada [email protected] Fax: (416)978-4107