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(h๐ฃโd ) 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].
1
Chapter 1. Discussion and plan
2
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 T๐ข , ๐ญT๐ข = I we have the same equality (1.1)
e ih
โ๐ฃ T
๐ขA
Q = e iT๐ข L Q
as before where ๐๐๐๐ q is disjoint from the boundary and only transversal to the boundary trajectories originate at ๐๐๐๐ q and L is h-pseudo-differential operator. Then (๏ธ )๏ธ t (1.2) Ftโhโ๐ฃ ๐ ๐( ฬ โ n)๐ Q๐ฃx u tQ๐คy = (๐ค๐h)๐ฃโd J(n) + O(h๐ฃโd+๐ฟ ) T๐ข with โซ๏ธ (1.3)
e inโ q๐ฃ q๐ค dยต๐
J(n) = ๐จ๐
as ๐ฬ is supported in (โ ๐ค๐ฅ , ๐ค๐ฅ ) and equals ๐ฃ in (โ ๐ฃ๐ฅ , ๐ฃ๐ฅ ) and n โ โค, |n| โค hโ๐ฟ . Here as before ๐จ๐ = {a(x, ๐) = ๐ } and ยต๐ = dxd๐ : da is the measure there, โ is the principal symbol of L. If โ is โvariable enoughโ on ๐จ๐ so the right-hand expression is decaying as n โ โ we can improve remainder estimate O(h๐ฃโd ) (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 ๐ข and ๐ for Neumann and Dirichlet boundary condition respectively; however more general boundary condition brings increment which depends on the point (x โฒ , ๐ โฒ ) โ 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
3
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) โซ๏ธ โฒ (1.4) J(n) = e inโโ|n|โ q๐ฃ q๐ค dยต๐ฃ.๐ ๐จ๐ฃ,๐
where ๐จ๐ฃ,๐ and ยต๐ฃ,๐ correspond to the first operator and โโฒ โฅ ๐ข 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) ยต๐ฃ,๐ {(x, ๐) โ ๐จ๐ฃ,๐ , โโฒ = ๐ข, โโ = ๐ข} = ๐ข we conclude that J(n) = o(๐ฃ) as n โ โ and we will be able to improve remainder estimate O(h๐ฃโd ) to o(h๐ฃโd ) and again there will be a non-Weyl correction term but it will depend also on โ and โโฒ . 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 ๐ฐ be an open connected subset in T * X disjoint from ๐X . Consider Hamiltonian billiards issued from ๐ฐ, let ๐ซt denote generalized Hamiltonian flow. Assume that (2.1)
|โx,๐ a(x, ๐)| โฅ ๐
(2.2)
|๐(x)| + |{a, ๐}(x, ๐)| โฅ ๐
(2.3)
๐ซt (x, ๐) = (x, ๐)
โt โ(x, ๐) โ ๐ซt (๐ฐ), โt โ(x, ๐) โ ๐ซt (๐ฐ),
with t = t(x, ๐) > ๐ข โ(x, ๐) โ ๐ฐ,
ฬ ), ๐ > ๐ข in X and ๐ = ๐ข on ๐X and where ๐ โ C โ (X (2.4) Along ๐ซt (๐ฐ) reflections are simple without branching, i.e. ๐โ๐ฃ ๐(y , ๐) โฉ {a(y , ๐) = ๐ } consists of two (disjoint) points (y , ๐ ยฑ ) such that ยฑ{a, ๐} > ๐ข as (y , ๐) โ ๐ซt (๐ฐ), y โ ๐X and ๐ = a(y , ๐).
Chapter 2. Simple Hamiltonian flow
4
Let us recall that ๐ : T * X |๐X โ T * ๐X is a natural map. Then exactly as in the case without boundary (2.5) One can select t(x, ๐) = T (a(x, ๐)) in ๐ฐ with T โ C โ (so period depends on energy level only2) and (2.6) As (x, ๐) โ ๐ฐ and ๐ข < t < T (x, ๐) ๐ซt (x, ๐) meats ๐X exactly (N โ ๐ฃ) times with N = ๐ผ๐๐๐๐; so as N โฅ ๐ฃ billiard consists of N segments with the ends on the boundary. We can consider instead of A with domain (2.7)
D(A) = {u โ H m (X ), รฐBu = ๐ข}
operator f (A) with โซ๏ธ (2.8)
f (๐ ) =
T โ๐ฃ (๐ ) d๐
โ๐ฃ
and introduce propagator e ih tf (A) so that the Hamiltonian flow ๐ญt of f (A) is periodic with period ๐ฃ. However now structure of f (A) near ๐X is a bit murky; but it is not really serious obstacle since we can consider problem (๏ธ )๏ธ โ๐ฃ (2.9) fฬ(hDt ) โ A e ih tf (A) = ๐ข, (2.10)
รฐBe ih
โ๐ฃ tf (A)
=๐ข
where fฬ 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 . โ๐ฃ Note that due to assumptions (2.1)โ(2.2) e ih tf (A) Q is a Fourier integral operator with the symplectomorphism ๐ญt provided Q is h-pseudo-differential operator with the symbol supported in ๐ฐ. Further, due to (2.3), (2.8) ๐ญ๐ฃ = I โ๐ฃ and therefore e ih f (A) Q is h-pseudo-differential operator. So, we arrive to But there could be subperiodic trajectories with periods nโ๐ฃ T (a(x, ๐)), n = ๐ค, ๐ฅ, ... . The structure of subperiodic trajectories described in subsection 6.2.5. 2)
Chapter 2. Simple Hamiltonian flow
5
Proposition 2.1. Let A be a Schrยจodinger operator with the scalar principal symbol a(x, ๐) satisfying (2.1)โ(2.3) and (2.8). Then (2.11)
e ih
โ๐ฃ f (A)โi๐
โก e ih
โ๐ฃ ๐L
in ๐ฐ
with ๐ = h and h-pseudo-differential operator L; here ๐
is Maslovโs constant and the principal symbol โ of L is defined by โ๏ธ โฒ (2.12) e iโ = e iโk e iโk ๐ฃโคkโคN
where โk corresponds to k-th segment (see theorem 6.2.3) and โโฒk corresponds to k-th reflection. Example 2.2. Consider Schrยจodinger operator. Assume that at the reflection point a(x, ๐) = ๐๐ฃ๐ค + a(x โฒ , ๐ โฒ ), b = ๐๐ฃ + i๐ฝ(๐ โฒ ). Then (2.13)
(๏ธ )๏ธโ๐ฃ (๏ธ )๏ธ ๐ฃ ๐ฃ โฒ e iโk = โ (๐ โ aโฒ ) ๐ค โ i๐ฝ (๐ โ aโฒ ) ๐ค + ฤฑ๐ฝ
with the right-hand expression calculated in the corresponding reflection point. Taking ๐ฝ = ๐ข or ๐ฝ = โ (formally) we get โโฒk = ๐ข and โโฒk = ๐ 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 โค ๐ โค h and derive local spectral asymptotics with the same precision (up to O(h๐คโd )) as there. Corollary 2.3. Let conditions of proposition 2.1 be fulfilled. Then asymptotics with the remainder o(h๐ฃโd ) holds provided (๏ธ )๏ธ (2.14) ยต๐ {(x, ๐) โ ๐จ๐ , โ๐จ๐ โ(x, ๐) = ๐ข} = ๐ข where โ๐จ๐ means a gradient along ๐จ๐ ; (ii) Asymptotics with the remainder O(h๐ฃโd+๐ฟ ) with small exponent ๐ฟ > ๐ข holds provided (๏ธ )๏ธ โฒ (2.15) ยต๐ {(x, ๐) โ ๐จ๐ , |โโฒ โ(x, ๐)| โค ๐} = o(๐๐ฟ ) with the small exponent ๐ฟ โฒ > ๐ข;
Chapter 2. Simple Hamiltonian flow
6
(iii) Furthermore asymptotics with the remainder o(h๐ฃโd ) holds as (2.1)โ(2.2) are replaced by their non-uniform versions (2.2)โฒ
|๐(x)| + |{a, ๐}(x, ๐)| > ๐ข
โt โ(x, ๐) โ ๐ซt (๐ฐ โฉ ๐จ๐ โ ๐ ๐ ),
where ยต๐ (๐ ๐ ) = ๐ข. We leave to the reader to formulate statement similar to (iii) but with the remainder estimate O(h๐ฃโd+๐ฟ ). Usually our purpose is the asymptotics with Q๐ฃx = ๐๐ฃ (x), Q๐คy = ๐๐ค (y ) where ๐j are smooth functions but not vanishing near the ๐X . Corollary 2.4. Asymptotics with the remainder estimate o(h๐ฃโd ) holds provided conditions (2.1), (2.2)โฒ , (2.3) and (2.14) are fulfilled in ๐ฐ = T * X , ฬ ) compactly supported. Qj = ๐j โ C โ (X Proof. Due to corollary 2.3(iii) contribution of the zone {x๐ฃ > ๐} to the remainder is o๐ (h๐ฃโd ) while contribution of zone {x๐ฃ โค ๐} to the remainder does not exceed O(๐h๐ฃโd ) as ๐ โฅ h. These arguments could be improved to ๐ = h๐ฟ and the remainder estimate could be improved to O(h๐ฃโd+๐ฟ ). 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 = {x๐ฃ > ๐ข} and ๐ฐ open subset in T * โdโ๐ฃ ร [๐ข, ๐]x๐ฃ ร โ๐ xโฒ where ๐ > ๐ข is a very small constant. We are interested in (๏ธ โ๐ฃ )๏ธ (2.16) Ftโhโ๐ฃ ๐ ๐ฬT (t) ๐ณ๐ e ih tf (A) Q with Q = Q(x, hDxโฒ , hDt ) with symbol supported in ๐ฐ; let us rewrite it as (๏ธ โ๐ฃ )๏ธ โ๐ฃ โ๐ฃ Ftโhโ๐ฃ ๐ ๐ฬT (t) ๐ณ๐(e ih (tโฬt )f (A) Q โฒ e ih ฬt f (A) ) = Ftโhโ๐ฃ ๐ ๐ฬT (t) ๐ณ๐ e ih tf (A) Q โฒ with Q โฒ = e ih
โ๐ฃ ฬ t f (A)
Qe โih
โ๐ฃ ฬ t f (A)
.
Chapter 2. Simple Hamiltonian flow
7
Note that if (2.1), (2.2) are fulfilled, ๐ > ๐ข and ฬt โฅ C ๐ are small enough then Q โฒ = Q โฒ (x, hDx , hDt ) is h-pseudo-differential operator with the symbol supported in {๐โฒฬt โค x๐ฃ โค C โฒฬt } and we can rewrite (๏ธ )๏ธ Q โฒ = Q โฒโฒ (x, hDx ) + Q โฒโฒโฒ (x, hDx , hDt ) hDt โ f (A) ; therefore we can rewrite (2.16) in the same form but with Q replaced by Q โฒโฒ . 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(h๐คโd ) 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 โฒ , ๐ โฒ , ๐ ) โ T * ๐X to (y โฒ , ๐ ยฑ ) โ ๐จ๐ . So, we need to analyze only near tangent zone {x๐ฃ โค ๐, |{a, x๐ฃ }| โค ๐}. 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)โฒ and (2.3) are fulfilled for all (x, ๐) โ T * X : ๐๐ฃ โค a(x, ๐) โค ๐๐ค with ๐๐ฃ < ๐๐ค . Then trajectories tangent to ๐X cannot penetrate into X and therefore x๐ฃ = ๐ข, ๐๐ฃ โค aโฒ (x, ๐ โฒ ) โค ๐๐ค is incompatible with axโฒ ๐ฃ (x, ๐) < ๐ข. Further, x๐ฃ = ๐ข, ๐๐ฃ โค aโฒ (x, ๐ โฒ ) โค ๐๐ค is incompatible with axโฒ ๐ฃ (x, ๐) > ๐ข as well; really otherwise almost tangent to boundary billiards make very small jumps which contradicts to periodicity after N reflections with fixed N. Therefore ๐X is bicharacteristically flat (2.17)
x๐ฃ = ๐ข, ๐๐ฃ โค aโฒ (x, ๐ โฒ ) โค ๐๐ค =โ axโฒ ๐ฃ (x, ๐) = ๐ข.
Further, (2.3) implies that (2.18) All solutions of equation (2.19)
z โฒโฒ + b(๐ญt (x โฒ , ๐ โฒ ))z = ๐ข
๐ฃ with b(x โฒ , ๐) = axโฒ ๐ฃ x๐ฃ (๐ข, x โฒ , ๐ โฒ ) ๐ค
are T๐ข -periodic for any (x โฒ , ๐ โฒ ) โ T * ๐X , ๐๐ฃ โค aโฒ (x, ๐ โฒ ) โค ๐๐ค . Therefore (2.3) implies that (2.20) ๐(x, ๐) โ ๐ญt โ ๐ with ๐ = x๐ฃ + |{a, x๐ฃ }| as ๐๐ฃ โค a(x, ๐) โค ๐๐ค .
Chapter 2. Simple Hamiltonian flow
8
Consider zone ๐ฐ๐ = {(x, ๐) : ๐(x, ๐) โ ๐}. Blowing up (x๐ฃ , ๐๐ฃ ) โ ๐โ๐ฃ (x๐ฃ , ๐๐ฃ ) we can prove easily that after this (๏ธ )๏ธ (2.21) ๐ญt (x, ๐) = ๐ญโฒt,๐ (x, ๐), ๐ญโฒโฒ(x โฒ ,๐โฒ ,t),๐ (x๐ฃ , ๐๐ฃ ) , with โ๏ธ (2.22) ๐n ๐ญโฒt,(n) (x, ๐), ๐ญโฒt,๐ (x, ๐) โผ ๐ญโฒt,๐ข (x โฒ , ๐ โฒ ) + nโฅ๐ค
(2.23)
๐ญโฒโฒ(x โฒ ,๐โฒ ,t),๐ (x๐ฃ , ๐๐ฃ ) โผ
โ๏ธ
๐n ๐ญโฒโฒ(x โฒ ,๐โฒ ,t),(n) (x๐ฃ , ๐๐ฃ )
nโฅ๐ฃ
where ๐ญโฒt,๐ข is the Hamiltonian flow on T * ๐X and ๐ญโฒโฒt,(๐ฃ) is the linearized with respect to (x๐ฃ , ๐๐ฃ ) billiard flow (described by (2.19)). One can prove easily that ๐ญโฒt,(n) and ๐ญโฒโฒ(x โฒ ,๐โฒ ,t),(n) are uniformly smooth. ๐ฃ
Then in blown-up coordinates (2.11) holds provided ๐ โฅ h ๐ค โ๐ฟ because โ = ๐โ๐ค h and then we can apply all the previous arguments and derive asymptotics; the remainder estimate is as sharp as O(h๐คโd ) provided (in not blown-up coordinates) (2.24)
|โ(x,๐) โ(x, ๐)| โฅ ๐๐ข
as x๐ฃ โ ๐(x, ๐) โค ๐๐ฃ ;
x| โ ๐(x, ๐) means exactly that x๐ฃ โฅ ๐|๐๐ฃ |. Therefore Proposition 2.5. Let A be a Schrยจ odinger operator with the principal symbol a(x, ๐) 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) : ๐(x, ๐) โค ๐๐ข }. ๐ฃ Then the contribution of zone {h ๐ค โ๐ฟ โค ๐(x, ๐) โค ๐๐ค } to the remainder is O(h๐คโd ). Since the contribution of zone {๐(x, ๐) โค ๐} to the remainder does not ๐ฃ exceed C ๐๐ค h๐ฃโd and we can take ๐ = h ๐ค โ๐ฟ we conclude that the contribution of zone {๐(x, ๐) โค ๐๐ค } to the remainder is O(h๐คโdโ๐ฟ ). 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) 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) : ๐(x, ๐) โค ๐๐ข }.
Chapter 2. Simple Hamiltonian flow
9
Then contribution of zone {๐(x, ๐) โค ๐๐ค } to the remainder is O(h๐คโdโ๐ฟ ) 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(h๐คโd ) 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 h๐ค ๐ on the standard hemisphere ๐d+ := {|x| = ๐ฃ, x๐ฃ > ๐ข} in โd+๐ฃ or harmonic oscillator h๐ค ๐ + |x|๐ค on the standard half-space โd+ = {x๐ฃ > ๐ข}. (i) Consider boundary operator of example 2.2. Then N = ๐ค and โ(z, ๐ ) = โโฒ (z, ๐ ) + โโฒ (๐(z), ๐ ) where z = (x โฒ , ๐ โฒ ), ๐(z) is the antipodal point and โโฒ is defined by (2.13). Then one can express conditions to โ as conditions to ๐ฝ; we leave exact statements to the reader; (ii) Consider Dirichlet or Neumann boundary conditions. Then as perturbation is hx๐ฃ one can calculate easily that โ = ๐
๐ + O(๐๐ค ) where ๐ is the incidence angle of trajectory and ๐
> ๐ข. 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 {๐๐ฃ โค a(x, ๐) โค ๐๐ค }. We assume that the phase space is connected: Problem 2.9. (i) So far our only examples are a standard hemisphere ๐d+ and a standard half-space โd+ (see figure below); are any other there really different examples? (ii) In particular, are there manifolds with N ฬธ= ๐ค (i.e. with N = ๐ฃ, ๐ฅ, ๐ฆ, ... )? We can consider N copies X๐ฃ , ... , XN of the standard quarter-sphere ๐๐ค โฉ {๐ข โค ๐ โค ๐๐ค } โฉ {๐ข โค ๐ โค ๐}, then glue {Xk , ๐ = ๐} with {Xk+๐ฃ , ๐ = ๐ข} for k = ๐ฃ, ... , N, where XN+๐ฃ := XN . Then for resulting manifold we have N reflections but for N ฬธ= ๐ค there will be singularity at North Pole.
Chapter 3. Branching Hamiltonian flow with โscatteringโ
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 ๐X which is not connected? (iv) In particular, are there manifolds with subperiodic billiards and (or) with subperiodic boundary trajectories?
3 Branching Hamiltonian flow with โscatteringโ Let us consider two manifolds X๐ฃ and X๐ค 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 โreflection-refractionโ. Further we are interested in the case when Hamiltonian flow on X๐ฃ is periodic while on X๐ค majority of trajectories are not periodic.
Chapter 3. Branching Hamiltonian flow with โscatteringโ
3.1
11
Analysis in X๐ค
Since we do not know any other examples of manifolds with periodic Hamiltonian flows with reflections but hemisphere or half-space and N = ๐ค then we assume that (3.1) For z โ T * Y ๐ซ๐ฃ,๐ (z) = ๐(z) 3) , ๐ is an antipodal map on T * Y . We also assume that ๐ซ๐ฃ,๐ and ๐ซ๐ค,๐ commute (3.2)
๐ซ๐ฃ,๐ โ ๐ซ๐ค,๐ = ๐ซ๐ค,๐ โ ๐ซ๐ฃ,๐ .
Then for points of ๐จ๐ค,๐ periodicity properties of the branching Hamiltonian flow with reflections ๐ญt coincide with those of ๐ญ๐ค,t . Now we need to impose condition to the boundary operator except that {A, B} is self-adjoint. Namely, consider a point z = (x โฒ , ๐ โฒ ) โ T * Y , and consider an auxiliary problem (3.3) (3.4)
aj (x โฒ , D๐ฃ , ๐ โฒ )uj = ๐ข
j = ๐ฃ, ๐ค.
รฐb(x โฒ , D๐ฃ , ๐ โฒ )u = ๐ข
(where u = (u๐ฃ , u๐ค )) and consider solutions which are combinations of exponents e i๐j x๐ฃ where for each j either ๐จ๐ ๐j > ๐ข or ๐จ๐ ๐j > ๐ข and {aj , x๐ฃ }(x โฒ , ๐j , ๐ โฒ ) > ๐ข. Denote by ๐ the set of points for which such non-trivial solutions exist. We assume that (3.5)
๐๐พ๐T * Y ๐ = ๐ข.
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 Q๐ฃ , Q๐ค be h-pseudo-differential operators with the symbols supported in T * X๐ค and (3.6)
ยต๐ค,๐ (๐ฅ๐ค,๐ โฉ ๐๐๐๐ q๐ฃ โฉ ๐๐๐๐ q๐ค ) = ๐ข
where ๐ฅ๐ค,๐ is the set of point (y , ๐) โ ๐จ๐ค,๐ periodic with respect to ๐ญ๐ค,t . Then for ๐(Q๐ฃx e(., ., ๐ ) tQ๐ฃy ) the standard Weyl two-term asymptotics holds with the remainder estimate o(h๐ฃโd ). We define ๐ซj,๐ in the following way: we lift z โ T * Y to z โ TjX |Y โฉ ๐จj,๐ with {aj , x๐ฃ }(z) > ๐ข, launch trajectory forward until it hits the boundary at (y โฒ , ๐) โ T * Xj |Y and then project to T * Y . 3)
Chapter 3. Branching Hamiltonian flow with โscatteringโ
12
Remark 3.2. (i) Note that (3.7)
ยตj,๐ (๐ j,๐ ) = ๐ข
where ๐ j,๐ = ๐ โฒj,๐ โช ๐ โฒโฒj,๐ , ๐ โฒj,๐ is the set of dead-end points of billiard (not generalized billiard) flow and ๐ โฒโฒj,๐ is the set of points (y , ๐) โ ๐จj,๐ such that ๐j ๐ญj,t (y , ๐) โ ๐ฃ๐ฅโj,๐ where ๐ฃk,๐ = ๐k {(x, ๐) โ ๐จk,๐ , {ak , x๐ฃ } = ๐ข. (ii) Note that the grows properties of the branching Hamiltonian flow with reflections ๐ญt coincide with those of ๐ญ๐ค,t as long as we ensure that along billiards we do not approach to glancing rays on X๐ฃ : (3.8) {a๐ฃ , x๐ฃ }(x, ๐) โฅ h๐ฟ๐ฃ
โ(x, ๐) โ ๐โ๐ฃ ๐ฃ ๐๐ค ๐ญ๐ค,t (y , ๐) โฉ ๐จ๐ฃ,๐
โ(y , ๐) โ ๐๐๐๐ q๐ค โt : ยฑt โ [๐ข, T ] ๐ญ๐ค,t (y , ๐) โ T * X๐ค |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)โ๐ฃ h๐ฃโd ) which is usually O(h๐ฃโd | ๐
๐๐ h|โ๐ฃ ) or O(h๐ฃโd+๐ฟ ) (depending on the growth and non-periodicity conditions to ๐ญ๐ค,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 we get set ๐ ๐ ,๐ and we need to impose condition ๐๐พ๐ ๐ ๐ ,๐ = o(๐๐ฟ ) or ๐๐พ๐ ๐ ๐ ,๐ = o(| ๐
๐๐ ๐|โ๐ฃ ) as ๐ โ +๐ข. (iv) In subsubsection โ7.4.3โ of subsection 7.4.3 we did not consider points z โ T * Y where some of operators aj are elliptic on ๐โ๐ฃ 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 a๐ค is cylindrically symmetric on T * X๐ค or if there no internal reflection for billiards from X๐ค at all: (3.9) |{a๐ฃ , x๐ฃ }(x, ๐)| โฅ ๐|{a๐ค , x๐ฃ }(x, ๐)|l โ(x, ๐) โ T * X๐ฃ |Y โฉ ๐จ๐ฃ,๐ โ(x, ๐) โ T * X๐ค |Y โฉ ๐จ๐ค,๐ : ๐๐ฃ (x, ๐) = ๐๐ค (x, ๐).
Chapter 3. Branching Hamiltonian flow with โscatteringโ
3.2
13
Analysis in X๐ฃ
However if operators Q๐ฃ and Q๐ค have symbols supported in X๐ฃ situation is very different: trajectories of a๐ฃ are periodic but the typical closed branching billiard is the one which does not contain segments of Hamiltonian trajectories of a๐ค . More precisely, due to conditions (3.10)
ยต๐ค,๐ (๐ฅ๐ค,๐ ) = ๐ข
and (3.7) we conclude that (3.11)
ยต๐ฃ,๐ (๐ฅโฒ๐ฃ,๐ ) = ๐ข
where ๐ฅโฒ๐ฃ,๐ is the set of points z โ ๐จ๐ฃ,๐ such that there exists a billiard trajectory of ๐ญt starting and ending in z and which is not entirely billiard of ๐ญ๐ฃ,t . Then for arbitrarily large T > ๐ข and arbitrarily small ๐ > ๐ข there exists a set ๐ฐ = ๐ฐT ,๐ โ T * X such that (3.12)
ยต๐ฃ,๐ (๐จ๐ฃ,๐ โ ๐ฐ) โค ๐,
(3.13)
๐ฝ๐๐๐(๐ฐ, ๐ฅโฒ๐ฃ,๐ ,T โช ๐ ๐ฃ,๐ ,T ) โฅ ๐พ
(3.14)
๐ฝ๐๐๐(Y , ฯx ๐ฐ) โฅ ๐พ
with ๐พ = ๐พ(T , ๐) > ๐ข were ๐ฅโฒ๐ฃ,๐ ,T s the set of points z โ ๐จ๐ฃ,๐ such that there exists a billiard trajectory of ๐ญt of the length not exceeding T , starting and ending in z and which is not entirely billiard of ๐ญ๐ฃ,t and ๐ ๐ฃ,๐ ,T is the set of points z โ ๐จ๐ฃ,๐ such that billiard trajectory of a๐ฃ hits the boundary at point of ๐ ๐ค,T ,๐ . As usual, these sets increase as T increases, their unions (with respect to T ) coincide with ๐ฅโฒ๐ฃ,๐ and ๐ ๐ฃ,๐ respectively and ๐ ๐ฃ,๐ ,T and ๐ ๐ฃ,๐ ,T โฉ ๐ฅโฒ๐ฃ,๐ ,T are closed sets. Therefore there exists operator Q = QT ,๐ such that ๐๐๐๐ q โ ๐ฐ and )๏ธ (๏ธ (3.15) |๐ Q๐ฃx e(., ., ๐ ) t((I โ Q)Q๐ค )y โ ๐
โฒ๐ข hโd โ ๐
โฒ๐ฃ h๐ฃโd | โค C๐ข ๐h๐ฃโd with ๐
โฒj = ๐
โฒj,Q๐ฃ ,Q๐ค ,Q (๐ ). This estimate holds for any (๏ธ )๏ธ fixed ๐ satisfying (3.10). Therefore we need to consider ๐ Q๐ฃx e(., ., ๐ ) tQ๐คy with ๐๐๐๐ q๐ค โ ๐ฐ. We can assume without any loss of the generality that (3.13), (3.14) are fulfilled for all ๐ . Then for ๐ โค |t| โค T (๏ธ )๏ธ (๏ธ )๏ธ (3.16) ๐ Q๐ฃx U(., ., t) tQ๐คy โก ๐ Q๐ฃx Z (., ., t) tQ๐คy
Chapter 3. Branching Hamiltonian flow with โscatteringโ
14
where as ยฑt > ๐ข Z (t) is Schwartz kernel of โapproximate propagation with scattering semigroupโ ๐ญ(t) constructed in the following way (for a sake of simplicity in notations we consider t > ๐ข: (i) We represent Q๐ค = Q๐ค,๐ฃ + ยท ยท ยท + Q๐ค,N where symbols Q๐ค,๐ have small supports; (ii) For each ๐ we select ๐ข = t๐ข,๐ < ยท ยท ยท < tM,๐ = t in such a way that for any z โ ๐๐๐๐ q๐ค,๐ ๐ญt โฒ (z) is disjoint from Y as t โฒ = tj,๐ and ๐ญt โฒ (z) hits Y no more than once as tj,๐ โค t โฒ โค tj+๐ฃ,๐ ; (iii) We set ๐ญ(t)Q๐ค,๐ =
(๏ธ
โ๏ธ
๐e ih
โ๐ฃ (t j+๐ฃ,๐ โtj,๐ )f (A)
)๏ธ
ยท Q๐ค,๐
๐ขโคjโคMโ๐ฃ
where ๐ โ C โ is supported in {x โ X๐ฃ , ๐ฝ๐๐๐(x, Y ) โฅ ๐พ โฒ } and equals ๐ฃ in {x โ X๐ฃ , ๐ฝ๐๐๐(x, Y ) โฅ ๐ค๐พ โฒ }. Then (3.17) and (3.18)
๐ญ(t๐ฃ + t๐ค )Q๐ค โก ๐ญ(t๐ค )๐ญ(t๐ฃ )Q๐ค
as ยฑ t๐ฃ > ๐ข, ยฑt๐ค > ๐ข
๐ญ(t)* โก ๐ญ(โt)
and due to periodicity of ๐ญ๐ฃ,t with period T๐ข ๐ญ(ยฑT๐ข ) 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 ๐ญ๐ฃ,t hits T * X |Y at points (x โฒ , ๐) with (x โฒ , ๐ โฒ ) โ / ๐๐ค ๐จ๐ค,๐ reflection coefficient ๐๐ฃ๐ฃ (x โฒ , ๐ โฒ , ๐ ) still satisfies |๐๐ฃ๐ฃ (x โฒ , ๐ โฒ , ๐ )| = ๐ฃ. (ii) However if ๐ญ๐ฃ,t hits T * X |Y at points (x โฒ ,(๏ธ๐) with (x โฒ ,)๏ธ๐ โฒ ) โ ๐๐ค ๐จ๐ค,๐ then reflection-refraction matrix ๐(x โฒ , ๐ โฒ , ๐ ) = ๐jk (x โฒ , ๐ โฒ , ๐ ) j,k=๐ฃ,๐ค is unitary but reflection coefficient ๐๐ฃ๐ฃ (x โฒ , ๐ โฒ , ๐ ) satisfies only |๐๐ฃ๐ฃ (x โฒ , ๐ โฒ , ๐ )| โค ๐ฃ and the equality means exactly that ๐๐ฃ๐ค (x โฒ , ๐ โฒ , ๐ ) = ๐๐ค๐ฃ (x โฒ , ๐ โฒ , ๐ ) = ๐ข (then |๐๐ค๐ค (x โฒ , ๐ โฒ , ๐ )| = ๐ฃ as well). So the principal symbol of ๐ญ(T๐ข ) is the product of e iโ๐ข (x,๐) where โ๐ข (x, ๐) is given by the usual formula for manifolds without boundary (but with
Chapter 3. Branching Hamiltonian flow with โscatteringโ
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 ๐ญ(T๐ข ) โก e i๐h
(3.19)
โ๐ฃ L
๐ญ(โ๐hโ๐ฃ T๐ข ) โก e โiL
,
*
with h-pseudo-differential operator L, L + L* โค ๐ข and ๐ = 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 ๐จ๐ โ = +โ. Then as ๐ฬ โ C๐ขโ [๐ โ ๐ฃ, ๐ฃ โ ๐] (๏ธ )๏ธ (3.20) Ftโhโ๐ฃ ๐ ๐ฬT๐ข (t โ nT๐ข )๐ Q๐ฃx Z (., ., t) tQ๐คy = ฬ๏ธฬ h๐ฃโd J(h, ๐, n, ๐ ) ยท ๐
(๏ธ )๏ธ (๏ธ ๐ )๏ธ + O h๐คโd , h
with (3.21)
๐ฃโd
โซ๏ธ
J(h, ๐, n, ๐ ) = (๐ค๐)
e ih
โ๐ฃ ((๐ ๐ฑ๐พ โโ๐ )n+๐ ๐จ๐ โ|n|)
dยต๐ฃ,๐ .
๐จ๐ฃ,๐
We can see easily that (3.22) J(h, ๐, n, ๐ ) = o(๐ฃ) as n โ โ provided (๏ธ{๏ธ }๏ธ)๏ธ (3.23) ยต๐ (x, ๐) โ ๐จ๐ : ๐จ๐ โ(x, ๐) = โ ๐ฑ๐พ โ(x, ๐) = ๐ข = ๐ข. On the other hand, note that (3.24)
โ๏ธ nโโคโ๐ข
e ih
โ๐ฃ (n๐ผโ|n|๐ฝ)
=
๐คe โ๐ฝ ๐ผ๐๐ ๐ผ โ ๐คe โ๐ค๐ฝ ๐ฃ โ ๐คe โ๐ฝ ๐ผ๐๐ ๐ผ + e โ๐ค๐ฝ
as ๐ฝ > ๐ข
and formally we can extend this for ๐ฝ = ๐ข as well. Then after summation with respect to n for ๐ฝ = ๐ข we get exactly ๐๐ผ ๐ช(๐ผ) with ๐ช defined by (6.2.63) and for ๐ฝ > ๐ข we get that the expression (3.24) (๏ธ )๏ธโ๐ฃ is equal to ๐๐ฝ ๐ช *๐ผ (๐ค๐)โ๐ฃ G (๐ผ, ๐ฝ) where G (๐ผ, ๐ฝ) = ๐ค๐ฝ ๐ผ๐ค + ๐ฝ ๐ค . Note that
Chapter 3. Branching Hamiltonian flow with โscatteringโ
16
(3.25) Function ๐ช(๐ผ, ๐ฝ) := ๐ช *๐ผ G (๐ผ, ๐ฝ)
(3.26)
is harmonic function as ๐ฝ > ๐ข, coincides with ๐ช(๐ผ) as ๐ฝ = ๐ข, is o(๐ฃ) as ๐ฝ โ +โ and is periodic with respect to ๐ผ. 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) ๐e(., ., ๐ ) = ๐
๐ข (๐ )hโd + โซ๏ธ (๏ธ )๏ธ (๏ธ )๏ธ )๏ธ ๐
๐ฃ (๐ ) + ๐ช hโ๐ฃ (๐ ๐ฑ๐พ โ โ ๐ ), โhโ๐ฃ ๐ ๐จ๐ โ dยต๐ dยต๐ + o(h๐ฃโd ) ๐จ๐
holds. Problem 3.5. Calculate all coefficients ๐jk for two Schrยจodinger operators A๐ฃ and A๐ค . We will solve this problem in the very special case: Example 3.6. Assume that (at the given point z โ T * Y ) aj = cj๐ค |๐|๐ค and the boundary condition is (3.28) (3.29)
u๐ค = ๐ผu๐ฃ , D๐ฃ u๐ค = โ๐ฝD๐ฃ u๐ฃ
(recall that x๐ฃ > ๐ข in both X๐ฃ , X๐ค ; in more standard notations of x๐ฃ < ๐ข in X๐ค one needs to skip โโโ). Then {A, B} to be self-adjoint requires ๐ผ๐ฝ โ = c๐ฃ๐ค c๐คโ๐ค . We do not consider gliding points since their measure is ๐ข. (i) Consider point ๐ โฒ with c๐ฃ๐ค |๐ โฒ |๐ค < ๐ < c๐ค๐ค |๐ โฒ |๐ค . One can prove easily that (3.30) with (3.31)
๐๐ฃ๐ฃ (๐ โฒ , ๐ ) = e ๐คi๐ (๏ธ (๏ธ )๏ธ ๐ฃ (๏ธ )๏ธโ ๐ฃ )๏ธ ๐(๐ โฒ , ๐ ) = ๐บ๐๐ผ๐๐บ๐ |๐ฝ|โ๐ค c๐คโ๐ค c๐ฃ๐ค c๐ค๐ค |๐ โฒ |๐ค โ ๐ ๐ค ๐ โ c๐ฃ๐ค |๐ โฒ | ๐ค
Chapter 4. Two periodic flows
17
and therefore assumption (3.23) is fulfilled provided c๐ฃ ฬธ= c๐ค , cj , ๐ผ, ๐ฝ are symmetric (i.e. cj ๐ = cj ) and the subprincipal symbol is ๐ข. If c๐ฃ > c๐ค this case does not appear. (ii) Consider point ๐ โฒ with cj๐ค |๐ โฒ |๐ค < ๐ for both j = ๐ฃ, ๐ค. One can prove easily that (๏ธ )๏ธ(๏ธ )๏ธโ๐ฃ (3.32) ๐๐ฃ๐ฃ (๐ โฒ , ๐ ) = ๐ โ ๐ฃ ๐ + ๐ฃ with (๏ธ )๏ธ ๐ฃ (๏ธ )๏ธโ ๐ฃ (3.33) ๐(๐ โฒ , ๐ ) = |๐ฝ|โ๐ค c๐คโ๐ค c๐ฃ๐ค ๐ โ c๐ค๐ค |๐ โฒ |๐ค ๐ค ๐ โ c๐ฃ๐ค |๐ โฒ | ๐ค and |๐๐ฃ๐ฃ | < ๐ฃ 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 ๐ญ๐ค,t and โ (see (2.15)) prove remainder estimate O(h๐ฃโd+๐ฟ ). Problem 3.8. Prove the same results as Hamiltonian billiards of a๐ฃ are assumed to be periodic only on one energy level ๐ rather than on neighboring levels.
4
Two periodic flows
Now we consider the case of both flows ๐ญ๐ฃt and ๐ญ๐คt generated by f (a๐ฃ ) and f (a๐ค ) 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 โd+ = โ+ ร โdโ๐ฃ with the principal symbols (4.1)
)๏ธ ๐ฃ (๏ธ aj (x, ๐) = ๐j๐ค |๐|๐ค + |x|๐ค | โ Ej ๐ค
intertwined through boundary conditions. Then Tj = ๐ค๐๐jโ๐ค .
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 ๐ญโ h (๐ข) of the negative eigenvalues of operator AB . In order to consider the eigenvalue counting function ๐ญโ h (๐) for arbitrary spectral parameter ๐ one needs to redefine (4.2)
Ej := Ej + ๐ค๐j๐ค ๐.
But ๐ญโ (๐ข) is also the maximal dimension of the negative subspace of ๐ฃ ๐ฃ the operator A which does not change if we replace A by J โ ๐ค AJ โ ๐ค with the positive self-adjoint operator J and the latter equals to ๐ญโ h (๐ข) for the ๐ค latter operator. Picking up J = J(x) equal ๐j in Xj we arrive to the case of ๐j = ๐ฃ and Tj = ๐ค๐. Therefore in this example one can make periods equal T๐ฃ = T๐ค . The dependence of ๐j 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 ๐j , Ej are imposed): Example 4.2. Consider two operators on hemisphere ๐d+ with the principal symbols (๏ธ )๏ธ (4.3) aj (x, ๐) = ๐j๐ค |๐|๐ค โ Ej
Chapter 4. Two periodic flows
19
where |๐| is calculated in the standard Riemannian metrics. ๐ฃ
The main obstacle here is that multiplying operator by J โ ๐ค 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, ๐ฝ๐๐๐(z, T * Y ) โค ๐๐ข } does not exceed C๐ข ๐๐ค๐ข h๐ฃโd and therefore one needs to consider the contribution of {z, ๐ฝ๐๐๐(z, T * Y ) โฅ ๐๐ข } with arbitrarily small but fixed ๐๐ข . In this zone one can consider U(x, y , t) as the solution of the Cauchy problem with respect to x๐ฃ (or y๐ฃ ) with data at {x๐ฃ = ๐ข} (or {y๐ฃ = ๐ข}). 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 โก ๐โฒ ๐ฌ(x โฒ , hDxโฒ , hDt , h)V with h-pseudo-differential operator R and V = รฐx,mโ๐ฃ รฐy ,mโ๐ฃ 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 ๐ฌ at (x โฒ , ๐, ๐ ) is defined as an averaging of Q along ๐ญjt (zj ) with t โ [๐ข, tj (zj )] where zj = ๐โ๐ฃ (x โฒ , ๐ โฒ ) โฉ ๐จj๐ and tj (zj ) is the time of the next hit of the boundary. โ๐ฃ Then instead of continuous family of Fourier integral operators e ih tA we can consider a discrete family โฑ n defined by the following way: consider vj = รฐmโ๐ฃ uj = (vjโ , vj+ ) with vjโ corresponding to incoming and outgoing solutions. We have two types of trajectories: those in ๐จj๐ which hit T * X |Y at points elliptic for (a๐ฅโj โ ๐ ) where j = ๐ฃ, ๐ค and those which hit T * X |Y at points hyperbolic for (ak โ ๐ ) for both k = ๐ฃ, ๐ค. The analysis of the former is of no different from what we have seen in the previous subsection: we just consider ๐ญjt and construct the (real-valued) symbol โ as long as we assume that (x โฒ , ๐) โ / 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 (๐jk )j,k=๐ฃ,๐ค which is a symbol defined on ๐ฐ ร (๐ โ ๐โฒ , ๐ + ๐โฒ ) where ๐ฐ is a a zone described above. This matrix is
Chapter 4. Two periodic flows
20
unitary in the norm โV โ =
(4.5)
(๏ธโ๏ธ(๏ธ
|Vj+ |๐ค
+
|Vjโ |๐ค
)๏ธ
|{aj , x๐ฃ }|x๐ฃ =๐ข
)๏ธ ๐ค๐ฃ
.
j
We also have diagonal unitary matrices M๐โฒ = ๐ฝ๐๐บ๐(e iโ๐ฃ๐ , e iโ๐ค๐ ) with โj๐ calculated along ๐-th leg of the closed trajectory of ๐ญjt with t = ๐ฃ๐ค T๐ข and ๐ = ๐ฃ, ๐ค. Consider closed branching billiard originated from point z โ T * Y ; first there are two trajectories (in X๐ฃ and X๐ค ) both hitting T * X |Y at ๐(z) and then there are two trajectories (in X๐ฃ and X๐ค ) returning to z. Let us recall that ๐(z) is an antipodal point. Therefore branching billiard is characterized by an unitary (in (4.5)-norm) matrix M(z, ๐ ) = ๐(z, ๐ ) M โฒ (z, ๐ ) ๐(๐(z), ๐ ) M โฒ (๐(z), ๐ ).
(4.6)
4.3
Analysis of the evolution
One can prove easily that modulo OT (h๐คโd ) (๏ธ )๏ธ (4.7) Ftโhโ๐ฃ ๐ ๐T (t)๐โฒ ๐ฌ(x โฒ , hDxโฒ , hDt , h)V โก โซ๏ธ โซ๏ธ โ๏ธ ๐ฃโd (๐ค๐h) ๐T (T๐ฃ + ยท ยท ยท + Tn )
โ๏ธ
Nฬธ=๐ฃ j=(j๐ฃ ,...,jN )โ{๐ฃ,๐ค}N
Mj๐ฃ j๐ค Mj๐ค j๐ฅ ยท ยท ยท MjNโ๐ฃ jN e โih
โ๐ฃ ๐ (T
j๐ฃ +...TjN )
๐ฌjN j๐ฃ dx โฒ d๐ โฒ
where ๐ โ C๐ขโ is supported in [ ๐ฃ๐ค , ๐ฃ] and symbols Mjk and ๐ฌ and Tj are calculated at (x โฒ , ๐ โฒ , ๐ ) where Tj = Tj (๐ ) since all trajectories are periodic on energy levels close to ๐ . One can rewrite terms with equal N as ๐ฃโd
(4.8) (๐ค๐h)
โซ๏ธ ๐(๐)e ฬ
iT โ๐ฃ ๐(Tj๐ฃ +ยทยทยท+TjN )
โซ๏ธ โซ๏ธ
โ๏ธ j=(j๐ฃ ,...,jN )โ{๐ฃ,๐ค}N
Mj๐ฃ j๐ค Mj๐ค j๐ฅ ยท ยท ยท MjNโ๐ฃ jN e โih
โ๐ฃ ๐ (T
j๐ฃ +...TjN )
๐ฌjN j๐ฃ dx โฒ d๐ โฒ d๐
Chapter 4. Two periodic flows
21
which in turn equals (4.9)
(๐ค๐h)
๐ฃโd
โซ๏ธ
โซ๏ธ โซ๏ธ ๐(๐) ฬ
(๏ธ )๏ธ ๐๐ S(๐, ๐ , x โฒ , ๐ โฒ )N ๐ฌ dx โฒ d๐ โฒ d๐
where (4.10)
S(๐T
โ๐ฃ
โฒ
(๏ธ
โฒ
, ๐, x , ๐ ) =
โ๐ฃ
โ๐ฃ
โ๐ฃ
โ๐ฃ
e i(๐T +๐ h )T๐ฃ m๐ฃ๐ฃ e i(๐T +๐ h )T๐ฃ m๐ฃ๐ค โ๐ฃ โ๐ฃ โ๐ฃ โ๐ฃ e i(๐T +๐ h )T๐ค m๐ค๐ฃ e i(๐T +๐ h )T๐ค m๐ค๐ค
)๏ธ
and M = (mjk ). Consider eigenvalues of this matrix; as T โ โ they tend to eigenvalues of the same matrix with ๐ = ๐ข. If (๏ธ )๏ธ (4.11) ๐๐พ๐{(x โฒ , ๐ โฒ ) : ๐ โ ๐ฒ๐๐พ๐ผ S(๐ข, ๐ , x โฒ , ๐ โฒ ) } = o(๐ฃ) โ๐ โ [ฬ ๐ โ ๐hT โ๐ฃ , ๐ฬ + ๐hT โ๐ฃ ]
as T โ โ
then this term is o(h๐ฃโd ) and therefore we estimated expression (4.7) by o(Th๐ฃโd ) which in the end of the day returns remainder estimate o(h๐ฃโd ) with the Tauberian main part. Note that instead of S we can consider matrix (๏ธ )๏ธ i๐ hโ๐ฃ T * e m m ๐ฃ๐ฃ ๐ฃ๐ค โฒ โฒ ฬ ,x ,๐ ) = (4.12) S(๐ โ๐ฃ * m๐ค๐ฃ e โi๐ h T g๐ค๐ค ๐ฃ
โ๐ฃ
with T * = ๐ฃ๐ค (T๐ฃ โT๐ค ) which is different from S(๐ข, ., ., .) by factor e ๐ค ih ๐ (T๐ฃ +T๐ค ) . Consider correction to the main part; we are interested at ๐ = ๐ข; so we integrate (4.7) with ๐ = ๐ฃ from ๐ = โโ to ๐ = ๐ข resulting in (after multiplication by hโ๐ฃ and modulo o(h๐ฃโd )) โซ๏ธ โซ๏ธ โ๏ธ โ๏ธ ๐ฃโd (4.13) (๐ค๐h) ๐T (T๐ฃ + ยท ยท ยท + Tn ) Nฬธ=๐ฃ jโ{๐ฃ,๐ค}N
Mj๐ฃ j๐ค Mj๐ค j๐ฅ ยท ยท ยท MjNโ๐ฃ jN (Tj๐ฃ + ... TjN )โ๐ฃ ๐ฌjN j๐ฃ dx โฒ d๐ โฒ where we plug ๐ = ๐ข. This latter equals to (4.14)
๐ฃโd
๐ฎ(๐ ) = (๐ค๐h)
โซ๏ธ โซ๏ธ
๐๐ ๐ช(L(x โฒ , ๐ โฒ , hโ๐ฃ ๐ )) ๐ฌ(x โฒ , ๐.๐ )dx โฒ d๐ โฒ
where L is defined by e iL = S and ๐ช is defined by (6.2.63).
Chapter 4. Two periodic flows
22
Theorem 4.3. Consider two Schrยจodinger operators A๐ฃ and A๐ค with all periodic trajectories on levels close to ๐ , satisfying (3.1). Further, let (3.5) and (4.11) be fulfilled. Then asymptotics (๏ธ )๏ธ (๏ธ )๏ธ (4.15) ๐๐ ๐ e(., ., ๐ ) = ๐๐ข (๐ )hโd + ๐๐ฃ (๐ ) + ๐ฎ(๐ ) h๐ฃโd + o(h๐ฃโd ) 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 (๐jk ). It follows from calculations of example 3.6 that ๐๐ค๐ค = โ๐๐ฃ๐ฃ ; thus we arrive to (๏ธ (4.16) ๐ = e ยฑi๐ , ๐ = ๐บ๐๐ผ๐ผ๐๐ ๐๐ฃ๐ฃ = ๐บ๐๐ผ๐ผ๐๐ (๐ โ ๐ฃ)(๐ + ๐ฃ)โ๐ฃ ; 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 of equation ๐ฝ๐พ๐(S โ ๐ฃ) = ๐ข. One can rewrite this equation as )๏ธ (๏ธ )๏ธ (๏ธ ๐ฃ ๐ฃ (4.17) ๐ผ๐๐ โ๐ + ๐ hโ๐ฃ (T๐ฃ + T๐ค ) = ๐ผ๐๐ ๐ผ ยท ๐ผ๐๐ โ๐ + ๐ hโ๐ฃ (T๐ฃ โ T๐ค ) ๐ค ๐ค (๏ธ i(๐+๐) )๏ธ e ๐ผ๐๐ ๐ผ e i(๐+๐) ๐๐๐ ๐ผ as M = which is the general form of the i(๐โ๐) โe ๐๐๐ ๐ผ e i(๐โ๐) ๐ผ๐๐ ๐ผ unitary matrix. The following problem seems to be a difficult one and worth of publication: Problem 4.6. Prove the same results as flows ๐ญj,t are assumed to be periodic only on one energy level ๐ 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 = ๐ฃ, ... , mโฒ with the all billiards periodic. (ii) Extend these results to the case when there are also manifolds Xj j = mโฒ + ๐ฃ, ... , 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