Eigenvalue Asymptotics for Fractional Laplacians

Transcription

1 Eigenvalue Asymptotics for Fractional Laplacians Partial Differential Equations and Analysis Seminar Mathematical Sciences Institute, Australian National University Victor Ivrii Department of Mathematics, University of Toronto Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

2 Table of Contents Table of Contents 1 Introduction 2 Operator 3 Results Main theorem 1D operator and κ 1 4 Proof sketched Simple rescaling method Seeley s method Discussion 5 Proof sketched. II Propagation near the boundary Propagation near the boundary. II Estimates in the boundary zone Contribution of the inner zone 6 Further discussion 7 References Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

3 Introduction Introduction At Spring 2016 my son Oleg mentioned in the Google chat that he would like to learn an analytic technique developed by one cool guy who actually refers to one of my papers. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

4 Introduction Introduction At Spring 2016 my son Oleg mentioned in the Google chat that he would like to learn an analytic technique developed by one cool guy who actually refers to one of my papers. The cool guy turned out to be Professor Rodrigo Bañuelos from Purdue University (and indeed, he is really cool) who is doing some probabilistic research and is interested in the spectral asymptotics for generators of some unitary group describing random walks in domain with boundary and and believing that it would be nice to get Ivrii-type results for such operators. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

5 Introduction Further, it turned out that there were other colleagues in the same field Mateusz Kwaśnicki from Wroc law University of Technology (and not only). Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

6 Introduction Further, it turned out that there were other colleagues in the same field Mateusz Kwaśnicki from Wroc law University of Technology (and not only). Finally, the colleague whom I knew (since his field was much closer to mine) Rupert Frank from Caltech and from whose paper with Leander Geisinger I was able to learn what exactly is this operator defined in the way I could work with. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

7 Introduction Like Smaug sensing Bilbo Baggins I opened my eye Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

8 Introduction Like Smaug sensing Bilbo Baggins I opened my eye Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

9 Introduction Like Smaug sensing Bilbo Baggins I opened my eye and I liked what I saw. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

10 Introduction Like Smaug sensing Bilbo Baggins I opened my eye and I liked what I saw. Yeah, I can do it! Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

11 Introduction Like Smaug sensing Bilbo Baggins I opened my eye and I liked what I saw. Yeah, I can do it! Actually, if I had known that people are interested in this I could do it long ago, may be 30 years ago... because all the necessary technique was already there. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

12 Introduction Like Smaug sensing Bilbo Baggins I opened my eye and I liked what I saw. Yeah, I can do it! Actually, if I had known that people are interested in this I could do it long ago, may be 30 years ago... because all the necessary technique was already there. But I have learned about this interest only now and rather occasionally. And probably the interest to such operators did not exist then. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

13 Introduction Like Smaug sensing Bilbo Baggins I opened my eye and I liked what I saw. Yeah, I can do it! Actually, if I had known that people are interested in this I could do it long ago, may be 30 years ago... because all the necessary technique was already there. But I have learned about this interest only now and rather occasionally. And probably the interest to such operators did not exist then. Here we go... Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

14 Operator Operator Let X R d, d 1 (but d = 1 has beed explored and there is no two term asymptotics here anyway) be a compact domain with the smooth boundary X C and let (g jk (x)) be a smooth non-degenerate Riemannian metrics defined in R d (actually how exactly we extend it from X does not matter). Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

15 Operator Operator Let X R d, d 1 (but d = 1 has beed explored and there is no two term asymptotics here anyway) be a compact domain with the smooth boundary X C and let (g jk (x)) be a smooth non-degenerate Riemannian metrics defined in R d (actually how exactly we extend it from X does not matter). Consider pseudo-differential operator Δ m/2 which is Weyl quantization of g(x, ξ) m/2, with m > 0 (so, it is self-adjoint) and consider the corresponding operator in X with the Dirichlet boundary condition: Λ m,x : H m/2 0 (X ) H m/2 Λ m,x u := ( Δ m u ) X (1) where H m/2 0 (X ) := {u H m/2 (R d ) : supp(u) X }, X is a closure of X and v X is a restriction of v to X. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

16 Operator Obviously, Λ m,x is a bounded operator from H m/2 0 (X ) to H m/2 := (H m/2 0 ) * but it also is a self-adjoint unbounded operator in L 2 (X ) with an appropriate domain D(Λ m,x ) := {u H m/2 0 (X ) : Λ m,x L 2 (X )} (2) Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

17 Operator Obviously, Λ m,x is a bounded operator from H m/2 0 (X ) to H m/2 := (H m/2 0 ) * but it also is a self-adjoint unbounded operator in L 2 (X ) with an appropriate domain D(Λ m,x ) := {u H m/2 0 (X ) : Λ m,x L 2 (X )} (2) and it differs from the operator Δ m/2 X which is a power of Δ X (a positive Laplacian with the Dirichlet boundary condition) and while difference is also of order m it is concentrated near X (more about this later) and contributes only to the second term in the asymptotics. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

18 Operator Remark As Δ m does not possess transmission property (unless m 2Z and this case is completely covered by an existing theory) neither Λ m,x nor Δ m X belong to L. Boutet De Monvel s algebra (and u C ( X ) D(Λ m,x u) does not imply Λ m,x u C ( X ) etc but it does not really matter for our analysis). Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

19 Operator Remark As Δ m does not possess transmission property (unless m 2Z and this case is completely covered by an existing theory) neither Λ m,x nor Δ m X belong to L. Boutet De Monvel s algebra (and u C ( X ) D(Λ m,x u) does not imply Λ m,x u C ( X ) etc but it does not really matter for our analysis). Still it possesses μ-trasmission property, see G. Grubb [G1, G2, G3]; Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

20 Operator Remark As Δ m does not possess transmission property (unless m 2Z and this case is completely covered by an existing theory) neither Λ m,x nor Δ m X belong to L. Boutet De Monvel s algebra (and u C ( X ) D(Λ m,x u) does not imply Λ m,x u C ( X ) etc but it does not really matter for our analysis). Still it possesses μ-trasmission property, see G. Grubb [G1, G2, G3]; however the calculus of such operators is not as developed as one of L. Boutet De Monvel s [BdM]. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

21 Operator Remark As Δ m does not possess transmission property (unless m 2Z and this case is completely covered by an existing theory) neither Λ m,x nor Δ m X belong to L. Boutet De Monvel s algebra (and u C ( X ) D(Λ m,x u) does not imply Λ m,x u C ( X ) etc but it does not really matter for our analysis). Still it possesses μ-trasmission property, see G. Grubb [G1, G2, G3]; however the calculus of such operators is not as developed as one of L. Boutet De Monvel s [BdM]. Conjecture Λ m,x Δ m/2 X is an integral operator with the Schwartz kernel K(x, y) which in the local coordinates such that X coincides with {x : x 1 > 0} and {y : y 1 > 0} satisfies D α x D β y K(x, y) C αβ x m 2 α 1 1 y m 2 β 1 1 (x 1 + y 1 + x y ) d α β. (3) Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

22 Results Main theorem Results As X is bounded the spectrum of Λ m,x is discrete tending to + and let N(λ) = N m,x (λ) be an eigenvalue counting function which is the number of eigenvalues of Λ m,x (counting multiplicities) lesser than λ. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

23 Results Main theorem Theorem 1 1 The following asymptotics holds: N(λ) = κ 0 λ d/m + O(λ (d 1)/m ) as λ +. (4) Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

24 Results Main theorem Theorem 1 1 The following asymptotics holds: N(λ) = κ 0 λ d/m + O(λ (d 1)/m ) as λ +. (4) 2 If d 2 and the measure of periodic geodesic billiards is zero then two-term asymptotics holds: N(λ) = κ 0 λ d/m + κ 1 λ (d 1)/m + o(λ (d 1)/m ) as λ + (5) Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

25 Results Main theorem Theorem 1 1 The following asymptotics holds: N(λ) = κ 0 λ d/m + O(λ (d 1)/m ) as λ +. (4) 2 If d 2 and the measure of periodic geodesic billiards is zero then two-term asymptotics holds: Here N(λ) = κ 0 λ d/m + κ 1 λ (d 1)/m + o(λ (d 1)/m ) as λ + (5) κ 0 = (2π) d ω d vol(x ) (6) is the standard coefficient, ω k is a volume of the unit ball in R k, vol(x ) is a Riemannian volume of X and coefficient κ 1 will be given by (8) (9). Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

26 Results 1D operator and κ 1 1D operator and κ 1 Let us consider 1-dimensional toy-model: X = R + is a half-line with the Euclidean metrics. Consider operator a m = ((D 2 x + 1) m/2 ) D (7) and denote by e m (x 1, y 1, λ) the Schwartz kernel of its spectral projector. Proposition 2 a m has absolutely continuos spectrum. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

27 Results 1D operator and κ 1 We can express κ 1 in terms of e m (x, y, λ): where κ 1,m = (2π) 1 d ω d 1 κ m vol d 1 ( X ) (8) κ m = = d 1 m 1 λ (d 1)/m 1( e m (x 1, x 1, λ) π 1 (λ 1) 1/m) dx 1 dλ (9) where π 1 (λ 1) 1/m is a Weyl approximation to e m,1 (x 1, x 1, λ). Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

28 Proof sketched Proof sketched Let us apply Schrödinger equation approach. We need to calculate the number N h (τ) of the eigenvalues not exceeding τ (as τ = 1) of the Schrödinger operator H = h m Λ m,x 1, h = λ 1/m. (10) Let e h (x, y, τ) be the Schwartz kernel of the spectral projector of H and u h (x, y, t) be the Schwartz kernel of the propagator e ih 1 th. Then Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

29 Proof sketched Proof sketched Let us apply Schrödinger equation approach. We need to calculate the number N h (τ) of the eigenvalues not exceeding τ (as τ = 1) of the Schrödinger operator H = h m Λ m,x 1, h = λ 1/m. (10) Let e h (x, y, τ) be the Schwartz kernel of the spectral projector of H and u h (x, y, t) be the Schwartz kernel of the propagator e ih 1tH. Then u(x, y, t) = and also e ih 1 tτ d τ e h (x, y, τ) (11) ( ih t + H)u = 0, (12) u t=0 = δ(x y) (13) where by default all operators are acting with respect to x. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

30 Proof sketched Simple rescaling method Simple rescaling method One can prove easily that e(x, y, τ) ch d as τ c. (14) Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

31 Proof sketched Simple rescaling method Simple rescaling method One can prove easily that e(x, y, τ) ch d as τ c. (14) On the other hand, as γ = γ(x) := 1 2 dist(x, X ) h the simple rescaling x x/γ, h h/γ of the inner ball B(x, γ) brings estimate e(x, x, τ) = κ(x)τ d/m h d + O(γ 1 h 1 d ) as τ 1. (15) with the standard coefficient κ(x). Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

32 Proof sketched Simple rescaling method Simple rescaling method One can prove easily that e(x, y, τ) ch d as τ c. (14) On the other hand, as γ = γ(x) := 1 2 dist(x, X ) h the simple rescaling x x/γ, h h/γ of the inner ball B(x, γ) brings estimate e(x, x, τ) = κ(x)τ d/m h d + O(γ 1 h 1 d ) as τ 1. (15) with the standard coefficient κ(x). Summation over {x : γ(x) h} of (15) and over {x : γ(x) h} of (14) brings N (1) = e(x, x, τ) dx = κ 0 h d + O(h 1 d log h) (16) which is not good! Remember, h = λ 1/m. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

33 Proof sketched Seeley s method Seeley s method The trouble is a factor γ 1 in the remainder estimate (15). However we can apply R. Seeley s method ([RS], 1978) simplified as we are interested only in asymptotics of e(x, x, τ) integrated with respect to x. Instead of (15) we can use ( e(x, x, τ) κ(x)τ d/m h d) ψ(x) dx = O(γ δ+d 1 h 1 d + γ d 2 h 2 d ) (17) as τ 1 and ψ C0 (B(y, γ)) is γ-admissible where B(y, γ) is a ball of radius γ with a center at y, γ = γ(y) and τ 1. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

34 Proof sketched Seeley s method Seeley s method The trouble is a factor γ 1 in the remainder estimate (15). However we can apply R. Seeley s method ([RS], 1978) simplified as we are interested only in asymptotics of e(x, x, τ) integrated with respect to x. Instead of (15) we can use ( e(x, x, τ) κ(x)τ d/m h d) ψ(x) dx = O(γ δ+d 1 h 1 d + γ d 2 h 2 d ) (17) as τ 1 and ψ C0 (B(y, γ)) is γ-admissible where B(y, γ) is a ball of radius γ with a center at y, γ = γ(y) and τ 1. To prove (17) we need to remember that γ 1 appears in the remainder estimate only because T γ is a time for which we can trace propagation of singularities for (12) and it should be T 1. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

35 Proof sketched Seeley s method Seeley s method The trouble is a factor γ 1 in the remainder estimate (15). However we can apply R. Seeley s method ([RS], 1978) simplified as we are interested only in asymptotics of e(x, x, τ) integrated with respect to x. Instead of (15) we can use ( e(x, x, τ) κ(x)τ d/m h d) ψ(x) dx = O(γ δ+d 1 h 1 d + γ d 2 h 2 d ) (17) as τ 1 and ψ C0 (B(y, γ)) is γ-admissible where B(y, γ) is a ball of radius γ with a center at y, γ = γ(y) and τ 1. To prove (17) we need to remember that γ 1 appears in the remainder estimate only because T γ is a time for which we can trace propagation of singularities for (12) and it should be T 1. And because there is mollify integration by x we can choose direction as we wish and we chose away from the boundary and can take as Seeley did T γ 1/2. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

36 Proof sketched Seeley s method Figure: Green rays are outgoing, red are incoming and blue are almost parallel to the boundary. On this picture we do not need to consider red rays! Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

37 Proof sketched Seeley s method Summation over {x : γ(x) h} of (16) and over {x : γ(x) h} of (14) brings remainder estimate O(h 1 d ) i.e. O(λ (d 1)/2m ) i.e. the first statement of Theorem 1. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

38 Proof sketched Seeley s method Summation over {x : γ(x) h} of (16) and over {x : γ(x) h} of (14) brings remainder estimate O(h 1 d ) i.e. O(λ (d 1)/2m ) i.e. the first statement of Theorem 1. But it is not our purpose as we want Ivrii-type results Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

39 Proof sketched Seeley s method Summation over {x : γ(x) h} of (16) and over {x : γ(x) h} of (14) brings remainder estimate O(h 1 d ) i.e. O(λ (d 1)/2m ) i.e. the first statement of Theorem 1. But it is not our purpose as we want Ivrii-type results and the remainder estimate O(λ (d 1)/m ) one could probably get by variational methods from the same results with m 2Z. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

40 Proof sketched Discussion For Ivrii-type results we need Ivrii-type methods. 35+ y.a. (before original Weyl conjecture, i.e. for m = 1) was proven the stumbling block were rays (almost) tangent to the boundary and R. Seley and V. Ivrii found different ways to overcome it (m Z + was an easy generalization). D. Vassiliev had shown a bit later that Seeley s method allows to derive the second term as well. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

41 Proof sketched Discussion For Ivrii-type results we need Ivrii-type methods. 35+ y.a. (before original Weyl conjecture, i.e. for m = 1) was proven the stumbling block were rays (almost) tangent to the boundary and R. Seley and V. Ivrii found different ways to overcome it (m Z + was an easy generalization). D. Vassiliev had shown a bit later that Seeley s method allows to derive the second term as well. Now those rays could be dealt in the same way but a new stumbling block are rays (almost) normal to the boundary because operator now does not have transmission property and even if v C ( X ) C m 0 ( X ) we have Λ m,x v is not smooth exactly on X in the normal direction. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

42 Proof sketched. II Propagation near the boundary Propagation near the boundary Notations Consider small vicinity of the boundary X ; we introduce coordinates x = (x 1 ; x ) = (x 1 ; x 2,..., x d ) in which x 1 = dist(x, X ) and g 1j = δ 1j for all j. Let D j = i j. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

43 Proof sketched. II Propagation near the boundary Propagation near the boundary Notations Consider small vicinity of the boundary X ; we introduce coordinates x = (x 1 ; x ) = (x 1 ; x 2,..., x d ) in which x 1 = dist(x, X ) and g 1j = δ 1j for all j. Let D j = i j. Then using positive commutator method (with L = L(x, hd )) we prove that if Q = Q(x, hd ) has a symbol supported in {(x, ξ ) : ξ ε} then singularities of φ(hd t )χ T (t)q(x, hd )ψ 0 (y)u h are supported in { x y T } as T [h 1 δ, ε 0 ]. Notations Here and below χ C 0 ([ 1, 1 2 ] [ 1 2, 1]), χ T (t) = χ(t/t ) and φ C (R), supp(φ) {τ c}, ψ 0 C (X ), supp(ψ 0 ) {x 1 ε 0 }, ε j are small constants and δ > 0 is an arbitrarily small exponent. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

44 Proof sketched. II Propagation near the boundary Then F t h 1 τ Γ x ( χt (t)q(x, hd )ψ 0 (y)u h ) = O(h s ) as τ c (18) where here and below Γ is an operator of the restriction to the diagonal {x = y}. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

45 Proof sketched. II Propagation near the boundary Then F t h 1 τ Γ x ( χt (t)q(x, hd )ψ 0 (y)u h ) = O(h s ) as τ c (18) where here and below Γ is an operator of the restriction to the diagonal {x = y}.this was a part of my approach of 1979 (reformulated several years later). Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

46 Proof sketched. II Propagation near the boundary. II Propagation near the boundary Next, using positive commutator method (with L = x 1 hd 1 ih/2)) we prove that for 1-dimensional operator at energy level 1 singularities cannot stall near x 1, that they are really reflecting from the boundary. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

47 Proof sketched. II Propagation near the boundary. II Propagation near the boundary Next, using positive commutator method (with L = x 1 hd 1 ih/2)) we prove that for 1-dimensional operator at energy level 1 singularities cannot stall near x 1, that they are really reflecting from the boundary. Then it is also true for d-dimensional case, Euclidean metrics, X = {x : x 1 0} after cut-off by Q = Q(x; hd ) with the symbol supported in { ξ (1 ε)} τ 1/m, τ 1 is an energy level. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

48 Proof sketched. II Propagation near the boundary. II Propagation near the boundary Next, using positive commutator method (with L = x 1 hd 1 ih/2)) we prove that for 1-dimensional operator at energy level 1 singularities cannot stall near x 1, that they are really reflecting from the boundary. Then it is also true for d-dimensional case, Euclidean metrics, X = {x : x 1 0} after cut-off by Q = Q(x; hd ) with the symbol supported in { ξ (1 ε)} τ 1/m, τ 1 is an energy level. Then using successive approximation method and propagation of singularities as x 1 h 1 δ we prove that this is also true in the general case as ξ is calculated according to metrics (g jk ). Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

49 Proof sketched. II Propagation near the boundary. II Then in this framework F t h 1 τ Γ ( χ T (t)q(x, hd )ψ 0 (y)u h ) = O(h s ) as τ 1 (19) where Γv = v(x, x, t) dx. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

50 Proof sketched. II Propagation near the boundary. II Then in this framework F t h 1 τ Γ ( χ T (t)q(x, hd )ψ 0 (y)u h ) = O(h s ) as τ 1 (19) where Γv = v(x, x, t) dx. How does it differ from (18)? Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

51 Proof sketched. II Propagation near the boundary. II Then in this framework F t h 1 τ Γ ( χ T (t)q(x, hd )ψ 0 (y)u h ) = O(h s ) as τ 1 (19) where Γv = v(x, x, t) dx. How does it differ from (18)? 1 First, now Q has a symbol supported in { ξ (1 ε) τ 1/m } rather than in { ξ ε τ 1/m }, Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

52 Proof sketched. II Propagation near the boundary. II Then in this framework F t h 1 τ Γ ( χ T (t)q(x, hd )ψ 0 (y)u h ) = O(h s ) as τ 1 (19) where Γv = v(x, x, t) dx. How does it differ from (18)? 1 First, now Q has a symbol supported in { ξ (1 ε) τ 1/m } rather than in { ξ ε τ 1/m }, 2 Second, now we need to apply Γ rather than Γ x. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

53 Proof sketched. II Propagation near the boundary. II Then in this framework F t h 1 τ Γ ( χ T (t)q(x, hd )ψ 0 (y)u h ) = O(h s ) as τ 1 (19) where Γv = v(x, x, t) dx. How does it differ from (18)? 1 First, now Q has a symbol supported in { ξ (1 ε) τ 1/m } rather than in { ξ ε τ 1/m }, 2 Second, now we need to apply Γ rather than Γ x. This analysis is new. As m Z + it was well known reflection of the transversal rays from the border. Now we cannot calculate reflected wave; we just know that it bounces from X. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

54 Proof sketched. II Estimates in the boundary zone Estimates in the boundary zone Combining (18) and (19) we conclude that (19) holds for Q = I. Then using Tauberian method we conclude that as supp(ψ 0 ) {x 1 2ε}, ψ 0 1 ( e(x, x, τ) e T (x, x, τ) ) ψ 0 (x) dx C 0 εh 1 d + o ε (h 1 d ) (20) where e T (x, x, τ) = h 1 τ ( Ft h 1 τ χ T (t)γ x u h ) dτ (21) is a Tauberian approximation; here and below ξ C0 ([ 1, 1]), χ = 1 on [ 1 2, 1 2 ] and we can take T = h1 δ ; C 0 does not depend on ε > 0 which is an arbitrarily small constant. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

55 Proof sketched. II Estimates in the boundary zone Next using successive approximation method we can prove ( e T (x, x, τ) τ d h d κ 0 (x) ) ψ 0 (x) dx τ d 1 h 1 d κ 1 (x )ψ 0 (x) dx where κ 1 (x ) = 0 X C 0 εh 1 d + o ε (h 1 d ) (22) e m,d (x, x 1 ; x, x 1 ; 1) dx 1 (23) and e m,d (x, x, τ) is a Schwartz kernel of the spectral projector of d-dimensional operator in the half-space {x : x 1 > 0}; as ψ 0 = 1 on X and τ = 1 the selected term coincides with κ 1 h 1 d defined by (8) (9). Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

56 Proof sketched. II Estimates in the boundary zone Combining (20) and (22) (23) we arrive to Proposition 3 Estimate (22) holds for e(x, x, τ) as well (not just for its Tauberian approximation). Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

57 Proof sketched. II Contribution of the inner zone Contribution of the inner zone Now we need to estimate ( e(x, x, τ) h d κ 0 (x) ) ψ 1 (x) dx 1 (24) where ψ 1 C 0 (X ) is supported in {x : x 1 ε/2}. Proposition 4 If the measure of periodic geodesic billiards is zero then expression (24) is o(h 1 d ). Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

58 Proof sketched. II Contribution of the inner zone Proof. The proof is standard (as in the original Weyl conjecture) as long as we know that singularities propagate along geodesic billiards transversal to the boundary. We do not need to consider now generalized billiards tangent to the boundary at some points. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

59 Proof sketched. II Contribution of the inner zone Proof. The proof is standard (as in the original Weyl conjecture) as long as we know that singularities propagate along geodesic billiards transversal to the boundary. We do not need to consider now generalized billiards tangent to the boundary at some points. But inside of X propagation goes along geodesics and near the boundary for transversal trajectories singularities bounced from the boundary as we discussed. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

60 Proof sketched. II Contribution of the inner zone Proof. The proof is standard (as in the original Weyl conjecture) as long as we know that singularities propagate along geodesic billiards transversal to the boundary. We do not need to consider now generalized billiards tangent to the boundary at some points. But inside of X propagation goes along geodesics and near the boundary for transversal trajectories singularities bounced from the boundary as we discussed. Proof of Theorem 1. Theorem 1 is due to Propositions 3 and 4. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

61 Further discussion Further discussion Remark 1 I do not have more explicit formula for κ 1 than (8) (9). I believe that κ m is a monotone decreasing function of m. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

62 Further discussion Further discussion Remark 1 I do not have more explicit formula for κ 1 than (8) (9). I believe that κ m is a monotone decreasing function of m. 2 It looks easy to generalize our results to domains described in [MonsterBook], Section 11.2 and satisfying dist(x, y) C 0 x y x, y X (25) (where dist(x, y) is a connected distance between x and y) with vertices, edges, conical points, spikes, thin cusps etc (see Subsection ). Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

63 Further discussion Further discussion Remark 1 I do not have more explicit formula for κ 1 than (8) (9). I believe that κ m is a monotone decreasing function of m. 2 It looks easy to generalize our results to domains described in [MonsterBook], Section 11.2 and satisfying dist(x, y) C 0 x y x, y X (25) (where dist(x, y) is a connected distance between x and y) with vertices, edges, conical points, spikes, thin cusps etc (see Subsection ). 3 Generalizations to domains not satisfying (25) with cuts and inward spikes is to be more challenging. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

64 Further discussion Further discussion Remark 1 I do not have more explicit formula for κ 1 than (8) (9). I believe that κ m is a monotone decreasing function of m. 2 It looks easy to generalize our results to domains described in [MonsterBook], Section 11.2 and satisfying dist(x, y) C 0 x y x, y X (25) (where dist(x, y) is a connected distance between x and y) with vertices, edges, conical points, spikes, thin cusps etc (see Subsection ). 3 Generalizations to domains not satisfying (25) with cuts and inward spikes is to be more challenging. 4 Generalizations to domains described in [MonsterBook], Section 12.1 with thick cusps seems to be a daunting task! Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

65 Further discussion Remark 1 G. Grubb in [G3] derived one-term asymptotics for more general operators. Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

66 Further discussion Remark 1 G. Grubb in [G3] derived one-term asymptotics for more general operators. 2 R. Frank and L. Geisinger in [FG] explored fractional Laplacians from the point of view of functional analysis and derived two-term asymptotics for Tr((τ Λ m,x ) + ) which does not require any conditions of the global geometric nature. Our methods provide τ 1 Tr((τ Λ m,x ) + ) = m d + m κ 0τ d/m m + d + m 1 κ 1τ (d 1)/m + O(τ (d 2)/m ) (26) Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

67 Further discussion Remark 1 G. Grubb in [G3] derived one-term asymptotics for more general operators. 2 R. Frank and L. Geisinger in [FG] explored fractional Laplacians from the point of view of functional analysis and derived two-term asymptotics for Tr((τ Λ m,x ) + ) which does not require any conditions of the global geometric nature. Our methods provide τ 1 Tr((τ Λ m,x ) + ) = 3 and under standard non-periodicity condition m d + m κ 0τ d/m m + d + m 1 κ 1τ (d 1)/m + O(τ (d 2)/m ) (26) τ 1 Tr((τ Λ m,x ) + ) = m d + m κ 0τ d/m m + d + m 1 κ 1τ (d 1)/m + m d + m 2 κ 2τ (d 2)/m + o(τ (d 2)/m ). (27) Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

68 References References I Bañuelos R. and Kulczycki T, Trace estimates for stable processes, Probab. Theory Related Fields 142: (2008). Bañuelos R., Kulczycki T. and Siudeja B., On the trace of symmetric stable processes on Lipschitz domains. J. Funct. Anal. 257(10): (2009). R. M. Blumenthal and R. K. Getoor, Sample functions of stochastic processes with stationary independent increments. J. Math. Mech., 10: (1961). L. Boutet de Monvel, Boundary problems for pseudodifferential operators, Acta Math., 126 (1-2):11 51 (1971). R. Frank and L. Geisinger, Refined Semiclassical Asymptotics for Fractional Powers of the Laplace Operator. arxiv (2011/2013). Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

69 References References II G. Grubb, Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators, Analysis and Part. Diff. Equats., 7(71): (2014). G. Grubb, Fractional Laplacians on domains, a development of Hörmander s theory of μ-transmission pseudodifferential operators. Adv. Math., 268: (2015). G. Grubb, Spectral results for mixed problems and fractional elliptic operators, J. Math. Anal. Appl., 421(2): (2015). M. Kwaśnicki., Eigenvalues of the fractional laplace operator in the interval. J. Funct. Anal., 262(5): (2012). R. Seeley, A sharp asymptotic estimate for the eigenvalues of the Laplacian in a domain of R 3. Advances in Math., 102 (3): (1978). Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32

70 References References III V. Ivrii, Second term of the spectral asymptotic expansion for the Laplace-Beltrami operator on manifold with boundary. Funct. Anal. Appl., 14 (2): (1980). V. Ivrii, Microlocal Analysis, Sharp Spectral, Asymptotics and Applications. Chapter 7. Standard Local Semiclassical Spectral Asymptotics near the Boundary. Section 8.5. Fractional Laplacians. Section Large eigenvalues for operators with weakly singular potentials. Section Operators in domains with thick cusps. V. Ivrii. 100 years of Weyl s law, Bulletin of Mathematical Sciences, Springer (2016). Victor Ivrii (Math., Toronto) Fractional Laplacians November 7, / 32