Estimates for the resolvent and spectral gaps for non self-adjoint operators. University Bordeaux

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1 for the resolvent and spectral gaps for non self-adjoint operators 1 / 29 Estimates for the resolvent and spectral gaps for non self-adjoint operators Vesselin Petkov University Bordeaux Mathematics Days in Sofia, July 9, 2014

2 for the resolvent and spectral gaps for non self-adjoint operators 2 / 29 Outline 1. Holomorphic functions without zeros in a strip. 2. Estimates for (I + B(z)) The cut-off resolvent and the scattering operator. 4. Spectral gap for generators of semigroupes. 5. Dynamical zeta function.

3 1. Holomorphic functions without zeros in a strip for the resolvent and spectral gaps for non self-adjoint operators 3 / Holomorphic functions without zeros in a strip We start with the following question. Let f (z) be a holomorphic function in C + = {z C : Im z > 0} such that for some α 0, C > 0, M N we have f (z) C(1 + z ) M e α Im z, z C +. (1) Assume that f (z) 0 in the strip 0 < Im z < 1 and consider z = x + i. What is the 2 optimal estimate for 1 f (x+i/2). Theorem 1 (Borichev, -P.) Let f (z) be a holomorphic function in C + satisfying (1). Assume that f (z) 0 for 0 < Im z < 1. Then log f (x + i/2) lim = 0. (2) x x 2 It is clear that (2) implies for every 0 < ɛ 1 the estimate 1 f (x + i/2) 2 Cɛeɛ x, x R.

4 1. Holomorphic functions without zeros in a strip for the resolvent and spectral gaps for non self-adjoint operators 4 / 29 Considering the function F (z) = f (z)eiαz C(z+i) M, we reduce the proof to the case when f (z) 1 in C +. Then G(z) = log(1/ f (z) ) is a positive harmonic function for 0 < Im z < 1. If for some x > 1 we have G(x + i/2) cx 2, c > 0, then by Harnack inequality we get Thus if G(x + t + i/2) c 1cx 2, 1 4 t 1, c1 > 0. 4 log f (x + i/2) lim inf = c < 0, x x 2 we can find a sequence of points x n R, x n+1 > x n + 1, n 0 so that Z xn+1/4 x n 1/4 1 dy log f (y + i/2) 1 + y c2c > 0 2 and then Z + 1 dy log f (y + i/2) 1 + y = +. 2 This contradicts the standard theorem for functions in H (C + ) and we obtain the result.

5 1. Holomorphic functions without zeros in a strip for the resolvent and spectral gaps for non self-adjoint operators 5 / 29 Remark 1 The assertion of Theorem 1 holds for holomorphic functions f (z) in C + for which we have f (z) 0 for 0 < Im z < 1 and In fact we can consider the function f (x) C(1 + x ) M, x R, f (z) Ce α z, α 0, z C +. g(z) = f (z)eiαz (z + i) M and apply the Phragmén-Lindelöf principle in the first and the the second quadrant of C to conclude that g(z) is bounded in C +.

6 1. Holomorphic functions without zeros in a strip for the resolvent and spectral gaps for non self-adjoint operators 6 / 29 To verify that the result of Theorem 1 is rather sharp, we establish the following Proposition 1 (Borichev, -P.) Let ρ(x) be a positive function such that lim x ρ(x) = 0. Then there exists a holomorphic function B(z) in C + without zeros in the domain {z C : 0 < Im z < 1} such that lim inf x log B(x + i/2) ρ(x)x 2 < 0. This estimate means that there exists α > 0 so that for some sequence x n, x n we have 1 B(x n + i/2) 2 Ceαρ(xn) xn.

7 1. Holomorphic functions without zeros in a strip for the resolvent and spectral gaps for non self-adjoint operators 7 / 29 Proof. We choose two sequences x n, x n 1 and k n N, n 1 so that k n ρ(x n)xn 2, n 1, (3) X k n <. (4) xn 2 n 1 Next we set z n = x n + i, n 1 and consider Blaschke product B(z) = Y z 2 n + 1 zn z zn kn. z z n n 1 The condition (4) guarantees the convergence of the infinite product. On the other hand, using (3) we get (xn + i/2) (xn + i) B(x n + i/2) kn = 3 kn e ρ(xn)x2 n. (x n + i/2) (x n i)

8 1. Holomorphic functions without zeros in a strip for the resolvent and spectral gaps for non self-adjoint operators 8 / 29 If we have a holomorphic function in C + with growth condition f (z) Ce z β, 1 < β < 2, z C +, and if f (z) 0 for 0 < Im z < 1, then we can prove that log f (x + i/2) lim inf >. x x β+1 This result is also sharp but we will not discuss this class of functions in this talk.

9 2. Estimates for (I + B(z)) 1 for the resolvent and spectral gaps for non self-adjoint operators 9 / Estimates for (I + B(z)) 1 Let H be Hilbert space with norm.. We denote also by. the norm of the operators in H and by L the space of bounded linear operators on H. Let B(z) : C + z L(H) be an operator valued holomorphic function in C +. Theorem 2 (Borichev, -P.) Let B(z) be holomorphic in C + and such that for some constants α 0, C > 0, M 0 we have B(z) C(1 + z ) M e α Im z, z C +. Assume that (I + B(z)) 1 L(H) for 0 < Im z < 1 and let Image B(z) V, V being a finite dimensional space of H independent of z C +. Then for every ɛ > 0 we have (I + B(x + i/2)) 1 C ɛe ɛ x 2. In particular, we cover the cases of matrix-valued operators. The argument is based on an estimate of the determinant

10 2. Estimates for (I + B(z)) 1 for the resolvent and spectral gaps for non self-adjoint operators 10 / 29 Now consider an operator valued holomorphic function A(z) : z C + T 1, where T 1 denotes the space of trace class operators in H with the norm 1. A compact operator B is trace class if P j λ j(b) <, λ j (B) being the eigenvalues of B. Recall that for every B T 1 we can define the determinant det(i + B) = Y j (1 + λ j (B)), and det(i + B) e B 1. Moreover, given B(z) T 1, we may consider the function F B (z) = ˆdet(I + B(z)) (I + B(z)) 1 which extends to an entire operator valued function in C such that F B (z) e B(z) 1 z, µ C. If B(z) 1 C z, an estimate for (det(i + B(z))) 1 implies an estimate for the resolvent (I + B(z)) 1, but for infinite dimensional spaces H this estimate in general is not optimal. It is an open problem to extend the above results for trace class operators B(z) holomorphic in C +.

11 3. The cut-off resolvent and the scattering operator for the resolvent and spectral gaps for non self-adjoint operators 11 / The cut-off resolvent and the scattering operator Consider a bounded connected domain K R n, n 2, with smooth boundary K. Set Ω = R n \ K and consider the Dirichlet problem 8 >< ( t 2 )u = 0 in R t Ω, u = 0 on R t K, (5) >: u(0, x) = f 0(x), u t(0, x) = f 1(x). Let D be the Dirichlet Laplacian in Ω which is a self-adjoint operator on H = L 2 (Ω) with domain D = H0 1 (Ω) H 2 (Ω). For Im λ > 0 the resolvent ( D λ 2 ) 1 is a bounded operator from H to D and, for all ϕ C0 (Ω), the cut-off resolvent R D (λ) = ϕ( D λ 2 ) 1 ϕ admits a meromorphic continuation in C for n odd and in C \ ir for n even with poles in C = {λ C : Im λ < 0} The poles of this continuation are independent on the choice of ϕ and they are called scattering resonances.

12 3. The cut-off resolvent and the scattering operator for the resolvent and spectral gaps for non self-adjoint operators 12 / 29 The estimates of the cut-off resolvent in the poles free regions play a crucial role for the decay of local energy and many problems in mathematical physics. In several important examples there exists a strip U δ = {λ C : δ Im λ 0, Re λ C 0 > 0}, δ > 0, where R D (λ) has no poles. In this case U δ is called spectral gap. This is the case for non-trapping obstacles, where R D (λ) is holomorphic in the region {λ C : Im λ C(log Re λ + 1), Re λ > C 0}. Moreover, in this region for non trapping obstacles we have a bound R D (λ) C(1 + λ ) M. The case of trapping obstacles is much more difficult since we can have different type trapping rays and the geometry of the obstacle can be rather complicated. Problem: Find an estimate of R D (λ) in a resonance free region U δ for trapping obstacles. There are many works studying obstacles with special geometry (example: several strictly convex disjoint obstacles (Figure 2)).

13 3. The cut-off resolvent and the scattering operator for the resolvent and spectral gaps for non self-adjoint operators 13 / 29 K x + ω ν (x + ) θ Figure: 1.Convex obstacle

14 3. The cut-off resolvent and the scattering operator for the resolvent and spectral gaps for non self-adjoint operators 14 / 29 K 1 K 2 K 3 Figure: 2. Trapping ray for three convex obstacles

15 3. The cut-off resolvent and the scattering operator (x, y) K Figure: 3. Trapping obstacle for the resolvent and spectral gaps for non self-adjoint operators 15 / 29

16 3. The cut-off resolvent and the scattering operator for the resolvent and spectral gaps for non self-adjoint operators 16 / 29 Conjecture. If K is a trapping obstacle and if U δ is a resonance free region, then for 0 < δ 1 < δ we have R D (z) L 2 L 2 C δ 1 e α z 2, α 0, z U δ1. (6) It is important to notice that in all known examples of trapping obstacles we have always a polynomial bound for R D (z) in the resonance free region U δ. Theorem 3 (J. F. Bony, -P.) Let K be a trapping obstacle and let U δ be a resonance free region. Then for dimension n = 2 one has the estimate while for n = 3 we have (6). R D (z) L 2 L 2 C δ 1 e α z, α 0, z U δ1, The proof is based on a semiclassical analysis of the operator h 2, 0 < h 1 combined with a fine complex dilatation depending on h and the examination of a Grushin problem. With this approach we can study also perturbations which are not compact. The importance of our method is that the result holds without any condition on the geometry.

17 3. The cut-off resolvent and the scattering operator for the resolvent and spectral gaps for non self-adjoint operators 17 / 29 Now we will discuss how we can reduce the problem of estimate of R D (z) in the spectral gap to that of the estimate of (I + B(z)) 1 with a trace class operator B(z). Consider the scattering operator S(λ) related to (5). S(z) an operator valued function S(z) : L 2 (S n 1 ) L 2 (S n 1 ), λ C, which has the form S(z) = I + B(z) with a trace class operator B(z). The kernel a(z, ω, θ) of K(z) is called scattering amplitude. The functions a(z, ω, θ) and S(z) are holomorphic for Im z 0 and they admit for n odd (resp. even) a meromorphic continuation in C (resp. C \ ir ) with poles z j, Im z j < 0, independent of ω, θ. These poles coincide with the resonances. For Im z 0 we have the estimate a(z, θ, ω) Ce α Im z (1 + z ) M, α 0 uniformly with respect to (ω, θ) S n 1 S n 1 and a similar estimate holds for B(z) L 2 L 2 for z C+.

18 3. The cut-off resolvent and the scattering operator for the resolvent and spectral gaps for non self-adjoint operators 18 / 29 The operator S(z) is unitary for z R and we have the equality S ( z) = S 1 (z), z C (7) if S(z) is invertible. This equality shows that the poles of S(z) in C are conjugated to the points z C + where S(z) is not invertible. If we have a spectral gap U δ, then S( z) as well as S ( z) will be holomorphic for z U δ and S(z) will be invertible for 0 < Im z δ. Moreover, an estimate of S 1 (z) implies an estimate for S( z) and by a result of Lax-Phillips we can obtain an estimate for R D ( z). However, we have not an optimal estimate for (I + B(z)) 1 and the problem mentioned in Section 2 remains open.

19 4. Spectral gap for generators of semigroups for the resolvent and spectral gaps for non self-adjoint operators 19 / Spectral gap for generators of semigroups Consider a semigroup of contractions Z(t) = e tb, t 0 in a Hilbert space H. Then the spectrum σ(b) of the generator B is included in the half plane {z C : Re z 0}. σ(z(t)) \ {0} e tσ(b), t 0. There are examples when σ(b) =, while σ(z(t)) is not trivial. Introduce the ground bound 1 ω 0 = lim log Z(t) t t and the spectral abscissa s(b) = sup{re λ : λ σ(b)}. We have a spectral gap if S(B) < ω 0, thus we may have ω 0 = 0 and s(b) < 0. For the stability and the decay of the solutions u(t, x) = e tb f (x), it is important to have an estimate Z(t) Ce αt, α > 0, t 0. (8)

20 4. Spectral gap for generators of semigroups for the resolvent and spectral gaps for non self-adjoint operators 20 / 29 If we have a spectral gap, the behavior of the resolvent (z B) 1 for s(b) < Re z ω 0, Im z plays a role in many arguments. It is natural to conjecture that this resolvent has at most an exponential growth O(e c Im z m ). However, recently, Mark Embree constructed an example of a semigroup Z(t) in the space l 2 such that Z(t) = 1, t 0 and S(B) = 1, while ( 1/2 + ik B) 1 2 n k, k N with an arbitrary increasing sequence n k. This is in contrast with the results for holomorphic functions with values in a finite dimensional space discussed in Section 2. In the construction, the operator B has only eigenvalues in its spectrum but their multiplicities are not bounded. In the situations which occur in the mathematical physics in general the multiplicities are bounded.

21 4. Spectral gap for generators of semigroups for the resolvent and spectral gaps for non self-adjoint operators 21 / 29 For the Dirichlet problem studied in Section 3 Lax and Phillips introduced a semigroup Z(t) = e tb whose behavior is closely realted to the decay of local energy. Moreover the spectrum of the generator B is formed by eigenvalues µ j with finite multiplicity determined as µ j = iλ j, where λ j are the scattering resonances. For non trapping obstacles we have the estimate (8). Next Lax-Phillips conjectured that if we have at least one trapping trajectory, then ω 0 = 0 and there exists t j to so that σ(z(t j )) S 1. This was proved by J. Ralston. In fact, we have a stronger result Theorem 4 (J.F.Bony - P.) Assume that σ(b) = a 0. Then if there exists at least one trapping trajectory in Ω we have {z C : e a z 1} σ(z(t)) for almost all t 0. Moreover, in the case when K is the union of two strictly convex disjoint obstacles we have {z C : z 1} = σ(z(t)) for almost all t 0. Ther second statement is surprising. We have only one periodic ray and we have a spectral gap. Nevertheless the spectrum of the semigroup Z(t) coincides with the unit disk. This means that the continuous spectrum of Z(t) is a very large set and the eigenvalues e tµ j, Re µ j < 0 of Z(t) are embedded in the unit disk.

22 5. Dynamical zeta function for the resolvent and spectral gaps for non self-adjoint operators 22 / Dynamical zeta function Let Ω R 3 be an open connected domain with C boundary and let We assume that K = R 3 \ Ω {x R 3 : x ρ 0}. K = Q j=1k j, K i K j =, i j, where K j, j = 1,..., Q, Q 3, are strictly convex closed obstacles satisfying the following condition introduced by M.Ikawa: (H) For every 1 i, j Q, i j, the convex hull of K i K j has no common points with K l, l i, l j. Given a periodic reflecting ray γ Ω with m γ reflections, denote by d γ the period (return time) of γ, by T γ the primitive period (length) of γ and by P γ the linear Poincaré map associated to γ. Let λ i,γ, i = 1, 2 be the eigenvalues of P γ with λ i,γ > 1.

23 5. Dynamical zeta function for the resolvent and spectral gaps for non self-adjoint operators 23 / 29 Let P be the set of primitive periodic rays. Set δ γ = 1 2 ln(λ1,γλ2,γ), rγ = ( 0 if m γ is even, 1 if m γ is odd, γ P. Introduce the zeta functions Z(s) = X m=1 ζ(s) = exp = Y γ P 1 m X ( 1) mrγ e m( stγ +δγ ), γ P X X m=1 γ P 1 m ( 1)mrγ m( stγ +δγ e ) 1 ( 1) rγ e stγ +δγ 1. There exists s 0 R such that for Re s > s 0 the series Z(s) is absolutely convergent and Z(s) is memorphic for Re s > s 0 a, a > 0. Following a recent result of L. Stoyanov for Q 3 under some condition which can be expressed by the curvatures and the distance between the obstacles there exists 0 < ɛ < a so that the dynamical zeta function Z(s) admits an analytic continuation for Re s s 0 ɛ.

24 5. Dynamical zeta function for the resolvent and spectral gaps for non self-adjoint operators 24 / 29 There are some similarities with the classical Riemann zeta function X 1 ζ(s) = n, Re s > 1 s and the inverse where µ(n) is the Mobius function It is well known that n=1 Z(s) = 1 ζ(s) = X n µ(n) n s, Re s > 1, µ(1) = 1, µ(n) = ( 1) k, k = p 1p 2...p k, p i p j, µ(n) = 0, n = p κ pκ k, max κ j 2, p i primes. log ζ(s) = X m X p k 1 mp = X p(ms) mp m. p(s) = X 1 p s m p prime

25 5. Dynamical zeta function for the resolvent and spectral gaps for non self-adjoint operators 25 / 29 b o s 2 s o Figure: 4. Line of absolute convergence and line of holomorphy

26 5. Dynamical zeta function for the resolvent and spectral gaps for non self-adjoint operators 26 / 29 In the physical literature and in several numerical analysis results the following conjecture is announced. Conjecture: The poles µ j, Re µ j < 0 of Z(s) determine the poles ( iµ j ) of R χ(z). More precisely, the poles z j of R χ(z) lie in a sufficiently small neighborhoods of iµ j and iµ j are called pseudo-poles of R χ(z). In the case of two strictly convex disjoint obstacles this conjecture is true (Ikawa, C. Gérard) and the poles are in a small neighborhoods of m π + iα d k, m Z, k N, where d > 0 is the distance between the obstacles and α k > 0 are determined by the eigenvalues of the Poincaré map related to the unique primitive periodic ray. Theorem 5 (Ikawa) Assume that Z(s) is absolutely convergent for Re s > s 0, s 0 < 0. Then the cut-off resolvent R D (z) is analytic for Im z > s 0, Re z C 0. For some time the physicists conjectured that the optimal strip {z C : δ Im z 0} where R D (z) is analytic is given by δ = s 0 by analogy with the case of two strictly convex obstacles (in this simple case Q = 2 the lines Re s = s 0 and Re s = s 2 coincide and we have a Dirichlet series with positive terms.)

27 5. Dynamical zeta function for the resolvent and spectral gaps for non self-adjoint operators 27 / 29 In fact the above conjecture is false if we have 3 and more strictly convex obstacles. In the later case the dynamical zeta function admits an analytic continuation for Re s > s 0 ɛ. The behavior of the cut-off resolvent for complex z is quite complicated and R D (z) cannot be expressed by Z( iz). Moreover, since we have perturbations with variable curvature there is no analog of the celebrate Selberg trace formula. Nevertheless, by a sophistical proof we have the following Theorem 6 (-P. Stoyanov) Let Q 3. Assume that there exists s 1 < s 0 < 0 such that Z(s) is analytic for Re s > s 1. Then for 0 < ɛ 1 the cut-off resolvent R D (z) is analytic for Im z > s 1 + ɛ, Re s C ɛ.

28 5. Dynamical zeta function for the resolvent and spectral gaps for non self-adjoint operators 28 / 29 The idea is to construct an approximation of the cut-off resolvent R D ( is) by a series involving complex phases and amplitudes. The leading term of this approximation is given by a series W (x, is) = X X e isϕj,n(x) a j,n (x, is) n=1 j. This series is neither absolutely convergent nor convergent for s 1 < Re s < s 0. To examine its analytical continuation, we must exploit a more deep and complete information related to the behavior of the stable and unstable manifolds in a neighborhood of the non-wandering set of the billiard flow. The non wandering set of the billiard flow has Lebesgue measure zero but its Hausdorff dimension is always different from 0 and appropriate analysis of the joint non integrability of the stable and unstable foliations plays a crucial role in the proof.

29 5. Dynamical zeta function for the resolvent and spectral gaps for non self-adjoint operators 29 / 29 Thank you!