Weyl Asymptotics and Random Perturbations in a One-Dimensional Semi-classical Case

We consider a simple model operator P in dimension 1 and show how random perturbations give rise to Weyl asymptotics in the interior of the range of P. We follow rather closely the work of Hager (Ann Henri Poincare 7(6):1035–1064, 2006) with some input also from Bordeaux Montrieux (Loi de Weyl presque sureet resolvante pour des operateurs differentiels nonautoadjoints, these, CMLS, Ecole Polytechnique, 2008) and Hager–Sjostrand (Math Ann 342(1):177–243, 2008). Some of the general ideas appear perhaps more clearly in this special situation.

The second Weyl coefficient for a first-order system

For a scalar elliptic self-adjoint operator on a compact manifold without boundary we have two-term asymptotics for the number of eigenvalues between 0 and λ when λ → ∞, under an additional dynamical condition. (See [3, Theorem 3.5] for an early result in this direction.) In the case of an elliptic system of first order, the existence of two-term asymptotics was also established quite early and as in the scalar case Fourier integral operators have been the crucial tool. The complete computation of the coefficient of the second term was obtained only in the 2013 paper [2]. In the present paper we simplify that calculation. The main observation is that with the existence of two-term asymptoti…

Interior Eigenvalue Density of Jordan Matrices with Random Perturbations

International audience; We study the eigenvalue distribution of a large Jordan block subject to a small random Gaussian perturbation. A result by E. B. Davies and M. Hager shows that as the dimension of the matrix gets large, with probability close to 1, most of the eigenvalues are close to a circle.We study the expected eigenvalue density of the perturbed Jordan block in the interior of that circle and give a precise asymptotic description.; Nous étudions la distribution de valeurs propres d’un grand bloc de Jordan soumis à une petite perturbation gaussienne aléatoire. Un résultat de E. B. Davies et M. Hager montre que quand la dimension de la matrice devient grande, alors avec probabilité…

Fractal Weyl law for open quantum chaotic maps

We study the semiclassical quantization of Poincar\'e maps arising in scattering problems with fractal hyperbolic trapped sets. The main application is the proof of a fractal Weyl upper bound for the number of resonances/scattering poles in small domains near the real axis. This result encompasses the case of several convex (hard) obstacles satisfying a no-eclipse condition.

Supersymmetric structures for second order differential operators

Necessary and sufficient conditions are obtained for a real semiclassical partial differential operator of order two to possess a supersymmetric structure. For the operator coming from a chain of oscillators, coupled to two heat baths, we show the non-existence of a smooth supersymmetric structure, for a suitable interaction potential, provided that the temperatures of the baths are different.

Spectra for Semiclassical Operators with Periodic Bicharacteristics in Dimension Two

We study the distribution of eigenvalues for selfadjoint $h$--pseudodifferential operators in dimension two, arising as perturbations of selfadjoint operators with a periodic classical flow. When the strength $\varepsilon$ of the perturbation is $\ll h$, the spectrum displays a cluster structure, and assuming that $\varepsilon \gg h^2$ (or sometimes $\gg h^{N_0}$, for $N_0 >1$ large), we obtain a complete asymptotic description of the individual eigenvalues inside subclusters, corresponding to the regular values of the leading symbol of the perturbation, averaged along the flow.

PT-symmetry and Schrödinger operators. The double well case

We study a class of $PT$-symmetric semiclassical Schrodinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double-well potential. In the simple well case, two of the authors have proved in [6] that, when the potential is analytic, the eigenvalues stay real for a perturbation of size $O(1)$. We show here, in the double-well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB-analysis, leading to a fairly explicit quantization condition.

Quasi-Modes and Spectral Instability in One Dimension

In this section we describe the general WKB construction of approximate “asymptotic” solutions to the ordinary differential equation $$\displaystyle P(x,hD_x)u=\sum _{k=0}^m b_k(x)(hD_x)^ku=0, $$ on an interval α < x < β, where we assume that the coefficients bk ∈ C∞(]α, β[). Here h ∈ ]0, h0] is a small parameter and we wish to solve (above equation) up to any power of h. We look for u in the form $$\displaystyle u(x;h)=a(x;h)e^{i\phi (x)/h}, $$ where ϕ ∈ C∞(]α, β[) is independent of h. The exponential factor describes the oscillations of u, and when ϕ is complex valued it also describes the exponential growth or decay; a(x;h) is the amplitude and should be of the form $$\displaystyle a(x;h…

Weyl symbols and boundedness of Toeplitz operators

International audience; We study Toeplitz operators on the Bargmann space, with Toeplitz symbols that are exponentials of inhomogeneous quadratic polynomials. It is shown that the boundedness of such operators is implied by the boundedness of the corresponding Weyl symbols.

Resolvent Estimates Near the Boundary of the Range of the Symbol

The purpose of this chapter is to give quite explicit bounds on the resolvent near the boundary of Σ(p) (or more generally, near certain “generic boundary-like” points.) The result is due (up to a small generalization) to Montrieux (Estimation de resolvante et construction de quasimode pres du bord du pseudospectre, 2013) and improves earlier results by Martinet (Sur les proprietes spectrales d’operateurs nonautoadjoints provenant de la mecanique des fluides, 2009) about upper and lower bounds for the norm of the resolvent of the complex Airy operator, which has empty spectrum (Almog, SIAM J Math Anal 40:824–850, 2008). There are more results about upper bounds, and some of them will be rec…

Semiclassical Gevrey operators and magnetic translations

We study semiclassical Gevrey pseudodifferential operators acting on the Bargmann space of entire functions with quadratic exponential weights. Using some ideas of the time frequency analysis, we show that such operators are uniformly bounded on a natural scale of exponentially weighted spaces of holomorphic functions, provided that the Gevrey index is $\geq 2$.

Proof II: Lower Bounds

In this chapter we give a lower bound on \(\ln \det S_{\delta ,z}\) which is valid with high probability, and then using also the upper bounds of Chap. 16, we conclude the proof of Theorem 15.3.1 with the help of Theorem 12.1.2.

Quasi-Modes in Higher Dimension

Recall that if a(x, ξ) and b(x, ξ) are two C1-functions defined on some domain in \({\mathbf {R}}^{2n}_{x,\xi }\), then we can define the Poisson bracket to be the C0-function on the same domain given by $$\displaystyle \{ a,b\} =a^{\prime }_\xi \cdot b^{\prime }_x-a^{\prime }_x \cdot b^{\prime }_\xi =H_a(b). $$ Here \(H_a=a^{\prime }_\xi \cdot \partial _x-a^{\prime }_x\cdot \partial _\xi \) denotes the Hamilton vector field of a. The following result is due to Zworski, who obtained it via a semi-classical reduction from the above mentioned result of Hormander. A direct proof was given in Dencker et al. and here we give a variant. We will assume some familiarity with symplectic geometry.

Weyl Asymptotics for the Damped Wave Equation

The damped wave equation is closely related to non-self-adjoint perturbations of a self-adjoint operator P of the form $$\displaystyle P_\epsilon =P+i\epsilon Q. $$ Here, P is a semi-classical pseudodifferential operator of order 0 on L2(X), where we consider two cases: X = Rn and P has the symbol P ∼ p(x, ξ) + hp1(x, ξ) + ⋯ . in S(m), as in Sect. 6.1, where the description is valid also in the case n > 1. We assume for simplicity that the order function m(x, ξ) tends to + ∞, when (x, ξ) tends to ∞. We also assume that P is formally self-adjoint. Then by elliptic theory (and the ellipticity assumption on P) we know that P is essentially self-adjoint with purely discrete spectrum. X is a com…

PT Symmetry and Weyl Asymptotics

For a class of PT-symmetric operators with small random perturbations, the eigenvalues obey Weyl asymptotics with probability close to 1. Consequently, when the principal symbol is nonreal, there are many nonreal eigenvalues.

Tunnel effect and symmetries for Kramers–Fokker–Planck type operators

AbstractWe study operators of Kramers–Fokker–Planck type in the semiclassical limit, assuming that the exponent of the associated Maxwellian is a Morse function with a finite number n0 of local minima. Under suitable additional assumptions, we show that the first n0 eigenvalues are real and exponentially small, and establish the complete semiclassical asymptotics for these eigenvalues.

Resolvent estimates for elliptic quadratic differential operators

Sharp resolvent bounds for non-selfadjoint semiclassical elliptic quadratic differential operators are established, in the interior of the range of the associated quadratic symbol.

Distribution of Eigenvalues for Semi-classical Elliptic Operators with Small Random Perturbations, Results and Outline

In this chapter we will state a result asserting that for elliptic semi-classical (pseudo-)differential operators the eigenvalues are distributed according to Weyl’s law “most of the time” in a probabilistic sense. The first three sections are devoted to the formulation of the results and in the last section we give an outline of the proof that will be carried out in Chaps. 16 and 17.

Resolvent Estimates for Non-Selfadjoint Operators via Semigroups

We consider a non-selfadjoint h-pseudodifferential operator P in the semiclassical limit (h → 0). If p is the leading symbol, then under suitable assumptions about the behavior of p at infinity, we know that the resolvent (z–P)–1 is uniformly bounded for z in any compact set not intersecting the closure of the range of p. Under a subellipticity condition, we show that the resolvent extends locally inside the range up to a distance \(\mathcal{O}(1)((h\ln \frac{1}{h})^{k/(k + 1)} )\) from certain boundary points, where \(k \in \{ 2,4, \ldots \} \). This is a slight improvement of a result by Dencker, Zworski, and the author, and it was recently obtained by W. Bordeaux Montrieux in a model sit…

Perturbations of Jordan Blocks

In this chapter we shall study the spectrum of a random perturbation of the large Jordan block A0, introduced in Sect. 2.4: $$\displaystyle A_0=\begin {pmatrix}0 &1 &0 &0 &\ldots &0\\ 0 &0 &1 &0 &\ldots &0\\ 0 &0 &0 &1 &\ldots &0\\ . &. &. &. &\ldots &.\\ 0 &0 &0 &0 &\ldots &1\\ 0 &0 &0 &0 &\ldots &0 \end {pmatrix}: {\mathbf {C}}^N\to {\mathbf {C}}^N. $$ Zworski noticed that for every z ∈ D(0, 1), there are associated exponentially accurate quasimodes when N →∞. Hence the open unit disc is a region of spectral instability. We have spectral stability (a good resolvent estimate) in \(\mathbf {C}\setminus \overline {D(0,1)}\), since ∥A0∥ = 1. σ(A0) = {0}.

Proof I: Upper Bounds

In this chapter we study upper bounds on singular values and determinants of certain operators related to Pδ. The bounds are not probabilistic; they only depend on a certain smallness of the perturbation.

Quadratic ${\mathcal P}{\mathcal T}$-symmetric operators with real spectrum and similarity to self-adjoint operators

It is established that a -symmetric elliptic quadratic differential operator with real spectrum is similar to a self-adjoint operator precisely when the associated fundamental matrix has no Jordan blocks.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

Analytic Bergman operators in the semiclassical limit

Transposing the Berezin quantization into the setting of analytic microlocal analysis, we construct approximate semiclassical Bergman projections on weighted $L^2$ spaces with analytic weights, and show that their kernel functions admit an asymptotic expansion in the class of analytic symbols. As a corollary, we obtain new estimates for asymptotic expansions of the Bergman kernel on $\mathbb{C}^n$ and for high powers of ample holomorphic line bundles over compact complex manifolds.

Spectrum and Pseudo-Spectrum

In this book all Hilbert spaces will be assumed to separable for simplicity. In this section we review some basic definitions and properties; we refer to Kato (Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132. Springer, New York, 1966), Reed and Simon (Methods of modern mathematical physics. I. Functional analysis, 2nd edn. Academic, New York, 1980; Methods of modern mathematical physics. II. Fourier analysis, self adjointness. Academic, New York, 1975; Methods of modern mathematical physics. IV. Analysis of operators. Academic, New York, 1978), Riesz and Sz.-Nagy (Lecons d’analyse fonctionnelle, Quatrieme edition. Academie des Sciences d…

From Resolvent Estimates to Semigroup Bounds

In Chap. 10 we saw a concrete example of how to get resolvent bounds from semigroup bounds. Naturally, one can go in the opposite direction and in this chapter we discuss some abstract results of that type, including the Hille–Yoshida and Gearhardt–Pruss–Hwang–Greiner theorems. As for the latter, we also give a result of Helffer and the author that provides a more precise bound on the semigroup.

Counting Zeros of Holomorphic Functions

In this chapter we will generalize Proposition 3.4.6 of Hager about counting the zeros of holomorphic functions of exponential growth. In Hager and Sjostrand (Math Ann 342(1):177–243, 2008. http://arxiv.org/abs/math/0601381) we obtained such a generalization, by weakening the regularity assumptions on the functions ϕ. However, due to some logarithmic losses, we were not quite able to recover Hager’s original result, and we still had a fixed domain Γ with smooth boundary.

Toeplitz band matrices with small random perturbations

We study the spectra of $N\times N$ Toeplitz band matrices perturbed by small complex Gaussian random matrices, in the regime $N\gg 1$. We prove a probabilistic Weyl law, which provides an precise asymptotic formula for the number of eigenvalues in certain domains, which may depend on $N$, with probability sub-exponentially (in $N$) close to $1$. We show that most eigenvalues of the perturbed Toeplitz matrix are at a distance of at most $\mathcal{O}(N^{-1+\varepsilon})$, for all $\varepsilon &gt;0$, to the curve in the complex plane given by the symbol of the unperturbed Toeplitz matrix.

The Complex WKB Method

In this chapter we shall study the exponential growth and asymptotic expansions of exact solutions of second-order differential equations in the semi-classical limit. As an application, we establish a Bohr-Sommerfeld quantization condition for Schrodinger operators with real-analytic complex-valued potentials.

Review of Classical Non-self-adjoint Spectral Theory

The first section of this chapter deals with Fredholm theory in the spirit of Appendix A in Helffer and Sjostrand (Mm Soc Math Fr (NS) 24–25:1–228, 1986), see also an appendix in Melin and Sjostrand (Asterique 284:181–244, 2003) and Sjostrand and Zworski (Ann Inst Fourier 57:2095–2141, 2007). The remaining sections give a brief account of the very beautiful classical theory of non-self-adjoint operators, taken from a section in Sjostrand (Lectures on Resonances) which is a brief account of parts of the classical book by Gohberg and Krein (Introduction to the Theory of Linear Non-Selfadjoint Operators. Translations of Mathematical Monographs, vol 18. AMS, Providence, 1969).

On the linearized local Calderón problem

Spectral Asymptotics for More General Operators in One Dimension

In this chapter, we generalize the results of Chap. 3. The results and the main ideas are close, but not identical, to the ones of Hager (Ann Henri Poincare 7(6):1035–1064, 2006). We will use some h-pseudodifferential machinery, see for instance Dimassi and Sjostrand (Spectral Asymptotics in the Semi-classical Limit, London Mathematical Society Lecture Note Series, vol 268. Cambridge University Press, Cambridge, 1999).

Distribution of Large Eigenvalues for Elliptic Operators

In this chapter we consider elliptic differential operators on a compact manifold and rather than taking the semi-classical limit (h →), we let h = 1 and study the distribution of large eigenvalues. Bordeaux Montrieux (Loi de Weyl presque sure et resolvante pour des operateurs differentiels non-autoadjoints, these, CMLS, Ecole Polytechnique, 2008. https://pastel.archives-ouvertes.fr/pastel-00005367, Ann Henri Poincare 12:173–204, 2011) studied elliptic systems of differential operators on S1 with random perturbations of the coefficients, and under some additional assumptions, he showed that the large eigenvalues obey the Weyl law almost surely. His analysis was based on a reduction to the s…

Resonances for nonanalytic potentials

We consider semiclassical Schr"odinger operators on $R^n$, with $C^infty$ potentials decaying polynomially at infinity. The usual theories of resonances do not apply in such a non-analytic framework. Here, under some additional conditions, we show that resonances are invariantly defined up to any power of their imaginary part. The theory is based on resolvent estimates for families of approximating distorted operators with potentials that are holomorphic in narrow complex sectors around $R^n$.

Spectral Asymptotics for $$\mathcal {P}\mathcal {T}$$ Symmetric Operators

\(\mathcal {P}\mathcal {T}\)-symmetry has been proposed as an alternative to self-adjointness in quantum physics, see Bender et al. (J Math Phys 40(5):2201–2229, 1999), Bender and Mannheim (Phys Lett A 374(15–16):1616–1620, 2010). Thus for instance, if we consider a Schrodinger operator on Rn, $$\displaystyle P=-h^2\Delta +V(x), $$ the usual assumption of self-adjointness (implying that the potential V is real valued) can be replaced by that of \(\mathcal {P}\mathcal {T}\)-symmetry: $$\displaystyle V\circ \iota =\overline {V}, $$ where ι : Rn →Rn is an isometry with ι2 = 1≠ι. If we introduce the parity operator \(\mathcal {P}_\iota u(x)=u(\iota (x))\) and the time reversal operator \(\mathc…

Semiclassical Gevrey operators on exponentially weighted spaces of holomorphic functions

We provide a general overview of the recent works [“Semiclassical Gevrey operators in the complex domain”, Ann. Inst. Fourier (to appear), arXiv:2009.09125 (opens in new tab); J. Spectr. Theory 12, No. 1, 53–82 (2022; Zbl 1486.30104)] by the authors, devoted to continuity properties of semiclassical Gevrey pseudodifferential operators acting on a natural scale of exponentially weighted spaces of entire holomorphic functions.

Resonances over a potential well in an island

In this paper we study the distribution of scattering resonances for a multidimensional semi-classical Schr\"odinger operator, associated to a potential well in an island at energies close to the maximal one that limits the separation of the well and the surrounding sea.

Return to Equilibrium, Non-self-adjointness and Symmetries, Recent Results with M. Hitrik and F. Hérau

In this talk we review some old and new results about the use of supersymmetric structures in semi-classical problems. Necessary and sufficient conditions are obtained for a real semiclassical partial differential operator of order two to possess a supersymmetric structure. For operators coming from a chain of oscillators, coupled to two heat baths, we show the non-existence of a smooth supersymmetric structure. The recent and new results all come from joint works with Michael Hitrik and Frederic Herau.

Positivity, complex FIOs, and Toeplitz operators

International audience; We establish a characterization of complex linear canonical transformations that are positive with respect to a pair of strictly plurisubharmonic quadratic weights. As an application, we show that the boundedness of a class of Toeplitz operators on the Bargmann space is implied by the boundedness of their Weyl symbols.