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Michael Hitrik

Publications and source records attributed to Michael Hitrik.

At least 37 records · Page 2Linked to original sources
arXiv2014

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.

math.SP↗
arXiv2012

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.

math-ph↗
arXiv2011

Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics

For a class of non-selfadjoint $h$--pseudodifferential operators with double characteristics, we give a precise description of the spectrum and establish accurate semiclassical resolvent estimates in a neighborhood of the origin. Specifically, assuming that the quadratic approximations of the principal symbol of the operator along the double characteristics enjoy a partial ellipticity property along a suitable subspace of the phase space, namely their singular space, we give a precise description of the spectrum of the operator in an ${\cal O}(h)$--neighborhood of the origin. Moreover, when all the singular spaces are reduced to zero, we establish accurate semiclassical resolvent estimates of subelliptic type, which depend directly on algebraic properties of the Hamilton maps associated to the quadratic approximations of the principal symbol.

math.AP↗
arXiv2011

Diophantine tori and Weyl laws for non-selfadjoint operators in dimension two

We study the distribution of eigenvalues for non-selfadjoint perturbations of selfadjoint semiclassical analytic pseudodifferential operators in dimension two, assuming that the classical flow of the unperturbed part is completely integrable. An asymptotic formula of Weyl type for the number of eigenvalues in a spectral band, bounded from above and from below by levels corresponding to Diophantine invariant Lagrangian tori, is established. The Weyl law is given in terms of the long time averages of the leading non-selfadjoint perturbation along the classical flow of the unperturbed part.

math.SP↗
arXiv2010

Tunnel effect and symmetries for Kramers Fokker-Planck type operators

We 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 $n_0$ of local minima. Under suitable additional assumptions, we show that the first $n_0$ eigenvalues are real and exponentially small, and establish the complete semiclassical asymptotics for these eigenvalues.

math.SP↗
arXiv2010

Transmission eigenvalues for elliptic operators

A reduction of the transmission eigenvalue problem for multiplicative sign-definite perturbations of elliptic operators with constant coefficients to an eigenvalue problem for a non-selfadjoint compact operator is given. Sufficient conditions for the existence of transmission eigenvalues and completeness of generalized eigenstates for the transmission eigenvalue problem are derived. In the trace class case, the generic existence of transmission eigenvalues is established.

math-ph↗
arXiv2010

Transmission eigenvalues for operators with constant coefficients

In this paper we study the interior transmission problem and transmission eigenvalues for multiplicative perturbations of linear partial differential operator of order $\ge 2$ with constant real coefficients. Under suitable growth conditions on the symbol of the operator and the perturbation, we show the discreteness of the set of transmission eigenvalues and derive sufficient conditions on the existence of transmission eigenvalues. We apply these techniques to the case of the biharmonic operator and the Dirac system. In the hypoelliptic case we present a connection to scattering theory.

math-ph↗
arXiv2009

Semiclassical hypoelliptic estimates for non-selfadjoint operators with double characteristics

For a class of non-selfadjoint semiclassical pseudodifferential operators with double characteristics, we study bounds for resolvents and estimates for low lying eigenvalues. Specifically, assuming that the quadratic approximations of the principal symbol of the operator along the double characteristics enjoy a partial ellipticity property along a suitable subspace of the phase space, namely their singular space, we establish semiclassical hypoelliptic a priori estimates with a loss of the full power of the semiclassical parameter, giving a localization for the low lying spectral values of the operator.

math.AP↗
arXiv2008

Tunnel effect for Kramers-Fokker-Planck type operators: return to equilibrium and applications

In the first part of this work, we consider second order supersymmetric differential operators in the semiclassical limit, including the Kramers-Fokker-Planck operator, such that the exponent of the associated Maxwellian $ϕ$ is a Morse function with two local minima and one saddle point. Under suitable additional assumptions of dynamical nature, we establish the long time convergence to the equilibrium for the associated heat semigroup, with the rate given by the first non-vanishing, exponentially small, eigenvalue. In the second part of the paper, we consider the case when the function $ϕ$ has precisely one local minimum and one saddle point. We also discuss further examples of supersymmetric operators, including the Witten Laplacian and the infinitesimal generator for the time evolution of a chain of classical anharmonic oscillators.

math.AP↗
arXiv2007

Spectra and semigroup smoothing for non-elliptic quadratic operators

We study non-elliptic quadratic differential operators. Quadratic differential operators are non-selfadjoint operators defined in the Weyl quantization by complex-valued quadratic symbols. When the real part of their Weyl symbols is a non-positive quadratic form, we point out the existence of a particular linear subspace in the phase space, intrinsically associated to the Weyl symbols, called a singular space, such that when the singular space has a symplectic structure, the associated heat semigroup is smoothing in every direction of its symplectic orthogonal complement. When the Weyl symbol of such an operator is elliptic on the singular space, this space is always symplectic and we prove that the spectrum of the operator is discrete and can be described as in the globally elliptic case. We also describe the large time behavior of contraction semigroups generated by these operators.

math.SP↗
arXiv2007

Tunnel effect for Kramers-Fokker-Planck type operators

We consider operators of Kramers-Fokker-Planck type in the semi-classical limit such that the exponent of the associated Maxwellian is a Morse function with two local minima and a saddle point. Under suitable additional assumptions we establish the complete asymptotics of the exponentially small splitting between the first two eigenvalues.

math.SP↗
arXiv2007

Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2

We study spectral asymptotics and resolvent bounds for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Spectral contributions coming from rational invariant Lagrangian tori are analyzed. Estimating the tunnel effect between strongly irrational (Diophantine) and rational tori, we obtain an accurate description of the spectrum in a suitable window in the complex spectral plane, provided that the strength of the non-selfadjoint perturbation $\gg h$ (or sometimes $\gg h^2$) is not too large.

math.SP↗
arXiv2005

Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions II. Vanishing Averages

This is the second in a series of works devoted to small non-selfadjoint perturbations of selfadjoint semiclassical pseudodifferential operators in dimension 2. As in our previous work, we consider the case when the classical flow of the unperturbed part is periodic. Under the assumption that the flow average of the leading perturbation vanishes identically, we show how to obtain a complete asymptotic description of the individual eigenvalues in certain domains in the complex plane, provided that the strength of the perturbation $ε$ is $\gg h^{1/2}$, or sometimes only $\gg h$, and enjoys the upper bound $ε={\cal O}(h^δ)$, for some $δ>0$.

math.SP↗
arXiv2005

Diophantine tori and spectral asymptotics for non-selfadjoint operators

We study spectral asymptotics for small non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part possesses several invariant Lagrangian tori enjoying a Diophantine property. We get complete asymptotic expansions for all eigenvalues in certain rectangles in the complex plane in two different cases: in the first case, we assume that the strength $ε$ of the perturbation is ${\cal O}(h^δ)$ for some $δ>0$ and is bounded from below by a fixed positive power of $h$. In the second case, $ε$ is assumed to be sufficiently small but independent of $h$, and we describe the eigenvalues completely in a fixed $h$-independent domain in the complex spectral plane.

math.SP↗