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Johannes Sjoestrand

Publications and source records attributed to Johannes Sjoestrand.

At least 37 records · Page 2Linked to original sources
arXiv2015

Rational invariant tori and band edge spectra for non-selfadjoint operators

We study semiclassical asymptotics for spectra of non-selfadjoint perturbations of selfadjoint analytic $h$-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Complete asymptotic expansions are established for all individual eigenvalues in suitable regions of the complex spectral plane, near the edges of the spectral band, coming from rational flow-invariant Lagrangian tori.

math.SP↗
arXiv2014

Interior eigenvalue density of Jordan matrices with random perturbations

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.

math.SP↗
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↗
arXiv2013

Weyl law for semi-classical resonances with randomly perturbed potentials

In this work we consider semi-classical Schrödinger operators with potentials supported in a bounded strictly convex subset ${\cal O}$ of ${\bf R}^n$ with smooth boundary. Letting $h$ denote the semi-classical parameter, we consider certain classes of small random perturbations and show that with probability very close to 1, the number of resonances in rectangles $[a,b]-i[0,ch^{2/3}[$, is equal to the number of eigenvalues in $[a,b]$ of the Dirichlet realization of the unperturbed operator in ${\cal O}$ up to a small remainder.

math.AP↗
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

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 non-real, there are many non-real eigenvalues.

math.SP↗
arXiv2011

Fractal Weyl law for open quantum chaotic maps

We study the semiclassical quantization of Poincaré 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.

math.AP↗
arXiv2011

From open quantum systems to open quantum maps

For a class of quantized open chaotic systems satisfying a natural dynamical assumption, we show that the study of the resolvent, and hence of scattering and resonances, can be reduced to the study of a family of open quantum maps, that is of finite dimensional operators obtained by quantizing the Poincaré map associated with the flow near the set of trapped trajectories.

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

Upper bound on the density of Ruelle resonances for Anosov flows

Using a semiclassical approach we show that the spectrum of a smooth Anosov vector field V on a compact manifold is discrete (in suitable anisotropic Sobolev spaces) and then we provide an upper bound for the density of eigenvalues of the operator (-i)V, called Ruelle resonances, close to the real axis and for large real parts.

math-ph↗
arXiv2010

Spectral properties of non-self-adjoint operators

This text is a slightly expanded version of my 6 hour mini-course at the PDE-meeting in Évian-les-Bains in June 2009. The first part gives some old and recent results on non-self-adjoint differential operators. The second part is devoted to recent results about Weyl distribution of eigenvalues of elliptic operators with small random perturbations.

math.SP↗
arXiv2010

From resolvent bounds to semigroup bounds

The purpose of this note is to revisit the proof of the Gearhardt-Prüss-Hwang-Greiner theorem for a semigroup S(t), following the general idea of the proofs that we have seen in the literature and to get an explicit estimate on the norm of S(t) in terms of bounds on the resolvent of the generator.

math.FA↗
arXiv2009

Counting zeros of holomorphic functions of exponential growth

We consider the number of zeros of holomorphic functions in a bounded domain that depend on a small parameter and satisfy an exponential upper bound near the boundary of the domain and similar lower bounds at finitely many points along the boundary. Roughly the number of such zeros is $(2πh)^{-1}$ times the integral over the domain of the laplacian of the exponent of the dominating exponential. Such results have already been obtained by M. Hager and by Hager and the author and they are of importance when studying the asymptotic distribution of eigenvalues of elliptic operators with small random perturbations. In this paper we generalize these results and arrive at geometrically natural statements and natural remainder estimates.

math.CV↗
arXiv2009

Resolvent estimates for non-self-adjoint operators via semi-groups

We consider a non-self-adjoint $h$-pseudodifferential operator $P$ in the semi-classical limit ($h\to 0$). If $p$ is the leading symbol, then under suitable assumptions about the behaviour 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 ${\cal O}(1)((h\ln \frac{1}{h})^{k/(k+1)})$ from certain boundary points, where $k\in \{2,4,...\}$. This is a slight improvement of a result by Dencker, Zworski and the author, and it has recently been obtained by W. Bordeaux Montrieux in a model situation where $k=2$. The method of proof is different from the one of Dencker et al, and is based on estimates of an associated semi-group.

math.SP↗