arXiv subjects

Johannes Sjöstrand

Publications and source records attributed to Johannes Sjöstrand.

12 recordsLinked to original sources

arXiv2026

Weyl laws for exponentially small singular values of the $\overline{\partial}$ operator

We study the number of exponentially small singular values of the semiclassical $\overline{\partial}$ operator on exponentially weighted $L^2$ spaces on a compact Riemann surface. Accurate upper and lower bounds on the number of such singular values are established in terms of auxiliary notions of upper and lower bound weights. Assuming that the Laplacian of the exponential weight changes sign along a curve, we construct optimal such weights by solving a free boundary problem, which yields Weyl asymptotics for the counting function of the singular values in an interval of the form $[0,\mathrm{e}^{-τ/h}]$, for $τ>0$ smaller than the oscillation of the weight. We also provide a precise description of the leading term in the Weyl asymptotics, in the regime of small $τ> 0$.

math.SP↗

arXiv2026

Stationary phase with Cauchy singularity. A critical point of signature $(+,-)$

Asymptotic expressions for an integral appearing in the solution of a d-bar problem are presented. The integral is a solid Cauchy transform of a function with a rapidly oscillating phase with a small parameter $h$, $0<h\ll 1$. Whereas standard steepest descent approaches can be applied to the case where the stationary points of the phase $ω_{k}$, $k=1,\ldots, N$ are far from the singularity $ζ$ of the integrand, a polarization approach is proposed for the case that $|ζ-ω_{k}|<\mathcal{O}(\sqrt{h})$ for some $k$. In this case the problem is studied in $\mathbb{C}^{2}$ ($\widetildeω:=\overlineω$ is treated as an independent variable) on steepest descent contours. An application of Stokes' theorem allows for a decomposition of the integral into three terms for which asymptotic expressions in terms of special functions are given.

math.AP↗

arXiv2025

Fractional anisotropic Calderón problem with external data

In this paper, we solve the fractional anisotropic Calderón problem with external data in the Euclidean space, in dimensions two and higher, for smooth Riemannian metrics that agree with the Euclidean metric outside a compact set. Specifically, we prove that the knowledge of the partial exterior Dirichlet--to--Neumann map for the fractional Laplace-Beltrami operator, given on arbitrary open nonempty sets in the exterior of the domain in the Euclidean space, determines the Riemannian metric up to diffeomorphism, fixing the exterior. We provide two proofs of this result: one relies on the heat semigroup representation of the fractional Laplacian and a pseudodifferential approach, while the other is based on a variable-coefficient elliptic extension interpretation of the fractional Laplacian.

math.AP↗

arXiv2023

Discussing semigroup bounds with resolvent estimates

The purpose of this paper is to revisit the proof of the Gearhart-Prüss-Huang-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 operator norm of $S(t)$ in terms of bounds on the resolvent of the generator. In \cite{HelSj} by the first two authors, this was done and some applications in semiclassical analysis were given. Some of these results have been subsequently published in two books written by the two first authors \cite{He1,Sj2}. A second work \cite{HeSj21} by the first two authors presents new improvements partially motivated by a paper of D. Wei \cite{W}. In this third paper, we continue the discussion on whether the aforementioned results are optimal, and whether one can improve these results through iteration. Numerical computations will illustrate some of the abstract results.

math.FA↗

arXiv2023

Tunneling for the $\overline{\partial}$-operator

We study the small singular values of the $2$-dimensional semiclassical differential operator $P = 2\,\mathrm{e}^{-ϕ/h}\circ hD_{\overline{z}}\circ \mathrm{e}^{ϕ/h}$ on $S^1+iS^1$ and on $S^1+i\mathbb{R}$ where $ϕ$ is given by $\sin y$ and by $y^3/3$, respectively. The key feature of this model is the fact that we can pinpoint precisely where in phase space the Poisson bracket $\{p,\overline{p}\}=0$, where $p$ is the semiclassical symbol of $P$. We give a precise asymptotic description of the exponentially small singular values of $P$ by studying the tunneling effects of an associated Witten complex. We use these asymptotics to determine a Weyl law for the exponentially small singular values of $P$.

math.SP↗

arXiv2022

Large $|k|$ behavior for the reflection coefficient for Davey-Stewartson II equations

The study of complex geometric optics solutions to a system of d-bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey-Stewartson II equations for large values of the spectral parameter $k$ in \cite{KlSjSt20} is extended to the reflection coefficient. For the case of potentials $q$ with compact support on some domain $Ω$ with smooth strictly convex boundary, improved asymptotic relations are provided.

math.AP↗

arXiv2021

Improving semi-groups bounds with resolvent estimates

The purpose of this paper 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 $\Vert S(t) \Vert$ in terms of bounds on the resolvent of the generator. A first version of this paper was presented by the two authors in ArXiv (2010) together with applications in semi-classical analysis and a part of these results has been published later in two books written by the authors. Our aim is to present new improvements, partially motivated by a paper of D. Wei. On the way we discuss optimization problems confirming the optimality of our results.

math.OC↗

arXiv2021

Resonances over a potential well in an island

In this paper we study the distribution of scattering resonances for a multidimensional semi-classical Schrödinger 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.

math.AP↗

arXiv2020

Large $|k|$ behavior of d-bar problems for domains with a smooth boundary

In a previous work on the large $|k|$ behavior of complex geometric optics solutions to a system of d-bar equations, we treated in detail the situation when a certain potential is the characteristic function of a strictly convex set with real-analytic boundary. We here extend the results to the case of sets with smooth boundary, by using almost holomorphic functions.

math.AP↗

arXiv1997

Nonlinear resonant tunneling in systems coupled to quantum reservoirs

An adiabatic approximation in terms of instantaneous resonances is developed to study the steady-state and time-dependent transport of interacting electrons in biased resonant tunneling heterostructures. The resulting model consists of quantum reservoirs coupled to regions where the system is described by nonlinear ordinary differential equations and has a general conceptual interest.

cond-mat↗

arXiv1996

Transport properties in resonant tunneling heterostructures

We use an adiabatic approximation in terms of instantaneous resonances to study the steady-state and time-dependent transport properties of interacting electrons in biased resonant tunneling heterostructures. This approach leads, in a natural way, to a transport model of large applicability consisting of reservoirs coupled to regions where the system is described by a nonlinear Schrödinger equation. From the mathematical point of view, this work is non-rigorous but may offer some fresh and interesting problems involving semiclassical approximation, adiabatic theory, non-linear Schrödinger equations and dynamical systems.

cond-mat↗

arXiv1995

On Schroedinger Equations with Concentrated Nonlinearities

Schrödinger equations with nonlinearities concentrated in some regions of space are good models of various physical situations and have interesting mathematical properties. We show that in the semiclassical limit it is possible to separate the relevant degrees of freedom by noticing that in the regions where the nonlinearities are effective all states are suppressed but the metastable ones (resonances). In this way the description of the nonlinear regions is reduced to ordinary differential equations weakly coupled to standard Schrödinger equations valid in the linear regions. The idea is illustrated through the study of a prototype equation recently proposed for resonant tunneling of electrons through a double barrier heterostructure and for which nonlinear oscillations have been numerically predicted.

cond-mat↗