We obtain bounds on the complex eigenvalues of non-self-adjoint Schrödinger operators with complex potentials, and also on the complex resonances of self-adjoint Schrödinger operators. Our bounds are compared with numerical results, and are seen to provide useful information.
We will discuss the asymptotic behaviour of the eigenvalues of Schrödinger operator with a matrix potential defined by Neumann boundary condition in $L_2^m(F)$, where $F$ is $d$-dimensional rectangle and the potential is a $m \times m$ matrix with $m\geq 2$, $d\geq 2$ , when the eigenvalues belong to the resonance domain, roughly speaking they lie near planes of diffraction. \textbf{Keywords:} Schrödinger operator, Neumann condition, Resonance eigenvalue, Perturbation theory. \textbf{AMS Subject Classifications:} 47F05, 35P15
We consider a general second-order elliptic differential operator on a domain with a cylindrical end. We impose Dirichlet boundary conditions on the boundary with the exception of a small set, where we impose Neumann boundary conditions. Shrinking this set to a point we calculate the asymptotic behaviour of eigenvalues and resonances.
In this paper we describe a simple method that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. We illustrate that even if the Neumann-to-Dirichlet map is obtained by a Finite Element Method (FEM) one can achieve good accuracy for the scattering matrix. We give various interesting examples of how this can be used to investigate the behaviour of resonances under conformal perturbations or when moving in Teichmüller space. For example, based on …
We study the dynamics of resonances of analytic perturbations of 0th order pseudodifferential operators $P(s)$. In particular, we prove a Fermi golden rule for resonances of $P(s)$ at embedded eigenvalues of $P=P(0)$. We also study the dynamics of eigenvalues of $P(t)=P+itΔ$ as the eigenvalues converge to simple eigenvalues of $P$. The 0th order pseudodifferential operators we consider satisfy natural dynamical assumptions and are used as microlocal models of internal waves.
We consider a Schrödinger Operator with a matrix potential defined in $L_2^m(F)$ by the differential expression\begin{equation*} L(ϕ(x))=(-Δ+V(x))ϕ(x) \end{equation*}and the Neumann boundary condition, where $F$ is the $d$ dimensional rectangle and $V$ is a martix potential, $m\geqslant 2, d\geqslant 2$. We obtain the asymptotic formulas of arbitrary order for the single resonance eigenvalues of the Schrödinger operator in $L_2^m(F)$.
We consider a Schroedinger operator on the axis with a bipartite potential consisting of two compactly supported complex-valued functions, whose supports are separated by a large distance. We show that this operator possesses a sequence of approximately equidistant complex-valued wavenumbers situated near the real axis. Depending on its imaginary part, each wavenumber corresponds to either a resonance or an eigenvalue. The obtained sequence of wavenumbers resembles transmission resonances in electromagnetic Fabry-Pérot interferometers formed by parallel mirrors. Our result has potential applic…
We consider the Schrödinger operator $H$ on the half-line with a periodic potential $p$ plus a compactly supported potential $q$. For generic $p$, its essential spectrum has an infinite sequence of open gaps. We determine the asymptotics of the resonance counting function and show that, for sufficiently high energy, each non-degenerate gap contains exactly one eigenvalue or antibound state, giving asymptotics for their positions. Conversely, for any potential $q$ and for any sequences $(\s_n)_{1}^\iy, \s_n\in \{0,1\}$, and $(\vk_n)_1^\iy\in \ell^2, \vk_n\ge 0$, there exists a potential $p$ suc…
We consider a parity-time ($\mathcal{PT}$-) symmetric waveguide consisting of a localized gain and loss elements separated by a variable distance. The situation is modelled by a Schrödiner operator with localized complex $\mathcal{PT}$-symmetric potential. Properties of the latter Hamiltonian are considered subject to the change of the gain-to-loss distance. Resonances, spectral singularities and eigenvalues are analyzed in detail and discussed in the context of the associated laser-absorber modes and $\mathcal{PT}$-symmetry breaking phase transition. Increasing gain-to-loss distance creates n…
Pollicott-Ruelle resonances for chaotic flows are the characteristic frequencies of correlations. They are typically defined as eigenvalues of the generator of the flow acting on specially designed functional spaces. We show that these resonances can be computed as viscosity limits of eigenvalues of second order elliptic operators. These eigenvalues are the characteristic frequencies of correlations for a stochastically perturbed flow.
We consider the Dirichlet Laplacian in a three-dimensional waveguide that is a small deformation of a periodically twisted tube. The deformation is given by a bending and an additional twisting of the tube, both parametrized by a coupling constant $δ$. We expand the resolvent of the perturbed operator near the bottom of its essential spectrum and we show the existence of exactly one resonance, in the asymptotic regime of $δ$ small. We are able to perform the asymptotic expansion of the resonance in $δ$, which in particular permits us to give a quantitative geometric criterion for the existence…
We study the existence of eigenvalues of the generalized Friedrichs model $H_μ(p)$, with a rank-one perturbation, depending on parameters $μ>0$ and $p\in\mathbb{T}^2$, and found an absolutely convergent expansions for eigenvalues at $μ(p)$, the coupling constant threshold. The expansions are highly dependent on that, whether the threshold $m(p)$ of the essential spectrum is: $(i)$ neither an threshold eigenvalue nor a threshold resonance; $(ii)$ a threshold resonance; $(iii)$ an threshold eigenvalue.
Let $Λ\subset \mathbb{R}^d$ be a domain consisting of several cylinders attached to a bounded center. One says that $Λ$ admits a threshold resonance if there exists a non-trivial bounded function $u$ solving $-Δu=νu$ in $Λ$ and vanishing at the boundary, where $ν$ is the bottom of the essential spectrum of the Dirichlet Laplacian in $Λ$. We derive a sufficient condition for the absence of threshold resonances in terms of the Laplacian eigenvalues on the center. The proof is elementary and is based on the min-max principle. Some two- and three-dimensional examples and applications to the study …
We investigate in a quantitative way the plasmon resonance at eigenvalues and the essential spectrum (the accumulation point of eigenvalues) of the Neumann-Poincaré operator on smooth domains. We first extend the symmetrization principle so that the single layer potential becomes a unitary operator from $H^{-1/2}$ onto $H^{1/2}$. We then show that the resonance at the essential spectrum is weaker than that at eigenvalues. It is shown that anomalous localized resonance occurs at the essential spectrum on ellipses, but cloaking does not occur on ellipses unlike the core-shell structure considere…
The paper is devoted to optimization of resonances in a 1-D open optical cavity. The cavity's structure is represented by its dielectric permittivity function e(s). It is assumed that e(s) takes values in the range 1 <= e_1 <= e(s) <= e_2. The problem is to design, for a given (real) frequency, a cavity having a resonance with the minimal possible decay rate. Restricting ourselves to resonances of a given frequency, we define cavities and resonant modes with locally extremal decay rate, and then study their properties. We show that such locally extremal cavities are 1-D photonic crystals consi…
We consider the Dirichlet Laplacian in a straight three dimensional waveguide with non-rotationally invariant cross section, perturbed by a twisting of small amplitude. It is well known that such a perturbation does not create eigenvalues below the essential spectrum. However, around the bottom of the spectrum, we provide a meromorphic extension of the weighted resolvent of the perturbed operator, and show the existence of exactly one resonance near this point. Moreover, we obtain the asymptotic behavior of this resonance as the size of the twisting goes to 0. We also extend the analysis to th…
The paper is devoted to optimization of resonances for Krein strings with total mass and statical moment constraints. The problem is to design for a given $α\in \R$ a string that has a resonance on the line $α+ ı\R$ with a minimal possible modulus of the imaginary part. We find optimal resonances and strings explicitly.
We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold $M = (0,\infty) \times Y$ whose rotation radius is constant outside some compact interval. The Laplacian on $M$ is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potentials on the half-line. We prove o Asymptotics of counting function of resonances at large radius o Inverse problem: The rotation radius is uniquely determined by its eigenvalues and resonances. Moreover, there exists an algorithm to recover the rotation radius from its eigenvalues…
We study resonances generated by rank one perturbations of selfadjoint operators with eigenvalues embedded in the continuous spectrum. Instability of these eigenvalues is analyzed and almost exponential decay for the associated resonant states is exhibited. We show how these results can be applied to Sturm-Liouville operators. Main tools are the Aronszajn-Donoghue theory for rank one perturbations, a reduction process of the resolvent based on Feshbach-Livsic formula, the Fermi golden rule and a careful analysis of the Fourier transform of quasi-Lorentzian functions. We relate these results to…
We consider the Hamiltonian $H$ of a 3D spinless non-relativistic quantum particle subject to parallel constant magnetic and non-constant electric field. The operator $H$ has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb $H$ by appropriate scalar potentials $V$ and investigate the transformation of these embedded eigenvalues into resonances. First, we assume that the electric potentials are dilation-analytic with respect to the variable along the magnetic field, and obtain an asymptotic expansion of the resonances as the coupling constant …
We discuss quantum graphs consisting of a compact part and semiinfinite leads. Such a system may have embedded eigenvalues if some edge lengths in the compact part are rationally related. If such a relation is perturbed these eigenvalues may turn into resonances; we analyze this effect both generally and in simple examples.
We consider the Schrödinger operator $Hy=-y"+(p+q)y$ with a periodic potential $p$ plus a compactly supported potential $q$ on the real line. The spectrum of $H$ consists of an absolutely continuous part plus a finite number of simple eigenvalues below the spectrum and in each spectral gap $\g_n\ne \es, n\ge1$. We prove the following results: 1) the distribution of resonances in the disk with large radius is determined, 2) the asymptotics of eigenvalues and antibound states are determined at high energy gaps, 3) if $H$ has infinitely many open gaps in the continuous spectrum, then for any sequ…
We study resonances for the generator of a diffusion with small noise in $R^d$ :$ L_ε= -εΔ+ \nabla F \cdot \nabla$, when the potential F grows slowly at infinity (typically as a square root of the norm). The case when F grows fast is well known, and under suitable conditions one can show that there exists a family of exponentially small eigenvalues, related to the wells of F . We show that, for an F with a slow growth, the spectrum is R+, but we can find a family of resonances whose real parts behave as the eigenvalues of the "quick growth" case, and whose imaginary parts are small.
We consider the Schrödinger operator on zigzag graphs with a periodic potential. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all compactly supported eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on some Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove that all resonances are real. We determine the asympto…
Eigenvalue behaviors of Schrödinger operator defined on $n$-dimensional lattice with $n+1$ delta potentials is studied. It can be shown that lower threshold eigenvalue and lower threshold resonance are appeared for $n\geq 2$, and lower super-threshold resonance appeared for $n=1$.
We discuss resonances for Schrödinger operators with compactly supported potentials on the line and the half-line. We estimate the sum of the negative power of all resonances and eigenvalues in terms of the norm of the potential and the diameter of its support. The proof is based on harmonic analysis and Carleson measures arguments.
We consider the 3D Schrödinger operator $H_0$ with constant magnetic field and subject to an electric potential $v_0$ depending only on the variable along the magnetic field $x_3$. The operator $H_0$ has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb $H_0$ by smooth scalar potentials $V=O((x_1,x_2)>^{-\de_\perp}x_3>^{-\de_\parallel})$, $\de_\perp>2, \de_\parallel>1$. We assume also that $V$ and $v_0$ have an analytic continuation, in the magnetic field direction, in a complex sector outside a compact set. We define the resonances of $H=H_0+…
We consider the periodic Jacobi operator $J$ with finitely supported perturbations on the half-lattice. We describe all eigenvalues and resonances of $J$ and give their properties. We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the Jost functions is one-to-one and onto, we show how the Jost functions can be reconstructed from the eigenvalues, resonances and the set of zeros of $S(ł)-1,$ where $S(ł)$ is the scattering matrix.
An important class of resonance problems involves the study of perturbations of systems having embedded eigenvalues in their continuous spectrum. Problems with this mathematical structure arise in the study of many physical systems, e.g. the coupling of an atom or molecule to a photon-radiation field, and Auger states of the helium atom, as well as in spectral geometry and number theory. We present a dynamic (time-dependent) theory of such quantum resonances. The key hypotheses are (i) a resonance condition which holds generically (non-vanishing of the {\it Fermi golden rule}) and (ii) local d…
This is a complement to our earlier work \cite{C10a} where a new eigenvalue concentration bound for multi-particle disordered quantum lattice systems was obtained. Here we show that the new bound leads to a simplified proof of multi-particle spectral and dynamical localization.
We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near infinity case the determinant is analyzed through the zeta-regularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order two with …
Using the method of complex scaling we show that scattering resonances of $ - Δ+ V $, $ V \in L^\infty_{\rm{c}} ( \mathbb R^n ) $, are limits of eigenvalues of $ - Δ+ V - i εx^2 $ as $ ε\to 0+ $. That justifies a method proposed in computational chemistry and reflects a general principle for resonances in other settings.
We consider the magnetic Schrödinger operator on the so-called zigzag periodic metric graph (a quasi 1D continuous model of zigzag nanotubes) with a periodic potential. The magnetic field (with the amplitude $B\in R$) is uniform and it is parallel to the axis of the nanotube. The spectrum of this operator consists of an absolutely continuous part (spectral bands separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all compactly supported eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on some Riemann surface…
This paper is mainly devoted to study time decay estimates of the higher-order Schrödinger type operator $H=(-Δ)^{m}+V(x)$ in $\mathbf{R}^{n}$ for $n>2m$ and $m\in\mathbf{N}$. For certain decay potentials $V(x)$, we first derive the asymptotic expansions of resolvent $R_{V}(z)$ near zero threshold with the presence of zero resonance or zero eigenvalue, as well identify the resonance space for each kind of zero resonance which displays different effects on time decay rate. Then we establish Kato-Jensen type estimates and local decay estimates for higher order Schrödinger propagator $e^{-itH}$ i…
For exterior dilation analytic potential, $V$, we use the method of complex scaling to show that the resonances of $ - Δ+ V $, in a conic neighbourhood of the real axis, are limits of eigenvalues of $ - Δ+ V - i εx^2 $ as $ ε\to 0+ $, if $V$ can be analytically extended from $\mathbb{R}^n$ to a truncated cone in $\mathbb{C}^n$.
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.
In this article, we prove the finiteness of the number of eigenvalues for a class of Schrödinger operators $H = -Δ+ V(x)$ with a complex-valued potential $V(x)$ on $\bR^n$, $n \ge 2$. If $\Im V$ is sufficiently small, $\Im V \le 0$ and $\Im V \neq 0$, we show that $N(V) = N(\Re V)+ k$, where $k$ is the multiplicity of the zero resonance of the selfadjoint operator $-Δ+ \Re V$ and $N(W)$ the number of eigenvalues of $-Δ+ W$, counted according to their algebraic multiplicity.
We are interested in diagonal perturbations of a periodic Jacobi operator that introduce embedded eigenvalues in its essential spectrum. Embedding multiple points in the essential spectrum has been known to be difficult, given that eigenvalues are destroyed easily by small perturbations. However, given a finite or countably infinite set of points within an absolutely continuous band of the original periodic operator (subject only to a very weak non-resonance condition) we are able to construct a diagonal perturbation that preserves the essential spectrum and places eigenvalues in all of those …
We study eigenvalues of quantum open baker's maps with trapped sets given by linear arithmetic Cantor sets of dimensions $δ\in (0,1)$. We show that the size of the spectral gap is strictly greater than the standard bound $\max(0,{1\over 2}-δ)$ for all values of $δ$, which is the first result of this kind. The size of the improvement is determined from a fractal uncertainty principle and can be computed for any given Cantor set. We next show a fractal Weyl upper bound for the number of eigenvalues in annuli, with exponent which depends on the inner radius of the annulus.
Necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range or exponentially decaying perturbation of a periodic Jacobi operator. As a corollary we can fully solve the inverse resonance problem: given resonances and eigenvalues we can recover the spectral measure of the Jacobi operator; we provide necessary and sufficient conditions under which such an operator exists and is unique; and we show that the inverse resonance problem is stable under small perturbations.
We show that the complex absorbing potential (CAP) method for computing scattering resonances applies to the case of exponentially decaying potentials. That means that the eigenvalues of $-Δ+ V - iεx^2$, $|V(x)|\leq C e^{-2γ|x|}$ converge, as $ ε\to 0+ $, to the poles of the meromorphic continuation of $ ( -Δ+ V -λ^2 )^{-1} $ uniformly on compact subsets of $\textrm{Re}\,λ>0$, $\textrm{Im}\,λ>-γ$, $\argλ> -π/8$.
A modified Fermi Golden Rule at threshold for 3D magnetic Schrödinger operators
Doppler Institute for Mathematical Physics and Applied Mathematics（多普勒应用数学物理研究所）；Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences（捷克科学院核物理研究所理论物理系）；Aalborg University（奥尔堡大学）；DICATAM, Sezione di Matematica, Università degli studi di Brescia（布雷西亚大学DICATAM数学系）
We consider a periodic Jacobi operator $H$ with finitely supported perturbations on ${\Bbb Z}.$ We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the scattering data: the inverse of the transmission coefficient and the Jost function on the right half-axis, is one-to-one and onto. We consider the problem of reconstruction of the scattering data from all eigenvalues, resonances and the set of zeros of $R_-(λ)+1,$ where $R_-$ is the reflection coefficient.
We investigate the existence of resonances for two-centers Coulomb systems with arbitrary charges in two dimensions, defining them in terms of generalised complex eigenvalues of a non-selfadjoint deformation of the two-centers Schrödinger operator. We construct the resolvent kernels of the operators and prove that they can be extended analytically to the second Riemann sheet. The resonances are then analysed by means of perturbation theory and numerical methods.
A model operator $H$ associated to a system of three-particles on the three dimensional lattice $\Z^3$ and interacting via pair non-local potentials is studied. The following results are proven: (i) the operator $H$ has infinitely many eigenvalues lying below the bottom of the essential spectrum and accumulating at this point, in the case, where both Friedrichs model operators $h_{μ_α}(0),α=1,2,$ have threshold resonances. (ii) the operator $H$ has a finite number of eigenvalues lying outside of the essential spectrum, in the case, where at least one of $h_{μ_α}(0), α=1,2,$ has a threshold eig…
We consider the Schrödinger operator with a periodic potential on quasi-1D models of armchair single-wall nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe the absolutely continuous spectrum of the Schrödinger operator: 1) the multiplicity, 2) endpoints of the gaps, they are given by periodic or antiperiodic eigenvalues or resonances (branch points of the Lyapunov function), 3) resonance gaps, where the Lyapunov function is non-real. We determine the asy…
