Asymptotics of resonances for a schrodinger operator with matrix values

Weyl-Titchmarsh M-Function Asymptotics for Matrix-Valued Schrödinger Operators

We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general matrix-valued Schrödinger operators on a half-line.

The IDS and Asymptotic of the Largest Eigenvalue of Random Schrödinger Operators with Decaying Random Potential

In this work we obtain the integrated density of states for the Schrödinger operators with decaying random potentials acting on $\ell^2(\mathbb{Z}^d)$. We also study the asymptotic of the largest and smallest eigenvalues of its finite volume approximation

Threshold of discrete Schrödinger operators with delta potentials on $n$-dimensional lattice

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 $n=1$.

Spectral asymptotics for magnetic Schrödinger operators in domains with corners

This paper is on magnetic Schrodinger operators in two dimensional domains with corners. Semiclassical formulas are obtained for the sum and number of eigenvalues. The obtained results extend former formulas for smooth domains in \cite{Fr, FK} to piecewise smooth domains.

Spectral asymptotics for the Schrödinger operator on the line with spreading and oscillating potentials

This study is devoted to the asymptotic spectral analysis of multiscale Schrödinger operators with oscillating and decaying electric potentials. Different regimes, related to scaling considerations, are distinguished. By means of a normal form filtrating the oscillations, a reduction to a non-oscillating effective Hamiltonian is performed.

Schrödinger算子的共振集

Resonances sets of Schrödinger operators

本文研究了Schrödinger算子的共振集，证明其可包含满足Blaschke条件的任意子集，并建立了更宽领域子集的充分条件。

Asymptotics and estimates of the discrete spectrum of the Schrodinger operator on a discrete periodic graph

The periodic Schrodinger operator $ H $ on a discrete periodic graph is considered. We estimate the discrete spectrum of the perturbed operator $ H _ {-} (t) = H-tV $, $ t> 0 $, where the potential $ V \ ge 0 $ is decreasing and $t>0$ is the coupling constant. If the potential has a power asymptotics at infinity, then we obtain asymptotics of the discrete spectrum of operator $ H_ {-} (t) $ with a large coupling constant.

Weyl asymptotics for magnetic Schrödinger operators and de Gennes' boundary condition

This paper is concerned with the discrete spectrum of the self-adjoint realization of the semi-classical Schrödinger operator with constant magnetic field and associated with the de Gennes (Fourier/Robin) boundary condition. We derive an asymptotic expansion of the number of eigenvalues below the essential spectrum (Weyl-type asymptotics). The methods of proof relies on results concerning the asymptotic behavior of the first eigenvalue obtained in a previous work [A. Kachmar, J. Math. Phys. Vol. 47 (7) 072106 (2006)].

The resonance counting function for Schrödinger operators with generic potentials

We show that the resonance counting function for a Schrödinger operator has maximal order of growth for generic sets of real-valued, or complex-valued, $L^\infty$-compactly supported potentials.

Resolvent estimates and resonance free domains for Schrödinger operators with matrix-valued potentials

We establish semiclassical resolvent estimates for Schrödinger operators with long-range matrix-valued potentials. As an application we prove resonance free domains both in trapping and non-trapping situations. Our results generalize the well-known results of \cite{Burq1, Martinez1} in the case of scalar Schrödinger operators.

On real resonances for three-dimensional Schrödinger operators with point interactions

We prove the absence of positive real resonances for Schrödinger operators with finitely many point interactions in $\mathbb{R}^3$ and we discuss such a property from the perspective of dispersive and scattering features of the associated Schrödinger propagator.

周期图上薛定谔算子的渐近等谱性

Asymptotic isospectrality of Schrödinger operators on periodic graphs

本文研究周期图上离散薛定谔算子在周期添加长边扰动下的谱性质，证明当新增边足够长时，扰动图与某个更高维无长边周期图渐近等谱，并给出恰好等谱的判据及谱带端点的渐近公式。

Semiclassical resonances for a two-level Schrödinger operator with a conical intersection

We study the resonant set of a two-level Schrödinger operator with a linear conical intersection. This model operator can be decomposed into a direct sum of first order systems on the real half-line. For these ordinary differential systems we locally construct exact WKB solutions, which are connected to global solutions, amongst which are resonant states. The main results are a generalized Bohr-Sommerfeld quantization condition and an asymptotic description of the set of resonances as a distorted lattice.

具有矩阵值复方势阱的一维 Schrödinger 算子的共振

Resonances for the one dimensional Schrödinger operator with the matrix-valued complex square-well potential

针对矩阵值复方势阱下的一维非自伴 Schrödinger 算子，本文通过显式 Jost 函数与复超越方程刻画共振，并给出其复平面分布、重数及数量的 Weyl 定律。

Lower bounds for resonance counting functions for Schrödinger operators with fixed sign potentials in even dimensions

If the dimension $d$ is even, the resonances of the Schrödinger operator $-Δ+V$ on ${\mathbb R}^d$ with $V$ bounded and compactly supported are points on $Λ$, the logarithmic cover of ${\mathbb C} \setminus \{0\}$. We show that for fixed sign potentials $V$ and for nonzero integers $m$, the resonance counting function for the $m$th sheet of $Λ$ has maximal order of growth.

Schrödinger operators with complex-valued potentials and no resonances

In dimension $d\geq 3$, we give examples of nontrivial, compactly supported, complex-valued potentials such that the associated Schrödinger operators have no resonances. If $d=2$, we show that there are potentials with no resonances away from the origin. These Schrödinger operators are isophasal and have the same scattering phase as the Laplacian on $\Real^d$. In odd dimensions $d\geq 3$ we study the fundamental solution of the wave equation perturbed by such a potential. If the space variables are held fixed, it is super-exponentially decaying in time.

Schrodinger operators and the distribution of resonances in sectors

The purpose of this paper is to give some refined results about the distribution of resonances in potential scattering. We use techniques and results from one and several complex variables, including properties of functions of completely regular growth. This enables us to find asymptotics for the distribution of resonances in sectors for certain potentials and for certain families of potentials.

Resonance free domain for a system of Schrödinger operators with energy-level crossings

We consider a $2\times 2$ system of 1D semiclassical differential operators with two Schrödinger operators in the diagonal part and small interactions of order $h^ν$ in the off-diagonal part, where $h$ is a semiclassical parameter and $ν$ is a constant larger than $1/2$. We study the absence of resonance near a non-trapping energy for both Schrödinger operators in the presence of crossings of their potentials. The width of resonances is estimated from below by $Mh\log(1/h)$ and the coefficient $M$ is given in terms of the directed cycles of the generalized bicharacteristics induced by two Hami…

离散 Schrödinger 算子共振的连续极限

Continuum limit of resonances for discrete Schrödinger operators

本文证明离散 Schrödinger 算子的复共振在连续极限下收敛到连续算子的共振，并借助推广的范数预解收敛与 Fourier 空间复畸变方法，为奇异势情形提供近似离散算子构造方案。

一维 Schrödinger 算子共振态的奇异性与渐近分布

Singularities and asymptotic distribution of resonances for Schrödinger operators in one dimension

研究一维 Schrödinger 算子共振态的高能分布，利用反射系数积分表示给出密度上界、扇形分布结果及多列共振态势的构造与渐近定位。

Asymptotic number of scattering resonances for generic Schrodinger operators

Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d>2 odd. Here V is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a. We show for generic potentials V that the number of resonances of -Delta+V with modulus less than r is approximatively equal to a constant times a^dr^d.

Resonance asymptotics for Schrodinger operators on hyperbolic space

We study the asymptotic distribution of resonances for scattering by compactly supported potentials in hyperbolic space. We first establish an upper bound for the resonance counting function that depends only on the dimension and the support of the potential. We then establish the sharpness of this estimate by proving the a Weyl law for the resonance counting function holds in the case of radial potentials vanishing to some finite order at the edge of the support. As an application of the existence of potentials that saturate the upper bound, we derive additional resonance asymptotics that hol…

Asymptotics of spectral quantities of Schrödinger operators

In this paper we provide new asymptotic estimates of the Floquet exponents of Schrödinger operators on the circle. By the same techniques, known asymptotic estimates of various others spectral quantities are improved.

Sharp Spectral Asymptotics for Magnetic Schrödinger Operator with Irregular Potential

Survey: In this paper I consider sharp spectral asymptotics for multidimensional magnetic Schrödinger operator with irregular coefficients with respect to two parameters -- semiclassical parameter $h$ and coupling parameter $μ$. There are few principally different cases, depending on dimension, rank of magnetic intensity matrix, relation between $h$ and $μ$ and some extra assumptions.

Large time behavior of solutions to Schrödinger equation with complex-valued potential

We study the large-time behavior of the solutions to the Schrödinger equation associated with a quickly decaying potential in dimension three. We establish the resolvent expansions at threshold zero and near positive resonances. The large-time expansions of solutions are obtained under different conditions, including the existence of positive resonances and zero resonance or/and zero eigenvalue.

Asymptotic expansion of the integrated density of states of a two-dimensional periodic Schrodinger operator

We prove the complete asymptotic expansion of the integrated density of states of a two-dimensional Schrodinger operator with a smooth periodic potential

Generic asymptotics of resonance counting function for Schrödinger point interactions

We study the leading coefficient in the asymptotical formula $ N (R) = \frac{W}π R + O (1) $, $ R \to \infty $, for the resonance counting function $ N (R)$ of Schrödinger Hamiltonians with point interactions. For such Hamiltonians, the Weyl-type and non-Weyl-type asymptotics of $N (R)$ was introduced recently in a paper by J. Lipovský and V. Lotoreichik (2017). In this note, we prove that the Weyl-type asymptotics is generic.

Asymptotics of the Weyl Function for Schrödinger Operators with Measure-Valued Potentials

We derive an asymptotic expansion for the Weyl function of a one-dimensional Schrödinger operator which generalizes the classical formula by Atkinson. Moreover, we show that the asymptotic formula can also be interpreted in the sense of distributions.

渐近双曲流形上薛定谔算子负特征值的分布

The distribution of negative eigenvalues of Schrödinger operators on asymptotically hyperbolic manifolds

本文研究渐近双曲流形上薛定谔算子负特征值的计数函数渐近性，给出势条件判定有限或无限个负特征值，并在无限情形下得到区间$(-\infty,-E)$内特征值计数当$E\to 0$时的渐近行为。

Eigenvalue asymptotics for Schrödinger operators on sparse graphs

We consider Schrödinger operators on sparse graphs. The geometric definition of sparseness turn out to be equivalent to a functional inequality for the Laplacian. In consequence, sparseness has in turn strong spectral and functional analytic consequences. Specifically, one consequence is that it allows to completely describe the form domain. Moreover, as another consequence it leads to a characterization for discreteness of the spectrum. In this case we determine the first order of the corresponding eigenvalue asymptotics.

Asymptotic behavior of eigenvalues of Schrödinger type operators with degenerate kinetic energy

We study the eigenvalues of Schrödinger type operators $T + λV$ and their asymptotic behavior in the small coupling limit $λ\to 0$, in the case where the symbol of the kinetic energy, $T(p)$, strongly degenerates on a non-trivial manifold of codimension one.

Sharp spectral asymptotics for four-dimensional Schrödinger operator with a strong degenerating magnetic field

I continue analysis of the Schrödinger operator with the strong degenerating magnetic field, started in \cite{IRO6}. Now I consider 4-dimensional case, assuming that magnetic field is generic degenerated and under certain conditions I derive spectral asymptotics with the principal part $\asymp h^{-4}$ and the remainder estimate $O(μ^{-1/2}h^{-3})$ where $μ\gg 1$ is the intensity of the field and $h\ll 1$ is the Plank constant; $μh\le 1$. These asymptotics can contain correction terms of magnitude $μ^{5/4}h^{-3/2}$ corresponding to the short periodic trajectories.

Sharp Spectral Asymptotics for 2-dimensional Schrödinger operator with a strong magnetic field. Note about forgotten generic case

I consider magnetic Schrödinger operator in dimension $d=2$ assuming that coefficients are smooth and magnetic field is non-degenerating. Then I extend the remainder estimate $O(μ^{-1}h^{-1}+1)$ derived in \cite{Ivr1} for the case when $V/F$ has no stationary points to the case when it has non-degenerating stationary points. If some of them are saddles and $μ^3h\ge 2$ then asymptotics contains correction terms of magnitude $μ^{-1}h^{-1}|\log μ^3 h|$.

Asymptotic behavior of subfunctions of the stationary Schrodinger operator

Subharmonic functions associated with the stationary Schrodinger operator are its weak subsolutions under appropriate assumptions on the potential of the operator. We prove for these functions analogs of several classical results on analytic and subharmonic functions, such as the Phragmen-Lindelof theorem with the precise growth rate, the Blaschke theorem on bounded analytic functions, the Carleman formula, the Hayman-Azarin theorem on asymptotic behavior of special subharmonic functions in many-dimensional cones.

Heat content and small time asymptotics for Schrödinger operators on $R^d$

This paper studies the heat content} for Schrödinger operators of the fractional Laplacian $(-Δ)^{α/2}$, $0<α\leq 2$ in $R^d$, $d\geq 1$. Employing probabilistic and analytic techniques, a small time asymptotic expansion formula is given and the "heat content invariants" are identified. These results are new even in the case of the Laplacian, $α=2$.

Resonances and Spectral Shift Function for a Magnetic SCHRÖdinger Operator

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+…

Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic Schrodinger operators

We prove the complete asymptotic expansion of the integrated density of states of a Schrodinger operator H = -Δ+ b acting in R^d when the potential b is either smooth periodic, or generic quasi-periodic (finite linear combination of exponentials), or belongs to a wide class of almost-periodic functions.

Spectral Asymptotics for Perturbed Spherical Schrödinger Operators and Applications to Quantum Scattering

We find the high energy asymptotics for the singular Weyl--Titchmarsh m-functions and the associated spectral measures of perturbed spherical Schrödinger operators (also known as Bessel operators). We apply this result to establish an improved local Borg-Marchenko theorem for Bessel operators as well as uniqueness theorems for the radial quantum scattering problem with nontrivial angular momentum.

Semiclassical asymptotics for a class of singular Schrödinger operators

Let $Ω\subset \mathbb{R}^d$ be bounded with $C^1$ boundary. In this paper we consider Schrödinger operators $-Δ+ W$ on $Ω$ with $W(x)\approx\mathrm{dist}(x, \partialΩ)^{-2}$ as $\mathrm{dist}(x, \partialΩ)\to 0$. Under weak assumptions on $W$ we derive a two-term asymptotic formula for the sum of the eigenvalues of such operators.

复特征值与共振的界

Bounds on complex eigenvalues and resonances

本文给出了复势非自伴薛定谔算子复特征值与自伴薛定谔算子复共振的界，并与数值结果比较，证明其有效性。

非自伴多通道离散薛定谔算子的共振簇

Clusters of resonances for a non-selfadjoint multichannel discrete Schrödinger operator

针对非自伴多通道离散薛定谔算子，研究其在未受扰算子谱阈值附近的共振分布，通过分析紧扰动项，计算共振精确数量并精确描述其在复平面特殊点周围的簇状分布位置。

The Lyapunov function for Schrödinger operators with a periodic 2x2 matrix potential

We consider the Schrödinger operator on the real line with a 2x2 matrix valued 1-periodic potential. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define a Lyapunov function which is analytic on a two sheeted 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 the existence of real as well as non-real resonances for specific potentials. We determine the asymptotics of the periodic and anti-periodic spectrum and of the resonances …

Spectral shift function and Resonances near the low ground state for Pauli and Schrödinger operators

We study the spectral shift function (SSF) $ξ(λ)$ and the resonances of the operator $H_V := \big( σ\cdot (-i\nabla - \textbf{A}) \big)^{2} + V$ in $L^2(\mathbb{R}^3)$ near the origin. Here $σ:= (σ_1,σ_2,σ_3)$ are the $2 \times 2$ Pauli matrices and $V$ is a hermitian potential decaying exponentially in the direction of the magnetic field $\textbf{B} := \text{curl} \hspace{0.6mm} \textbf{A}$. We give a representation of the derivative of the SSF as a sum of the imaginary part of a holomorphic function and a harmonic measure related to the resonances of $H_V$. This representation warrant the Br…

无限圆柱及其他乘积空间上 Schrödinger 算子的共振

Resonances for Schrödinger operators on infinite cylinders and other products

本文研究 $\mathbb{R}^d\times\mathbb{S}^1$ 上紧支位势 Schrödinger 算子的高能量共振，证明其可由圆平均位势及 $\mathbb{R}^d$ 上的共振刻画，并给出局域化、预解逼近与 $d=1$ 时的渐近和刚性结果。

Semiclassical asymptotics and gaps in the spectra of periodic Schrödinger operators with magnetic wells

We show that, under some very weak assumption of effective variation for the magnetic field, a periodic Schrödinger operator with magnetic wells on a noncompact Riemannian manifold $M$ such that $H^1(M, \R)=0$ equipped with a properly disconnected, cocompact action of a finitely generated, discrete group of isometries has an arbitrarily large number of spectral gaps in the semi-classical limit.

阈值处三维磁性Schrödinger算子的修正Fermi黄金定则

A modified Fermi Golden Rule at threshold for 3D magnetic Schrödinger operators

本文研究了三维Schrödinger算子在阈值特征值处引入小磁场后转变为共振的现象，求得了共振虚部渐近展开的主项，并揭示了其与弱电场诱导共振的主要差异。

Singular Schrödinger operators and Robin billiards. Spectral properties and asymptotic expansions

This paper summarizes the contents of a plenary talk at the Pan African Congress of Mathematics held in Rabat in July 2017. We provide a survey of recent results on spectral properties of Schrödinger operators with singular interactions supported by manifolds of codimension one and of Robin billiards with the focus on the geometrically induced discrete spectrum and its asymptotic expansions in term of the model parameters.

无完全正则性条件下 $\mathbb{R}^d$ 上局部磁 Schrödinger 算子的锐利半经典谱渐近

Sharp semiclassical spectral asymptotics for local magnetic Schrödinger operators on $\mathbb{R}^d$ without full regularity

本文研究 $d\geq3$ 时局部磁 Schrödinger 算子，在磁势光滑、电势五阶可微且五阶导数 Hölder 连续的较弱条件下，证明局部化计数函数和 Riesz 平均的锐利半经典谱渐近。

Asymptotics for Christoffel functions associated to continuum Schrödinger operators

We prove asymptotics of the Christoffel function, $λ_L(ξ)$, of a continuum Schrödinger operator for points in the interior of the essential spectrum under some mild conditions on the spectral measure. It is shown that $Lλ_L(ξ)$ has a limit and that this limit is given by the Radon--Nikodym derivative of the spectral measure with respect to the Martin measure. Combining this with a recently developed local criterion for universality limits at scale $λ_L(ξ)$, we compute universality limits for continuum Schrödinger operators at scale $L$ and obtain clock spacing of the eigenvalues of the finite …

Schrödinger operators on zigzag graphs

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…

Semiclassical spectral asymptotics for a magnetic Schrödinger operator with non-vanishing magnetic field

We consider a magnetic Schrödinger operator $H^h$, depending on a semiclassical parameter $h>0$, on a compact Riemannian manifold. We assume that there is no electric field. We suppose that the minimal value $b_0$ of the intensity of the magnetic field $b$ is strictly positive. We give a survey of the results on asymptotic behavior of the eigenvalues of the operator $H^h$ in the semiclassical limit.

径向薛定谔算子的逆 Steklov 与 Calderón 问题中的局部 Hölder 稳定性及量化共振

Local Hölder Stability in the Inverse Steklov and Calderón Problems for Radial Schrödinger operators and Quantified Resonances

本文针对单位球上特殊 $L^2$ 径向势的薛定谔算子，利用 Steklov 频谱差的 Laplace 变换公式与矩稳定性估计，获得了逆 Steklov 与 Calderón 问题的局部 Hölder 稳定性，改进了先前的对数稳定性结果。