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.
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.
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.
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.
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.
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.
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…
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…
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.
We study L-system realizations of the original Weyl-Titchmarsh functions $(-m_α(z))$. In the case when the minimal symmetric Schrödinger operator is non-negative, we describe the Schrödinger L-systems that realize inverse Stieltjes functions $(-m_α(z))$. This approach allows to derive a necessary and sufficient conditions for the functions $(-m_α(z))$ to be inverse Stieltjes. In particular, the criteria when $(-m_\infty(z))$ is an inverse Stieltjes function is provided. Moreover, the value $m_\infty(-0)$ and parameter $α$ allow us to describe the geometric structure of the realizing $(-m_α(z))…
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.
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 …
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 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…
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 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…
Using a unified approach employing a homogeneous Lippmann-Schwinger-type equation satisfied by resonance functions and basic facts on Riesz potentials, we discuss the absence of threshold resonances for Dirac and Schrodinger operators with sufficiently short-range interactions in general space dimensions. More specifically, assuming a sufficient power law decay of potentials, we derive the absence of zero-energy resonances for massless Dirac operators in space dimensions $n \geq 3$, the absence of resonances at $\pm m$ for massive Dirac operators (with mass $m > 0$) in dimensions $n \geq 5$, a…
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 all 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. In example we show the existence of real and co…
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 prove that the resonances of long range perturbations of the (semiclassical) Laplacian are the zeroes of natural perturbation determinants. We more precisely obtain factorizations of these determinants of the form $ \prod_{w = {\rm resonances}}(z-w) \exp (φ_p(z,h)) $ and give semiclassical bounds on $ \partial_z φ_p $ as well as a representation of Koplienko's regularized spectral shift function. Here the index $ p \geq 1 $ depends on the decay rate at infinity of the perturbation.
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.
We discover that the distribution of (frequency and phase) resonances plays a role in determining the spectral type of supercritical quasi-periodic Schrödinger operators. In particular, we disprove the second spectral transition line conjecture of Jitomirskaya in the early 1990s.
We prove that a compactly supported perturbation of a rational or simply periodic algebro-geometric potential of the one-dimensional Schrödinger equation on the half line is uniquely determined by the location of its Dirichlet eigenvalues and resonances.
This paper studies the resonances of Schrödinger operators with bounded, compactly supported, real-valued potentials on d-dimensional Euclidean space, where d is even. If the potential V is non-trivial and d is not 4 then the meromorphic continuation of the resolvent of the Schrödinger operator has infinitely many poles, with a quantitative lower bound on their density. A somewhat weaker statement holds if d =4. We prove several inverse-type results. If the meromorphic continuations of the resolvents of two Schrödinger operators $-Δ+V_1$ and $-Δ+V_2$ have the same poles, with both potentials b…
We consider the Schrödinger operator on the real line with a $N\ts N$ matrix valued periodic potential, N>1. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define the Lyapunov function, which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. The Lyapunov function has (real or complex) branch points, which we call resonances. We determine the asymptotics of the periodic, anti-periodic spectrum and of the resonances at high en…
We turn back to the well known problem of interpretation of the Schrodinger operator with the pseudopotential being the first derivative of the Dirac function. We show that the problem in its conventional formulation contains hidden parameters and the choice of the proper selfadjoint operator is ambiguously determined. We study the asymptotic behavior of spectra and eigenvectors of the Hamiltonians with increasing smooth potentials perturbed by short-range potentials. Appropriate solvable models are constructed and the corresponding approximation theorems are proved. We introduce the concepts …
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 prove that radial, monotonic, superexponentially decaying potentials in R^n, n greater than or equal to 1 odd, are determined by the resonances of the associated semiclassical Schrodinger operator among all superexponentially decaying potentials in R^n.
We construct a semiclassical Schrödinger operator such that the imaginary part of its resonances closest to the real axis changes by a term of size $h$ when a real compactly supported potential of size $o ( h )$ is added.
We show that for a one-dimensional Schrödinger operator with a potential whose first moment is integrable the scattering matrix is in the unital Wiener algebra of functions with integrable Fourier transforms. Then we use this to derive dispersion estimates for solutions of the associated Schrödinger and Klein-Gordon equations. In particular, we remove the additional decay conditions in the case where a resonance is present at the edge of the continuous spectrum.
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 prove compactness of a restricted set of real-valued, compactly supported potentials $V$ for which the corresponding Schrödinger operators $H_V$ have the same resonances, including multiplicities. More specifically, let $B_R(0)$ be the ball of radius $R > 0$ about the origin in $R^d$, for $d=1,3$. Let $\mathcal{I}_R (V_0)$ be the set of real-valued potentials in $C_0^\infty( \overline{B}_R(0); R)$ so that the corresponding Schrödinger operators have the same resonances, including multiplicities, as $H_{V_0}$. We prove that the set $\mathcal{I}_R (V_0)$ is a compact subset of $C_0^\infty (\o…
We prove a sharp Hörmander multiplier theorem for Schrödinger operators $H=-Δ+V$ on $\mathbb{R}^n$. The result is obtained under certain condition on a weighted $L^\infty$ estimate, coupled with a weighted $L^2$ estimate for $H$, which is a weaker condition than that for nonnegative operators via the heat kernel approach. Our approach is elaborated in one dimension with potential $V$ belonging to certain critical weighted $L^1$ class. Namely, we assume that $\int (1+|x|) |V(x)|dx$ is finite and $H$ has no resonance at zero. In the resonance case we assume $\int (1+|x|^2) |V(x)| dx$ is finite.
For a selfadjoint Schrödinger operator on the half line with a real-valued, integrable, and compactly-supported potential, it is investigated whether the boundary parameter at the origin and the potential can uniquely be determined by the scattering matrix or by the absolute value of the Jost function known at positive energies, without having the bound-state information. It is proved that, except in one special case where the scattering matrix has no bound states and its value is $+1$ at zero energy, the determination by the scattering matrix is unique. In the special case, it is shown that t…
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 solutions of the eigenvalue equation at zero energy for a class of non-local Schrödinger operators with potentials decreasing to zero at infinity. Using a path integral approach, we obtain detailed results on the spatial decay of both $L^2$ and resonance solutions at infinity. We highlight the interplay of the kinetic term and the potential in these decay behaviours, and identify the decay mechanisms resulting from specific balances of global lifetimes with or without the potential.
