共查询到20条相似文献,搜索用时 15 毫秒
1.
We study discrete Schrödinger operators with compactly supported potentials on Z d . Constructing spectral representations and representing S-matrices by the generalized eigenfunctions, we show that the potential is uniquely reconstructed from the S-matrix of all energies. We also study the spectral shift function \({\xi(\lambda)}\) for the trace class potentials, and estimate the discrete spectrum in terms of the moments of \({\xi(\lambda)}\) and the potential. 相似文献
2.
Jonathan Eckhardt 《Complex Analysis and Operator Theory》2014,8(1):37-50
We utilize the theory of de Branges spaces to show when certain Schrödinger operators with strongly singular potentials are uniquely determined by their associated spectral measure. The results are applied to obtain an inverse uniqueness theorem for perturbed spherical Schrödinger operators. 相似文献
3.
Hironobu Sasaki 《偏微分方程通讯》2013,38(7):1175-1197
We study the inverse scattering problem for the three dimensional nonlinear Schrödinger equation with the Yukawa potential. The nonlinearity of the equation is nonlocal. We reconstruct the potential and the nonlinearity by the knowledge of the scattering states. Our result is applicable to reconstructing the nonlinearity of the semi-relativistic Hartree equation. 相似文献
4.
Valeria Banica Rémi Carles Gigliola Staffilani 《Geometric And Functional Analysis》2008,18(2):367-399
We study the long-time behavior of radial solutions to nonlinear Schr?dinger equations on hyperbolic space. We show that the
usual distinction between short-range and long-range nonlinearity is modified: the geometry of the hyperbolic space makes
every power-like nonlinearity short range. The proofs rely on weighted Strichartz estimates, which imply Strichartz estimates
for a broader family of admissible pairs, and on Morawetz-type inequalities. The latter are established without symmetry assumptions.
Received: July 2006, Revision: April 2007, Accepted: April 2007 相似文献
5.
We propose a modification of the standard inverse scattering transform for the focusing nonlinear Schrödinger equation (also other equations by natural generalization) formulated with nonzero boundary conditions at infinity. The purpose is to deal with arbitrary-order poles and potentially severe spectral singularities in a simple and unified way. As an application, we use the modified transform to place the Peregrine solution and related higher-order “rogue wave” solutions in an inverse-scattering context for the first time. This allows one to directly study properties of these solutions such as their dynamical or structural stability, or their asymptotic behavior in the limit of high order. The modified transform method also allows rogue waves to be generated on top of other structures by elementary Darboux transformations rather than the generalized Darboux transformations in the literature or other related limit processes. © 2019 Wiley Periodicals, Inc. 相似文献
6.
This is the second in a series of papers on scattering theory for one-dimensional Schrödinger operators with Miura potentials admitting a Riccati representation of the form q = u′ + u 2 for some u ∈ L 2(?). We consider potentials for which there exist ‘left’ and ‘right’ Riccati representatives with prescribed integrability on half-lines. This class includes all Faddeev–Marchenko potentials in L 1(?, (1 + |x|)dx) generating positive Schrödinger operators as well as many distributional potentials with Dirac delta-functions and Coulomb-like singularities. We completely describe the corresponding set of reflection coefficients r and justify the algorithm reconstructing q from r. 相似文献
7.
We propose a new approach for deriving nonlinear evolution equations solvable by the inverse scattering transform. The starting point of this approach is consideration of the evolution equations for the scattering data generated by solutions of an arbitrary nonlinear evolution equation that rapidly decrease as x±. Using this approach, we find all nonlinear evolution equations whose integration reduces to investigation of the scattering-data evolution equations that are differential equations (in either ordinary or partial derivatives). In this case, the evolution equations for the scattering data themselves are linear and moreover solvable in the finite form. 相似文献
8.
We study inverse scattering problems at a fixed energy for radial Schrödinger operators on \({\mathbb{R}^n}\), \({n \geq 2}\). First, we consider the class \({\mathcal{A}}\) of potentials q(r) which can be extended analytically in \({\Re z \geq 0}\) such that \({\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-\rho}}\), \({\rho > \frac{3}{2}}\). If q and \({\tilde{q}}\) are two such potentials and if the corresponding phase shifts \({\delta_l}\) and \({\tilde{\delta}_l}\) are super-exponentially close, then \({q=\tilde{q}}\). Second, we study the class of potentials q(r) which can be split into q(r) = q 1(r) + q 2(r) such that q 1(r) has compact support and \({q_2 (r) \in \mathcal{A}}\). If q and \({\tilde{q}}\) are two such potentials, we show that for any fixed \({a>0, {\delta_l - \tilde{\delta}_l \ = \ o \left(\frac{1}{l^{n-3}}\ \left({\frac{ae}{2l}}\right)^{2l}\right)}}\) when \({l \rightarrow +\infty}\) if and only if \({q(r)=\tilde{q}(r)}\) for almost all \({r \geq a}\). The proofs are close in spirit with the celebrated Borg–Marchenko uniqueness theorem, and rely heavily on the localization of the Regge poles that could be defined as the resonances in the complexified angular momentum plane. We show that for a non-zero super-exponentially decreasing potential, the number of Regge poles is always infinite and moreover, the Regge poles are not contained in any vertical strip in the right-half plane. For potentials with compact support, we are able to give explicitly their asymptotics. At last, for potentials which can be extended analytically in \({\Re z \geq 0}\) with \({\mid q(z)\mid \leq C (1+ \mid z \mid)^{-\rho}}\), \({\rho >1}\), we show that the Regge poles are confined in a vertical strip in the complex plane. 相似文献
9.
We consider the system of three quantum particles (two are bosons and the third is arbitrary) interacting by attractive pair contact potentials on a three-dimensional lattice. The essential spectrum is described. The existence of the Efimov effect is proved in the case where either two or three two-particle subsystems of the three-particle system have virtual levels at the left edge of the three-particle essential spectrum for zero total quasimomentum (K=0). We also show that for small values of the total quasimomentum (K0), the number of bound states is finite. 相似文献
10.
Ukrainian Mathematical Journal - We consider a one-dimensional Schrödinger equation on the entire axis whose potential rapidly decreases on the left-hand side and infinitely increases on the... 相似文献
11.
Using the concept of an intrinsic metric on a locally finite weighted graph, we give sufficient conditions for the magnetic Schrödinger operator to be essentially self-adjoint. The present paper is an extension of some recent results proven in the context of graphs of bounded degree. 相似文献
12.
Applications of Mathematics - A special type of Jacobi matrices, discrete Schrödinger operators, is found to play an important role in quantum physics. In this paper, we show that given the... 相似文献
13.
We consider the Hamiltonian H
(K) of a system consisting of three bosons that interact through attractive pair contact potentials on a three-dimensional integer lattice. We obtain an asymptotic value for the number N(K,z) of eigenvalues of the operator H0(K) lying below z0 with respect to the total quasimomentum K0 and the spectral parameter z–0. 相似文献
14.
Schrödinger Operators on Zigzag Nanotubes 总被引:1,自引:0,他引:1
We consider the Schr?dinger operator with a periodic potential on quasi-1D models of zigzag single-wall carbon 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 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 asymptotics of the periodic and antiperiodic spectrum and of the resonances at high energy. We show that there exist two
types of gaps: i) stable gaps, where the endpoints are periodic and anti-periodic eigenvalues, ii) unstable (resonance) gaps,
where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We describe all finite gap potentials.
We show that the mapping: potential
all eigenvalues is a real analytic isomorphism for some class of potentials.
Submitted: October 5, 2006. Accepted: December 15, 2006. 相似文献
15.
M. J. Ablowitz Bao-Feng Feng Xu-Dan Luo Z. H. Musslimani 《Theoretical and Mathematical Physics》2018,196(3):1241-1267
Nonlocal reverse space–time equations of the nonlinear Schrödinger (NLS) type were recently introduced. They were shown to be integrable infinite-dimensional dynamical systems, and the inverse scattering transform (IST) for rapidly decaying initial conditions was constructed. Here, we present the IST for the reverse space–time NLS equation with nonzero boundary conditions (NZBCs) at infinity. The NZBC problem is more complicated because the branching structure of the associated linear eigenfunctions is complicated. We analyze two cases, which correspond to two different values of the phase at infinity. We discuss special soliton solutions and find explicit one-soliton and two-soliton solutions. We also consider spatially dependent boundary conditions. 相似文献
16.
Haruya Mizutani 《偏微分方程通讯》2013,38(2):169-224
The present article is concerned with Schrödinger equations on non-compact Riemannian manifolds with asymptotically conic ends. It is shown that, for any admissible pair (including the endpoint), local in time Strichartz estimates outside a large compact set are centered at origin hold. Moreover, we prove global in space Strichartz estimates under the nontrapping condition on the metric. 相似文献
17.
We prove existence of modified wave operators for one-dimensional Schrödinger equations with potential in
If in addition the potential is conditionally integrable, then the usual Möller wave operators exist. We also prove asymptotic completeness of these wave operators for some classes of random potentials, and for almost every boundary condition for any given potential. 相似文献
18.
Daniel Lenz Carsten Schubert Peter Stollmann 《Integral Equations and Operator Theory》2008,62(4):541-553
We construct an expansion in generalized eigenfunctions for Schr?dinger operators on metric graphs. We require rather minimal
assumptions concerning the graph structure and the boundary conditions at the vertices.
相似文献
19.
Nikolay Tzvetkov 《偏微分方程通讯》2013,38(1):125-135
We consider the cubic nonlinear Schrödinger equation, posed on ? n × M, where M is a compact Riemannian manifold and n ≥ 2. We prove that under a suitable smallness in Sobolev spaces condition on the data there exists a unique global solution which scatters to a free solution for large times. 相似文献
20.
Let \(x \in \mathbb {R}^{d}\), d ≥ 3, and \(f: \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ i, j ≤ d, let \(a_{ij} : \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated towhere \(J: \mathbb {R}^{d} \times \mathbb {R}^{d} \rightarrow \mathbb {R}\) is a symmetric measurable function. Let \(q: \mathbb {R}^{d} \rightarrow \mathbb {R}.\) We specify assumptions on a, q, and J so that non-negative bounded solutions to satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to \(\mathcal {L}f = 0.\)
相似文献
$$\mathcal{L}f(x) = \frac12 \sum\limits_{i=1}^{d} \sum\limits_{j=1}^{d} \frac{\partial}{\partial x_{i}} \left( a_{ij}(\cdot) \frac{\partial f}{\partial x_{j}}\right)(x) + {\int}_{\mathbb{R}^{d}\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy $$
$$\mathcal{L}f + qf = 0 $$