Related papers: Trace formulas for Schr\"odinger operators on the …

We show how a matrix version of the Buslaev-Faddeev-Zakharov trace formulae for a one-dimensional Schr\"odinger operator leads to Lieb-Thirring inequalities with sharp constants $L^{cl}_{\gamma,d}$ with $\gamma\ge 3/2$ and arbitrary $d\ge…

We show that for a one-dimensional Schr\"odinger 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…

We consider the operators $H_0=M_0^{-1}(x) P(D)$ and $H =M^{-1} (x) P(D)$ where $M_0 (x)$ and $M (x)$ are positively definite bounded matrix-valued functions and $P(D)$ is an elliptic differential operator. Our main result is that the wave…

In this article, we consider the nonlinear Schr\"odinger equation on the cylinder $\mathbb{R}^d\times \mathbb{T}$. In the long range case, we show there is no linear scattering state of the nonlinear Schr\"odinger equation on $\mathbb{R}^d…

Let $\Sigma\subset\mathbb{R}^d$ be a $C^\infty$-smooth closed compact hypersurface, which splits the Euclidean space $\mathbb{R}^d$ into two domains $\Omega_\pm$. In this note self-adjoint Schr\"odinger operators with $\delta$ and…

Superoscillations are a phenomenon in physics, where linear combinations of low-frequency plane waves interfere almost destructively in such a way that the resulting wave has a higher frequency than any of the individual waves. The…

We consider the spectrum of the discrete Schr\"odinger equation with one-dimensional perturbation. We obtain the explicit form of scattering matrix and find the exact condition of absence of singular part of the spectrum. We calculated also…

We investigate the soliton dynamics for the fractional nonlinear Schrodinger equation by a suitable modulational inequality. In the semiclassical limit, the solution concentrates along a trajectory determined by a Newtonian equation…

We consider decaying oscillatory perturbations of periodic Schr\"odinger operators on the half line. More precisely, the perturbations we study satisfy a generalized bounded variation condition at infinity and an $L^p$ decay condition. We…

In this paper, we consider the existence and the asymptotic completeness of the wave operators for Schrodinger equations with time-dependent potentials which are short-range in space.

The inherently homogeneous stationary-state and time-dependent Schroedinger equations are often recast into inhomogeneous form in order to resolve their solution nonuniqueness. The inhomogeneous term can impose an initial condition or, for…

In this paper, we study a Schr\"odinger-type equation featuring a derivative in the nonlinear term and incorporating diffusion effects. This type of equation arises in various physical applications, such as modeling low-order magnetization…

We prove an error estimate for a Lie-Trotter splitting operator associated to the Schrodinger-Poisson equation in the semiclassical regime, when the WKB approximation is valid. In finite time, and so long as the solution to a compressible…

We describe the application of the quantum mechanical bootstrap to the solution of one-dimensional scattering problems. By fixing a boundary and modulating the Robin parameter of the boundary conditions we are able to extract the reflection…

We investigate the large time behavior of the solutions to the nonlinear focusing Schr\"odinger equation with a time-dependent damping in the energy sub-critical regime. Under non classical assumptions on the unsteady damping term, we prove…

In the paper, we experimentally study the inverse problem with the resonant scattering determinant. We analyze the structure of characteristics of perturbed linear waves. Assuming there is the common part of potential perturbation…

The linear system of differential equations for determination of transmission and reflection amplytudes of scattered electron in the field of one dimensional arbitrary potential is obtained. It is shown that in general the scattering…

This paper establishes trace-formulae for a class of operators defined in terms of the functional calculus for the Laplace operator on divergence-free vector fields with relative and absolute boundary conditions on Lipschitz domains in…

In this paper, we investigate the existence and characterizations of the Fr\'echet derivatives of the solution to time-harmonic elastic scattering problems with respect to the boundary of the obstacle. Our analysis is based on a technique -…

The inverse scattering problem for the Schr$\mathrm{\ddot{o}}$dinger operators on the line is considered when the potential is real valued and integrable and has a finite first moment. It is shown that the potential on the line is uniquely…