Related papers: Determining a random Schr\"odinger operator: both …
Quantum scattering by a one-dimensional odd potential proportional to the square of the distance to the origin is considered. The Schr\"odinger equation is solved exactly and explicit algebraic expressions of the wavefunction are given. A…
Spectral components of one-dimensional Schr\"odinger operator with complex potential are investigated. An effective upper bound for the total number of eigenvalues and spectral singularities is established. For dissipative Schr\"odinger…
We review recent and give some new results on the spectral properties of Schroedinger operators with a random potential of alloy type. Our point of interest is the so called Wegner estimate in the case where the single site potentials…
We study the scattering properties of Schr\"{o}dinger operators with potentials that have short-range decay along a collection of rays in $\bbR^d$. This generalizes the classical setting of short-range scattering in which the potential is…
The quantum-mechanical scattering on a compact Riemannian manifold with semi-axes attached to it (hedgehog-shaped manifold) is considered. The complete description of the spectral structure of Schroedinger operators on such a manifold is…
The irregular solutions of the stationary Schr\"odinger equation are important for the fundamental formal development of scattering theory. They are also necessary for the analytical properties of the Green function, which in practice can…
We are concerned with time-dependent inverse source problems in elastodynamics. The source term is supposed to be the product of a spatial function and a temporal function with compact support. We present frequency-domain and time-domain…
In a previous work of 2014 on a quantum system governed by the repulsive Hamiltonian, the author proved uniqueness for short-range interactions described by a scattering operator consisting of regular and singular parts. In this paper, the…
We study half-line Schr\"odinger operators with locally $H^{-1}$ potentials. In the first part, we focus on a general spectral theoretic framework for such operators, including a Last--Simon-type description of the absolutely continuous…
This paper concerns inverse problems for strongly coupled Schr\"odinger equations. The purpose of this inverse problem is to retrieve a stationary potential in the strongly coupled Schr\"odinger equations from either boundary or internal…
We consider the fixed angle inverse scattering problem and show that a compactly supported potential is uniquely determined by its scattering amplitude for two opposite fixed angles. We also show that almost symmetric or horizontally…
We prove upper bounds on the number of resonances and eigenvalues of Schr\"odinger operators $-\Delta+V$ with complex-valued potentials, where $d\geq 3$ is odd. The novel feature of our upper bounds is that they are \emph{effective}, in the…
In this paper, we study the inverse scattering problem for energy-dependent Schr\"{o}dinger equations on the half-line with energy-dependent boundary conditions at the origin. Under certain positivity and very mild regularity assumptions,…
We study localisation effects of strong disorder on the spectral and dynamical properties of (matrix and scalar) Schroedinger operators with non-monotone random potentials, on the d-dimensional lattice. Our results include dynamical…
Mathematically rigorous inversion method is developed to recover compactly supported potentials from the fixed-energy scattering data in three dimensions. Error estimates are given for the solution. An algorithm for inversion of noisy…
We give formulas and equations for finding generalized scattering data for the Schr\"odinger equation in open bounded domain at fixed energy from the impedance boundary map (or Robin-to-Robin map). Combining these results with results of…
We extend methods of Ding and Smart from their breakthrough paper in 2020 which showed Anderson localization for certain random Schr\"odinger operators on $\ell^2(\mathbb{Z}^2)$ via a quantitative unique continuation principle and Wegner…
We study the inverse scattering for Schr{\"o}dinger operators on locally perturbed periodic lattices. We show that the associated scattering matrix is equivalent to the Dirichlet-to-Neumann map for a boundary value problem on a finite part…
A review of the author's results is given. Inversion formulas and stability estimates for the solutions to 3D inverse scattering problems with fixed-energy data are obtained. Inversions of exact and noisy data are stidied. The inverse…
We consider several inverse problems for elliptic equations whose coefficients are random, without imposing a special probabilistic structure on the randomness. The main body treats the Schr\"odinger equation. We compare what can be…