Related papers: Zero energy asymptotics of the resolvent for a cla…
We prove unique continuation properties for linear variable coefficient Schr\"odinger equations with bounded real potentials. Under certain smallness conditions on the leading coefficients, we prove that solutions decaying faster than any…
We consider discrete one-dimensional Schroedinger operators whose potentials decay asymptotically like an inverse square. In the super-critical case, where there are infinitely many discrete eigenvalues, we compute precise asymptotics of…
Consider a Schr\"odinger operator on an asymptotically Euclidean manifold $X$ of dimension at least two, and suppose that the potential is of attractive Coulomb-like type. Using Vasy's second 2nd-microlocal approach, "the Lagrangian…
We study solutions to the linear wave equation on the cosmological region of Schwarzschild-de Sitter spacetimes. We show that all sufficiently regular finite-energy solutions to the linear equation possess a particular finite-order…
We consider the long time asymptotics of (not necessarily small) odd solutions to the nonlinear Schr\"odinger equation with semi-linear and nonlocal Hartree nonlinearities, in one dimension of space. We assume data in the energy space…
We consider a discrete Schroedinger operator whose potential is the sum of a Wigner-von Neumann term and a summable term. The essential spectrum of this operator equals to the interval [-2,2]. Inside this interval, there are two critical…
We show absence of positive eigenvalues for generalized 2-body hard- core Schroedinger operators under the condition of bounded strictly convex obstacles. A scheme for showing absence of positive eigenvalues for generalized N -body…
On this short note, we apply the Mourre theory of the limiting absorption with {\it difference} type conditions on the potential, instead of conditions on the derivatives. In order that we modify the definition of the conjugate operator,…
For Schr\"odinger operators with potentials that are asymptotically homogeneous of degree $-2$, the size of the coupling determines whether it has finite or infinitely many negative eigenvalues. In the latter case the asymptotic…
We prove that if a solution of the discrete time-dependent Schr\"odinger equation with bounded real potential decays fast at two distinct times then the solution is trivial. For the free Shr\"odinger operator and for operators with…
We study the asymptotic behavior of the counting function of negative eigenvalues of Schr\"odinger operators with real valued potentials on asymptotically hyperbolic manifolds. We establish conditions on the potential that determine if…
Recent work by Hintz--Vasy provides a partial asymptotic analysis of the low-energy limit of scattering for Schr\"odinger operators with a short-range potential. Using a slight refinement of Hintz's algorithm, we complete the asymptotic…
In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schr\"odinger equation with indefinite potential \begin{equation*} \left\{\begin{array}{l} i \partial_t u+\Delta u-V(x)u…
We prove necessary and sufficient conditions for lattice Schr\"{o}dinger operators to have a zero energy bound state in arbitrary dimension. The two criteria are sharp, complementary, and depend crucially on both the dimension and…
Using the recent analysis of the output of the low-energy resolvent of Schr\"odinger operators on asymptotically conic manifolds (including Euclidean space) when the potential is short-range, we produce detailed asymptotic expansions for…
The zero range potential is constructed for a system of two particles interacting via the Coulomb potential. The singular part of the asymptote of the wave function at the origin which is caused by the common effect of the zero range…
We present a conditionally exactly solvable singular potential for the one-dimensional Schr\"odinger equation which involves the exactly solvable inverse square root potential. Each of the two fundamental solutions that compose the general…
Commutator methods are applied to get limiting absorption principles for the discrete standard and Molchanov-Vainberg Schr\"odinger operators $H_{\mathrm{std}}= \Delta+V$ and $H_{\mathrm{MV}} = D+V$ on $\ell^2(\mathbb{Z}^d)$, with emphasis…
We prove necessary and sufficient conditions for the Schr\"odinger operators to have zero-energy bound states at the threshold of the essential spectrum such that they have bounded $k$-th moment. This result is the extension of the results…
We prove low frequency resolvent estimates and local energy decay for the Schr{\"o}dinger equation in an asymptotically Euclidean setting. More precisely, we go beyond the optimal estimates by comparing the resolvent of the perturbed…