Related papers: Resolvent bounds for repulsive potentials
We investigate quantitative (or effective) versions of the limiting absorption principle, for the Schr\"odinger operator on asymptotically conic manifolds with short-range potentials, and in particular consider estimates of the form $$ \|…
For spherically symmetric repulsive Hamiltonians we prove the Besov bound, the radiation condition bounds and the limiting absorption principle. The Sommerfeld uniqueness result also follows as a corollary of these. In particular, the…
We study the asymptotic behaviour of solutions to semi-classical nonlinear Schrodinger equations with a potential, for concentrating and oscillating initial data, when the nonlinearity is repulsive and the potential is a polynomial of…
It is well known that the resolvent of the free Schr\"odinger operator on weighted $L^2$ spaces has norm decaying like $\lambda^{-\frac{1}{2}}$ at energy $\lambda$. There are several works proving analogous high-frequency estimates for…
We prove some local smoothing estimates for the Schr\"{o}dinger initial value problem with data in $L^2(\mathbb{R}^d)$, $d \geq 2$ and a general class of potentials. In the repulsive setting we have to assume just a power like decay…
We obtain a new bound on the location of eigenvalues for a non-self-adjoint Schr\"odinger operator with complex-valued potentials by obtaining a weighted $L^2$ estimate for the resolvent of the Laplacian.
On a simple model $V(x,y)=A\,x^2+B\,y^2+C\,x^2y^2+D\,(x^2y^4+x^4y^2)$ we demonstrate that even in a classically repulsive regime (i.e., at couplings which make the potential decreasing to $-\infty$ in some directions) quantum mechanics may…
We consider the cubic nonlinear Schr\"odinger equation with an exceptional potential. We obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions. We prove the existence of…
A revision of the recursive method proposed by S.A. Shakir [Am. J.Phys. \textbf{52}, 845 (1984)] to solve bound eigenvalues of the Schr\"odinger equation is presented. Equations are further simplified and generalized for computing wave…
In this paper we consider magnetic Schr\"odinger operators in R^n, n \ge 3. Under almost optimal conditions on the potentials in terms of decay and regularity we prove smoothing and Strichartz estimates, as well as a limiting absorption…
We study the asymptotics of the Schr\"odinger equation with time-dependent potential in dimension one. Assuming that the potential decays sufficiently rapidly as $|x| \to \infty$, we prove that the solution can be written as the sum of a…
In this note we give explicit solutions to the wave equation associated to the Schr\"odinger operator with three-inverse square potential on $R_+^3$
We relax the regularity condition on potentials of the Schr\"odinger equation in uniqueness results on the inverse boundary value problem which were recently proved in [11] and [5].
We adapt the arguments in the recent work of Duyckaerts, Landoulsi, and Roudenko to establish a scattering result at the sharp threshold for the $3d$ focusing cubic NLS with a repulsive potential. We treat both the case of short-range…
We study the long-time behavior of solutions to the Schr\"odinger equation with a repulsive Coulomb potential on $\mathbb{R}^3$ for spherically symmetric initial data. Our approach involves computing the distorted Fourier transform of the…
We derive analytic expressions of the recursive solutions to the Schr\"{o}dinger's equation by means of a cutoff potential technique for one-dimensional piecewise constant potentials. These solutions provide a method for accurately…
We consider repulsive potential energies $\E_q(\Sigma)$, whose integrand measures tangent-point interactions, on a large class of non-smooth $m$-dimensional sets $\Sigma$ in $\R^n.$ Finiteness of the energy $\E_q(\Sigma)$ has three sorts of…
Approximate analytical bound state solutions of the radial Schr\"odinger equation are studied for a two-term diatomic molecular potential in terms of the hypergeometric functions for the cases where $q\geq1$ and $q=0$. The energy…
We consider eigenvalue sums of Schr\"odinger operators $-\Delta+V$ on $L^2(\R^d)$ with complex radial potentials $V\in L^q(\R^d)$, $q<d$. We prove quantitative bounds on the distribution of the eigenvlaues in terms of the $L^q$ norm of $V$.…
We find positive non-radial solutions for a system of Schr\"odinger equations in a weak fully attractive or repulsive regime in presence of an external radial trapping potential that exhibits a maximum or a minimum at infinity.