Related papers: Schr\"odinger equation with delta potential in sup…
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 paper, a nonlinear Schr\"odinger equation with an attractive (focusing) delta potential and a repulsive (defocusing) double power nonlinearity in one spatial dimension is considered. It is shown, via explicit construction, that both…
We study the one-dimensional nonlinear Schr\"odinger equation with the cubic-quintic combination of attractive and repulsive nonlinearities, and a trapping potential represented by a delta-function. We determine all bound states with a…
In this work, we study the Schr\"odinger equation $i\partial_t\psi=-\Delta\psi+\eta(t)\sum_{j=1}^J\delta_{x=a_j(t)}\psi$ on $L^2((0,1),C)$ where $\eta:[0,T]\longrightarrow R^+$ and $a_j:[0,T]\longrightarrow (0,1)$, $j=1,...,J$. We show how…
This paper investigates the nonlinear Schr\"{o}dinger equation with a singular convolution potential. It demonstrates the local well-posedness of this equation in a modified Sobolev space linked to the energy. Additionally, we derive…
In this paper, the space-fractional Schr\"{o}dinger equations with singular potentials are studied. Delta-like or even higher-order singularities are allowed. By using the regularising techniques, we introduce a family of 'weakened'…
The angular part of the Schrodinger equation for a central potential is brought to the one-dimensional 'Schrodinger form' where one has a kinetic energy plus potential energy terms. The resulting polar potential is seen to be a family of…
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…
We examine the one-dimensional quantum dynamics of a Schroedinger particle in a potential represented by a generalized function of the form $U(x) = -\alpha \delta (x) + \beta d(\delta (x))/dx$ superposed on a well behaved potential $V(x)$.…
We study the spectral properties of a Schr\"{o}dinger operator $H_0$ modified by $\delta$ interactions and show explicitly how the poles of the new Green's function are rearranged relative to the poles of original Green's function of $H_0$.…
The relation between the Poisson and Schr\"odinger equation in one dimension is obtained through a simple transformation. It is pointed out that this analogy between both equations can be only applied for potentials that involve a…
Using Mathematica 3.0, the Schroedinger equation for bound states is solved. The method of solution is based on a numerical integration procedure together with convexity arguments and the nodal theorem for wave functions. The interaction…
Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schr\"odinger's equation. This paper, in contrast, investigates the integral form of Schr\"odinger's equation. While both forms are…
We introduce two potentials explicitly given by the Lambert-W function for which the exact solution of the one-dimensional stationary Schr\"odinger equation is written through the first derivative of a double-confluent Heun function. One of…
We study the Cauchy problem for the non-linear Schr\"odinger equation with singular potentials. For point-mass potential and nonperiodic case, we prove existence and asymptotic stability of global solutions in weak-L^{p} spaces. Specific…
We obtain exact solutions of the 2D Schr\"odinger equation with the Singular Even-Power and Inverse-Power Potentials in non-commutative complex space, using the Power-series expansion method. Hence we can say that the Schr\"odinger equation…
We consider a Schr\"odinger operator $H=-\Delta+V(\vec x)$ in dimension two with a quasi-periodic potential $V(\vec x)$. We prove that the absolutely continuous spectrum of $H$ contains a semiaxis and there is a family of generalized…
We consider the cubic nonlinear Schrodinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, and does not have bound states, we obtain the long time asymptotic…
We consider the asymptotic behavior in time of solutions to the cubic nonlinear Schr"odinger equation with repulsive delta potential (delta-NLS). We shall prove that for a given asymptotic profile, there exists a solution to (delta-NLS)…
The Schroedinger equation with one and two dimensional potentials are solved in the frame work of the sl(2) Lie algebra. Eigenfunctions of the Schroedinger equation for various asymmetric double-well potentials have been determined and the…