Related papers: On two perturbation series for short-range attract…
Analytical solutions to the time-dependent Shr\"{o}dinger equation in one dimension are developed for time-independent potentials, one consisting of an infinite wall and a repulsive delta function. An exact solution is obtained by means of…
The Self-Adjoint Extension in the Schrodinger equation for potentials behaved as an attractive inverse square at the origin is critically reviewed. Original results are also presented. It is shown that the additional solutions must be…
We propose a new approach to the Rayleigh-Schr\"{o}dinger perturbation expansions of bound states in quantum mechanics. We are inspired by the enormous flexibility of solvable interactions with several (N) discontinuities. Their standard…
In this letter, we consider a Schrodinger equation for a well potential with varying width. We solve one dimensional time-dependent Schrodinger equation subject to time-dependent boundary conditions for a spinless particle inside infinite…
We revisit the resolution of the one-dimensional Schr\"odinger hamiltonian with a Coulomb $\lambda$/|x| potential. We examine among its self-adjoint extensions those which are compatible with physical conservation laws. In the…
We present one dimensional potentials $V(x)= V_0[e^{2|x|/a}-1]$ as solvable models of a well $(V_0>0)$ and a barrier ($V_0<0$). Apart from being new addition to solvable models, these models are instructive for finding bound and scattering…
We show that infinitely differentiable solutions to parabolic and hyperbolic equations, whose right-hand sides are analytical in time, are also analytical in time at each fixed point of the space. These solutions are given in the form of…
We discuss a finite rectangular well as a perturbation for the infinite one with a depth $\lambda^2$ of the former as a perturbation parameter. In particular we consider a behaviour of energy levels in the well as functions of complex…
We study the Schroedinger equation of a class of two-level systems under the action of a periodic time-dependent external field in the situation where the energy difference 2epsilon between the free energy levels is sufficiently small with…
Analytic solutions and their formal asymptotic expansions for a family of the singularly perturbed $q-$difference-differential equations in the complex domain are constructed. They stand for a $q-$analog of the singularly perturbed partial…
We derive uniform in time $L^\infty$-bound for solutions to an aggregation-diffusion model with attractive-repulsive potentials or fully attractive potentials. We analyze two cases: either the repulsive nonlocal term dominates over the…
A new type of perturbative expansion is built in order to give a rigorous derivation and to clarify the range of validity of some commonly used model equations. This model describes the evolution of the modulation of two short and localized…
A semilinear reaction-diffusion two-point boundary value problem, whose second-order derivative is multiplied by a small positive parameter $\eps^2$, is considered. It can have multiple solutions. An asymptotic expansion is constructed for…
We present doubly-periodic solutions of the infinitely extended nonlinear Schrodinger equation with an arbitrary number of higher-order terms and corresponding free real parameters. Solutions have one additional free variable parameter that…
In theoretical physics, we sometimes have two perturbative expansions of physical quantity around different two points in parameter space. In terms of the two perturbative expansions, we introduce a new type of smooth interpolating function…
Motivated by recent experimental studies of matter-waves and optical beams in double well potentials, we study the solutions of the nonlinear Schr\"{o}dinger equation in such a context. Using a Galerkin-type approach, we obtain a detailed…
We consider an equidistant array of disjoint potential wells in $\mathbb{R}^\nu,\: \nu\ge 2$, built over a straight line, and show that, under a restriction on the potential support aspect ratio, a perturbation consisting of longitudinal…
Treating the nonlinear term of the Gross-Pitaevskii nonlinear Schrodinger equation as a perturbation of an isolated discrete eigenvalue of the linear problem one obtains a Rayleigh-Schrodinger power series. This power series is proved to be…
We have obtained the perturbative expressions up to sixth order for the energy of the bound state in a one dimensional, arbitrarily weak, short range finite well, applying a method originally developed by Gat and Rosenstein Ref. [3]. The…
The parameter dependence of the various attractive solutions of the three variable nonlinear Lorenz model equations for thermal convection in Rayleigh-B\'enard flow is studied. Its bifurcation structure has commonly been investigated as a…