Related papers: On one-dimensional Schroedinger problems allowing …
A comprehensive review of the discrete quantum mechanics with the pure imaginary shifts and the real shifts is presented in parallel with the corresponding results in the ordinary quantum mechanics. The main subjects to be covered are the…
From the algebraic treatment of the quasi-solvable systems, and a q-deformation of the associated $su(2)$ algebra, we obtain exact solutions for the q-deformed Schrodinger equation with a 3-dimensional q-deformed harmonic oscillator…
We discuss a new approach to solve the low lying states of the Schroedinger equation. For a fairly large class of problems, this new approach leads to convergent iterative solutions, in contrast to perturbative series expansions. These…
Using the elementary axioms of special relativity and quantum mechanics we construct a wave equation which generalizes the Schrodinger equation. We also solve the general second and some higher order differential equations.
In this paper, we classify the fundamental solutions for a class of Schrodinger operators.
We prove dispersive estimates for linear Schroedinger equations in two space dimensions. The potential is assumed to be real-valued with some polynomial decay (faster than a negative third power), and zero energy is assumed to be a regular…
A duality between an electrostatic problem in a three dimensional world and a quantum mechanical problem in a one dimensional world which allows one to obtain the ground state solution of the Schr\"odinger equation by using electrostatic…
Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular $j$ function. We show general cases in which these systems have solutions, and then we look at certain situations in…
The asymptotic iteration method is used to find exact and approximate solutions of Schroedinger's equation for a number of one-dimensional trigonometric potentials (sine-squared, double-cosine, tangent-squared, and complex cotangent).…
We investigate the existence, non-existence, and multiplicity of positive solutions to a class of quasilinear Schrodinger equations with a prescribed mass condition in higher dimensions. Using the dual approach, the equation is transformed…
The analytical solutions of the N-dimensional Schrodinger equation with position-dependent mass for a general class of central potentials is obtained via the series expansion method. The position-dependent mass is expanded in series about…
The generalized Schrodinger equation deduced in the earlier papers is compared with conventional constructions of quantum field theory. In particular, it yields the usual Schrodinger equation of quantum field theory written without normal…
In \cite{poiret}, we explain how we can construct global solutions for the cubic Schr\"odinger equation in three dimensional with initial data in $ L^2(\mathds{R}^3) $. The main ingredient of this proof is the existence of the bilinear…
We present a new six-parameter family of potentials whose solutions are expressed in terms of the hypergeometric functions 3F2, 2F2 and 1F2. Both the scattering data and the bound states of these potentials are explicitly computed and the…
We construct a variety of new exactly-solvable quantum systems, the potentials of which are given in terms of Lambert-W functions. In particular, we generate Schr\"odinger models with energy-dependent potentials, conventional Schr\"odinger…
We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the…
For a given length and a given degree and an arbitrary partition of the positive integers, there always is a cell containing a polynomial progression of that length and that degree; moreover, the coefficients of the generating polynomial…
The independent solutions of the one-dimensional Schr\"odinger equation are approximated by means of the explicit summation of the leading constituent WKB series. The continuous matching of the particular solutions gives the uniformly valid…
In this paper, we obtain pointwise convergence of solutions to the Schrodinger equation along a class of curves in $\mathbb{R}^{2}$ by the polynomial partitioning.
A set of exactly solvable one-dimensional quantum mechanical potentials is described. It is defined by a finite-difference-differential equation generating in the limiting cases the Rosen-Morse, harmonic, and P\"oschl-Teller potentials.…