We prove the theorem announced in Phys. Rev. Lett. 85:5022, 2001 concerning the existence and properties of fractal states for the Schrödinger equation in the infinite one-dimensional well.

In this paper, we investigate the one-dimensional discrete Schrödinger equation with general, symmetric boundary conditions. Our results primarily concern the number of energy states lying in the wells.

Using the method of shape invariant potentials, a number of exact solutions of one dimensional effective mass Schrödinger equation are obtained. The solutions with equi-spaced spectrum are discussed in detail.

We characterise the set of periods for which number theoretical obstructions prevent us from solving the Schrödinger equation on a two dimensional torus as well as the asymptotic occurrence of possible resonances.

Using conformal coordinates associated with projective conformal relativity we obtain a conformal Klein-Gordon partial differential equation. As a particular case we present and discuss a conformal ‘radial’ d’Alembert-like equation. As a by-product we show that this ‘radial’ equation can be identified with a one-dimensional Schrödinger-like equation in which the potential is exactly the second ...

We derive a perturbed two-dimensional nonlinear Schrödinger equation which describes the propagation of gap-soliton bullets in nonlinear periodic waveguides at frequencies close to the gap for Bragg reflection. Analysis and simulations of this equation show that the bullets amplitude undergoes stable focusing–defocusing cycles. © 2003 Elsevier B.V. All rights reserved.

We use Lie point symmetries of the 2+1-dimensional cubic Schrödinger equation to obtain new analytic solutions in a systematic manner. We present an analysis of the reduced ODEs, and in particular show that although the original equation is not integrable they typically can belong to the class of Painlevé type equations.

We consider the two-dimensional, time-dependent Schrödinger equation discretized with the Crank-Nicolson finite difference scheme. For this difference equation we derive discrete transparent boundary conditions (DTBCs) in order to get highly accurate solutions for open boundary problems. We apply inhomogeneous DTBCs to the transient simulation of quantum waveguides with a prescribed electron in...

In one-dimensional quantum mechanics, or the Sturm-Liouville theory, Crum’s theorem describes the relationship between the original and the associated Hamiltonian systems, which are iso-spectral except for the lowest energy state. Its counterpart in ‘discrete’ quantum mechanics is formulated algebraically, elucidating the basic structure of the discrete quantum mechanics, whose Schrödinger equa...

We develop an algebraic approach to studying the spectral properties of the stationary Schrödinger equation in one dimension based on its high order conditional symmetries. This approach makes it possible to obtain in explicit form representations of the Schrödinger operator by n×n matrices for any n ∈ N and, thus, to reduce a spectral problem to a purely algebraic one of finding eigenvalues of...