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 a traveling wave reduction technique, we have shown that Maccari equation, (2?1)-dimensional nonlinear Schrödinger equation, medium equal width equation, (3?1)-dimensional modified KdV–Zakharov– Kuznetsev equation, (2?1)-dimensional long wave-short wave resonance interaction equation, perturbed nonlinear Schrödinger equation can be reduced to the same family of auxiliary elliptic-like equ...

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.

In the case of tight transverse confinement (cigar-shaped trap) the threedimensional (3D) nonlinear Schrödinger equation, describing superfluid Fermi atoms at unitarity (infinite scattering length |a| → ∞), is reduced to an effective one-dimensional form by averaging over the transverse coordinates. The resultant effective equation is a 1D nonpolynomial Schrodinger equation, which produces resu...

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...

The two-dimensional self-dual Chern-Simons equations are equivalent to the conditions for static, zero-energy vortex-like solutions of the (2+1) dimensional gauged nonlinear Schrödinger equation with Chern-Simons matter-gauge coupling. The finite charge vacuum states in the Chern-Simons theory are shown to be gauge equivalent to the finite action solutions to the two-dimensional chiral model (o...

We derive a dispersion estimate for one-dimensional perturbed radial Schrödinger operators. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum.