Heriot-Watt Mathematics Report Series
HWM01-1, January 2001
Multidimensional Schrödinger equations with Abelian potentials
V M Buchstaber, J C Eilbeck, V Z Enolskii, D V Leykin and M Salerno
Abstract
We consider the 2D complex Schrödinger equation with an Abelian
potential and a fixed energy level. The potential wave function and
the spectral Bloch variety are calculated in terms of the Kleinian
hyperelliptic functions associated with a genus two hyperelliptic
curve. In the special case when the curve covers two elliptic
curves, an intrinsic elliptic 2D exactly solvable Schrödinger
equation is constructed. The solutions obtained are illustrated by a
number of plots. Brief details of the 3D case are also given.
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