Supersymmetric quantum mechanics living

on topologically non-trivial Riemann surfaces

MILOSLAV ZNOJIL^1,* and VÍT JAKUBSKݲ

¹Nuclear Physics Institute ASCR, 250 68 Řež, Czech Republic

²Departamento de Física, Universidad de Santiago de Chile,

Casilla 307, Santiago 2, Chile

^*Corresponding author

E-mail: znojil@ujf.cas.cz; jakub@ujf.cas.cz

Abstract. Supersymmetric quantum mechanics is constructed in a new

non-Hermitian representation. Firstly, the map between the partner

operators $H^{(\pm)}$ is chosen antilinear. Secondly, both these

components of a super-Hamiltonian ${\cal H}$ are defined along

certain topologically non-trivial complex curves $r^{(\pm)}(x)$

which spread over several Riemann sheets of the wave function. The

non-uniqueness of our choice of the map ${\cal T}$ between

`tobogganic' partner curves $r^{(+)}(x)$ and $r^{(-)}(x)$ is

emphasized.

Keywords. Supersymmetry; Schr\"{o}dinger equation; complexified

coordinates.

PACS Nos 11.30.Pb; 03.65.Fd; 93.65.Db