Sturm--Schrödinger equations: Formula for metric

MILOSLAV ZNOJIL^1,* and HENDRIK B GEYER²

¹Nuclear Physics Institute ASCR, 250 68 \v{R}e\v{z}, Czech Republic

²Institute of Theoretical Physics, University of Stellenbosch and

Stellenbosch Institute for Advanced Study, 7600 Stellenbosch, South Africa

^*Corresponding author

E-mail: znojil@ujf.cas.cz; hbg@sun.ac.za

Abstract. Sturm--Schr\"{o}dinger equations $H\psi=E\,W\psi$ with

$H\neq H^\dagger$ and $W\neq W^\dagger \neq I$ are considered, with

a weak point of the theory lying in the purely numerical

matrix-inversion form of the double-series definition of the

necessary metric operator $\Theta$ in the physical Hilbert space of

states [M Znojil, {\it J. Phys. A: Math. Theor.} {\bf 41}, 215304 (2008)]. This

shortcoming is removed here via an amended, single-series definition

of $\Theta$.

Keywords. Crypto-Hermiticity; Sturm--Schr\"{o}dinger bound-state problem.

PACS Nos 03.65.Bz; 03.65.Ca