Schrödinger equations with power potentials

ABSTRACT

General formulae for solutions of the Schrödinger equation with power potentials are derived. The wavefunctions are expressed as products of the asymptotic factors and special forms of the Hessenberg determinants, in general, of infinite order. Conditions under which the order of the determinants becomes finite are determined. It is shown that solutions represented by the finite-order determinants may exist only if the highest power of the radial variable in the potential function is even.

GRAPHICAL ABSTRACT

KEYWORDS:

• Schrödinger equation
• quasi-exactly solvable model
• recurrence relation
• Hessenberg determinant

Acknowledgements

The authors thank Andreas Savin for several discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Jacek Karwowski is grateful for a research grant from the National Chiao Tung University. Financial support was also provided by the National Science Council of Taiwan [grant number NSC 102-2113-M-009-015-MY3]; and the Ministry of Education [MOE-ATU project].