On the rational monodromy-free potentials with sextic growth

TL;DR

This paper explores rational potentials with sextic growth that are monodromy-free in the Schrödinger equation, identifying new classes beyond Darboux transformations and relating them to algebraic varieties and Calogero-Moser solutions.

Contribution

It characterizes monodromy-free sextic growth potentials, including those not obtainable via Darboux transformations, and connects them to algebraic structures and integrable systems.

Findings

Existence of monodromy-free potentials with quasi-rational eigenfunctions outside Darboux construction.

Relations established between algebraic varieties of potentials and integrable Calogero-Moser systems.

Elementary solutions provided for the Calogero-Moser problem with sextic external field.

Abstract

We study the rational potentials $V(x)$, with sextic growth at infinity, such that the corresponding one-dimensional \Sch equation has no monodromy in the complex domain for all values of the spectral parameter. We investigate in detail the subclass of such potentials which can be constructed by the Darboux transformations from the well-known class of quasi-exactly solvable potentials $$V= x^6 - \nu x^2 +\frac{l(l+1)}{x^2}.$$ We show that, in contrast with the case of quadratic growth, there are monodromy-free potentials which have quasi-rational eigenfunctions, but which can not be given by this construction. We discuss the relations between the corresponding algebraic varieties, and present some elementary solutions of the Calogero-Moser problem in the external field with sextic potential.