On a class of non self-adjoint quantum non-linear oscillators with real spectrum, J.Nonlinear Math.Phys

This article is part of the special issue published in honour of Francesco Calogero on the occasion of his 70th birthday We prove reality of the spectrum for a class of PT − symmetric, non self-adjoint quantum nonlinear oscillators of the form H = p2 + P (q) + igQ(q). Here P (q) is an even polynomial of degree 2p positive at infinity, Q(q) an odd polynomial of degree 2r − 1, and the conditions p> 2r, |g | < R for some R> 0 hold. 1 Introduction and statement of the results Quantum nonlinear oscillators exhibiting remarkable ambiguities in the quantization pro-cedure have recently drawn considerable attention from Francesco Calogero[1], [2], [3]. In this paper we deal with another remarkable phenomenon taking place in a different class of quantum non linear oscillators (the PT-symmetric ones, see below) namely the realit