Spectral asymptotics of the harmonic oscillator perturbed by bounded potentials (Q2574903)

This paper concerns the operator \(T= -{d^2\over dx^2}+ x^2+ q(x)\) in \(L^2(\mathbb{R})\), where \(q\) belongs to the Banach space \(B\) given by \[ B= \{r\in L^\infty(\mathbb{R});\,\| r\|_B=\| r\|_\infty+ \| q'\|_\infty+\| q_1\|_\infty< +\infty\} \] and \(q_1(x)= \int^x_0 q(t)\,dt\). The main goal of this paper is to determine the asymptotics of \(\mu_n-(2n- 1)\) \((n\geq 0)\), as \(n\to+\infty\), where \(\mu_0< \mu_1<\mu_2<\cdots\) are the eigenvalues of \(T\).