The spectrum of a Harmonic Oscillator Operator Perturbed by Point Interactions

Abstract

We consider the operator $L=-(d/dx{)}^2+x^{2}y+w(x)y,y\in L^2(\mathbb{R})$, where $w(x)=s[\delta (x-b)-\delta (x+b)],b\ne 0,$ real, $s\in \mathbb{C}$. This operator has a discrete spectrum: eventually the eigenvalues are simple and ${\lambda }_n=(2n+1)+s^2(\kappa (n)/n)+\rho (n)$, where $\kappa (n)=\frac{1}{2\pi }[(-1{)}^{n+1}\sin (2b\sqrt{2n})-\frac{1}{2}\sin (4b\sqrt{2n})]$ and $|\rho (n)|\le C(\log n)/(n^{3/2})$ If $s=i\gamma$, $\gamma$ real, the number $T(\gamma )$ of non-real eigenvalues is finite, and $T(\gamma )\le [C(1+|\gamma |)\log (e+|\gamma |){]}^2.$ The analogue of the above equations is given in the case of any two-point interaction perturbation $w(x)=c_{+}\delta (x-b)+c_{-}\delta (x+b),c_{+},c_{-}\in \mathbb{C}.$

65 pages