The regularized trace of a two-dimensional oscillator (Q1326066)

This article deals with the spectral properties of the singular differential operator \[ {\mathcal L} u = - \Delta u(x,y) + (r^ 2 + q(r)) u(x,y), \qquad r = (x^ 2 + y^ 2)^{1/2}. \] The main result is a formula for the regularized trace of this operator under the condition that \(q(r)\) is a real continuous function vanishing outside of a segment \([\varepsilon,a]\) \((0 < \varepsilon < a)\).