Perturbations of non self-adjoint Sturm-Liouville problems, with applications to harmonic oscillators (Q864420)

The author considers the semiclassical Schrödinger operator \[ H^h=h^2\frac {d^2}{dx^2}+V(x) \] on \(L^2([-1,1])\) with Dirichlet boundary conditions and complex potential \(V\) which is assumed to be the restriction of an analytic function. Replacing \(V(x)\) with \(V_{\delta ,\beta }(x)=V(x)+i\delta \operatorname{sgn}(x-\beta )\), the resulting operator is denoted by \(H^h_{\delta ,\beta }\). In this paper, the pseudospectrum \[ {\lim}_h\text{Sp}(H_{\alpha ,\beta }^h)=\bigcap_{h>0} \overline{\bigcup_{0<t<h}\sigma (H^h_{\delta ,\beta })} \] of \(H^h_{\delta ,\beta }\) is considered. First, for \(\delta =0\) and fixed \(h\neq0\), the asymptotics of the large eigenvalues are found, whereas for \(\delta >0\), it is shown in Theorem 6 that for large \(E_0\) with small imaginary part, \(E_0\in \lim_h\)Sp\((H^h_{\alpha ,\beta })\) if an only if \(\Re(\int_{-1}^\beta \sqrt{V-E_0}\,dx)=0\) or \(\Re(\int_\beta^1 \sqrt{V-E_0}\,dx)=0\). The paper concludes with a detailed discussion of the pseudospectrum for the harmonic oscillator \(V(x)=ix^2\).