A remark on spectral properties of certain non-selfadjoint Schrödinger operators (Q2444947)

This paper studies the non-selfadjoint Schrödinger operator \(H_\epsilon=-\partial_x^2+x^{2m}+i\epsilon^{-1}f(x)\) on \(L^2(R)\), where \(m\) is a natural number and \(\epsilon>0\) is a small parameter. Bounds are obtained for the norm of the resolvent on the imaginary axis \(\Psi(\epsilon)=\sup_{\lambda\in R}\|(H_\epsilon-i\lambda)^{-1}\|\), and on \(\Sigma(\epsilon)=\inf \mathrm{Re}(\sigma(H_\epsilon))\). Assuming that \(f\) is a real-valued Morse function satisfying \(\displaystyle|\partial_x^j(f(x)-|x|^{-k})|\leq C|x|^{-k-l-1}\) for \(l=0,1,2,3\) and large \(|x|\), it is proved that \(\displaystyle C^{-1}\epsilon^{-\nu(m)}\leq \Psi(\epsilon)\leq C\epsilon^{-\nu(m)}\) and \(\Sigma(\epsilon)\geq C^{-1}\epsilon^{-\nu(m)}\), where \(\nu(m)=\min\left(\frac{2m}{k+3m+1},\frac{1}{2}\right)\). This result generalizes previous results in the case \(m=1\) [\textit{I. Gallagher} et al., Int. Math. Res. Not. 2009, No. 12, 2147--2199 (2009; Zbl 1180.35383)]. Another result shows that \(\Sigma(\epsilon)\) can be much larger than \(\Psi(\epsilon)\): when \(f(x)=(1+x^2)^{-\frac{k}{2}}\), \(k>0\), it is proved that \(\displaystyle\Sigma(\epsilon)\geq C \epsilon^{-\nu'(m)}\) with \(\nu'(m)=\min\left(\frac{1}{2},\frac{2m}{k+2m}\right)\).