A Remark on Spectral Properties of Certain Non-selfadjoint Schrödinger Operators

Abstract

In this paper, we study the spectral and pseudospectral properties of the differential operator $H_{\varepsilon} = -\partial_{x}^{2} + x^{2m} +i\varepsilon^{-1}f(x)$ on $L^{2}(\mathbb{R})$, where $\varepsilon > 0$ is a small parameter, $m \in \mathbb{N}$ and $f$ is a real-valued Morse function which satisfies $| \partial^{l}_{x} ( f(x) - |x|^{-k} ) | \le C|x|^{-k-l-1}$ for $l = 0,1,2,3$ and large $|x|$. We show that $\Psi(\varepsilon) = ( \sup_{\lambda \in \mathbb{R} } \| ( H_{\varepsilon} - i\lambda )^{-1} \| )^{-1}$ and $\Sigma(\varepsilon) = \inf \Re (\sigma(H_{\varepsilon}))$ satisfy $C^{-1} \varepsilon^{-\nu(m)} \le \Psi(\varepsilon) \le C \varepsilon^{-\nu(m)}$ and $\Sigma(\varepsilon) \ge C^{-1} \varepsilon^{-\nu(m)}$, $\nu(m) = \min \left\{ \frac{2m}{k+3m+1}, \frac{1}{2} \right\}$. This extends the result of I.~Gallagher, T.~Gallay and F.~Nier [3] (2009) for the case $m=1$ to general $m \in \mathbb{N}$.