Tokyo Journal of Mathematics

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

Daisuke AIBA

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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 $m \in \mathbb{N}$.

Article information

Source
Tokyo J. Math., Volume 36, Number 2 (2013), 337-345.

Dates
First available in Project Euclid: 31 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1391177974

Digital Object Identifier
doi:10.3836/tjm/1391177974

Mathematical Reviews number (MathSciNet)
MR3161561

Zentralblatt MATH identifier
1293.35182

Subjects
Primary: 35P15: Estimation of eigenvalues, upper and lower bounds

Citation

AIBA, Daisuke. A Remark on Spectral Properties of Certain Non-selfadjoint Schrödinger Operators. Tokyo J. Math. 36 (2013), no. 2, 337--345. doi:10.3836/tjm/1391177974. https://projecteuclid.org/euclid.tjm/1391177974


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References

  • C. Villani, Hypocoercivity, Memoirs of the AMS.
  • C. Villani, Hypocoercive diffusion operators, International Congress of Mathematicians, vol. 3, European Mathematical Society, Zürich, (2006), 473–498.
  • I. Gallagher, T. Gallay and F. Nier, Spectral Asymptotics for Large Skew-Symmetric Perturbations of the Harmonic Oscillator, International Mathematics Research Notices (2009), 2147–2199.
  • J. Aguilar and J. M. Combes, A class of analytic perturbations for one-body Schrödinger Hamiltonians, Communications in Mathematical Physics 22 (1971), 269–279.
  • J. H. Schenker, Estimating Complex Eigenvalues of Non-Self Adjoint Schrödinger Operators via Complex Dilations, Mathematical Research Letters 18 (2011), no 04, 755–765.
  • T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften 132, Berlin, Springer, 1966.