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Non self-adjoint harmonic oscillator compact semigroups and pseudospectra. (English) Zbl 1034.34099

The author deals with the complex harmonic oscillator \[ H_cf(x)=-\frac{d^2}{dx^2} f(x)+cx^2f(x) \] in \(L^2({\mathbb R})\) with Dirichlet boundary conditions and \(c\in {\mathbb C}\), \(\text{Im}\,c>0\), \(\text{Re} \, c>0\).
The main results of the paper extend those given by E. B. Davies in [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, 585–599 (1999; Zbl 0931.70016) and Commun. Math. Phys. 200, 35–41 (1999; Zbl 0921.47060)]; they concerns resolvent norm-estimates on \(H_c\) defined above.
More precisely, in section 3 it is shown that, for all \(b>0\), \(1/2<p<3\) fixed, \[ \| (H_c-(b\eta+c\eta^p))^{-1}\| \to \infty \quad \text{as} \quad \eta\to \infty. \] In section 4, the author provides an explicit formula for the heat kernel of \(H_c\), which implies that the bounded holomorphic semigroup of contractions generated by \(-H_c\) is compact in a maximal angular sector. Using this result, the author obtains (section 5) estimates having as a consequence that, for every \(b>0\), there exists \(M_b>0\) such that \[ \lim_{\eta\to \infty} \|(H_c-(\eta+\text{i }b))^{-1}\| \leq M_b, \qquad \lim_{\eta\to \infty} \|(H_c-(c\eta-\text{i }b))^{-1}\| \leq M_b . \] The existence of a set that encloses the \(\varepsilon\)-pseudospecta of \(H_c\), \[ \text{Spec}_\varepsilon(H_c):= \text{Spec }H_c\cup \{z\in {\mathbb C}:\| (H_c-z)^{-1}\geq 1/\varepsilon\}, \] for \(\varepsilon>0\), follows from the estimates above.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47A75 Eigenvalue problems for linear operators
47D06 One-parameter semigroups and linear evolution equations
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