Spectral projections of the complex cubic oscillator

Abstract

We prove the spectral instability of the complex cubic oscillator $-\frac{d^2}{d{x}^2}+i{x}^3+i\alpha x$ for non-negative values of the parameter $\alpha$, by getting the exponential growth rate of $\Vert {\Pi }_n(\alpha )\Vert$, where ${\Pi }_n(\alpha )$ is the spectral projection associated with the $n$-th eigenvalue of the operator. More precisely, we show that for all non-negative $\alpha$ \[ \lim\limits_{n\to+\infty}\frac{1}{n}\log\|Π_n(α)\| = \fracπ{\sqrt{3}}. \]

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Cited by in corpus (11)

• Mathematical and physical aspects of complex symmetric operators
• Pseudospectra in non-Hermitian quantum mechanics
• Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator
• Local form-subordination condition and Riesz basisness of root systems
• Pseudomodes for non-self-adjoint Dirac operators
• A Bound on the Pseudospectrum of the Harmonic Oscillator with Imaginary Cubic Potential
• Pseudospectra of the Schroedinger operator with a discontinuous complex potential
• Bifurcation of nonlinear eigenvalues in problems with antilinear symmetry
• Computation and Verification of Spectra for Non-Hermitian Systems
• The pseudospectrum of an operator with Bessel-type singularities
• Spectral properties of complex Airy operator on the semi-axis