Root system of singular perturbations of the harmonic oscillator type operators

Abstract

We analyze perturbations of the harmonic oscillator type operators in a Hilbert space H, i.e. of the self-adjoint operator with simple positive eigenvalues ${\mu }_k$ satisfying ${\mu }_{k+1}-{\mu }_k\ge \Delta >0$. Perturbations are considered in the sense of quadratic forms. Under a local subordination assumption, the eigenvalues of the perturbed operator become eventually simple and the root system forms a Riesz basis.

16 pages; extended Section 5; published version

References in corpus (4)

• Differential operators admitting various rates of spectral projection growth
• Contributions of Issai Schur to Analysis
• The spectrum of a Harmonic Oscillator Operator Perturbed by Point Interactions
• Eigenvalue Asymptotics of Perturbed Self-adjoint Operators

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• The Birman-Schwinger Operator for a Parabolic Quantum Well in a Zero-Thickness Layer in the Presence of a Two-Dimensional Attractive Gaussian Impurity
• The two lowest eigenvalues of the harmonic oscillator in the presence of a Gaussian perturbation
• $\mathcal{P}\mathcal{T}$-symmetric currents of a Bose-Einstein condensate in a triple well
• Bifurcation of nonlinear eigenvalues in problems with antilinear symmetry
• A Bose-Einstein Condensate with $\mathcal{P}\mathcal{T}$-Symmetric Double-Delta Function Loss and Gain in a Harmonic Trap: A Test of Rigorous Estimates
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• Some results on the rotated infinitely deep potential and its coherent states
• Spectral properties of complex Airy operator on the semi-axis
• Non-self-adjoint Toeplitz matrices whose principal submatrices have real spectrum
• Local form subordination without a power decay and a criterion of Riesz basesness