Eigensystem of an L
                     2-perturbed harmonic oscillator is an unconditional basis

Adduci, James; Mityagin, Boris

doi:10.2478/s11533-011-0139-3

Cited by 26 publications

(61 citation statements)

References 38 publications

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“…separation distance of eigenvalues tends to infinity, that is used for the proof of the existence of similarity transformations [13]. Recent results on basis properties for perturbations of harmonic oscillator type operators [1,41,3] give a possibility to investigate the structure of similarity transformation in these cases as well. Another step is to consider e.g.…”

Section: Bounded Perturbationsmentioning

confidence: 99%

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On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators

Krejčiřı́k

Siegl

Železný

2013

Complex Anal. Oper. Theory

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We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations in detail. We show that they can be expressed as the sum of the identity and an integral Hilbert-Schmidt operator. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive the similar self-adjoint operator and also find the associated "charge conjugation" operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.

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Section: Bounded Perturbationsmentioning

confidence: 99%

“…On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators 1 …”

mentioning

confidence: 99%

On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators

Krejčiřı́k

Siegl

Železný

2013

Complex Anal. Oper. Theory

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“…It would be natural to expect that Hamiltonians whose imaginary parts are integrable functions on phase space will not lead to similar divergences, and, in the presence of P T -symmetry, a bounded mapping to a Hermitian counterpart would be expected. It has been proven in [31][32][33], for example, that certain well-behaved types of non-Hermitian perturbations of Hermitian operators with "harmonic oscillator type" spectra will lead to Riesz bases, an no finite time divergences would be expected. The systematic investigation of the connection between such mathematical constraints and their physical interpretation would be an interesting topic for further studies.…”

Section: Discussionmentioning

confidence: 99%

Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator

Graefe¹,

Korsch²,

Rushton³

et al. 2015

J. Phys. A: Math. Theor.

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Abstract. The non-Hermitian quadratic oscillator known as Swanson oscillator is one of the popular P T -symmetric model systems. Here a full classical description of its dynamics is derived using recently developed metriplectic flow equations, which combine the classical symplectic flow for Hermitian systems with a dissipative metric flow for the anti-Hermitian part. Closed form expressions for the metric and phasespace trajectories are presented which are found to be periodic in time. Since the Hamiltonian is only quadratic the classical dynamics exactly describes the quantum dynamics of Gaussian wave packets. It is shown that the classical metric and trajectories as well as the quantum wave functions can diverge in finite time even though the P T -symmetry is unbroken, i.e., the eigenvalues are purely real.

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“…In H we consider a matrix differential operator L αβ , where α and β are the two-component vectors, α = α (1) , α (2) , β = β (1) , β (2) . This operator is defined by matrix differential expression (4.8) on the following domain: In the sequel, it is convenient to represent the operator L αβ as the sum, L αβ = L αβ + M, where L αβ coincides with L αβ if one sets C 1 = C 2 = 0.…”

Section: Double-walled Carbon Nanotube Modelmentioning

confidence: 99%

“…Our main result of [41] is the following statement: Theorem 4.4 Assume that the structural parameters are such thatk 1 =k 2 =k andk 1 =k 2 =k. Assume that the boundary parameters α (i) and β (i) , i = 1, 2, are such that the following conditions hold: α (1) = α (2) , β (1) = β (2) ,…”

Section: Double-walled Carbon Nanotube Modelmentioning

confidence: 99%

On the completeness of root vectors generated by systems of coupled hyperbolic equations

Shubov

2014

Mathematische Nachrichten

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The paper is the second in a set of two papers, which are devoted to a unified approach to the problem of completeness of the generalized eigenvectors (the root vectors) for a specific class of linear non‐selfadjoint unbounded matrix differential operators. The list of the problems for which such operators are the dynamics generators includes the following: (a) initial boundary‐value problem (IBVP) for a non‐homogeneous string with both distributed and boundary damping; (b) IBVP for small vibrations of an ideal filament with a one‐parameter family of dissipative boundary conditions at one end and with a heavy load at the other end; this filament problem is treated for two cases of the boundary parameter: non‐singular and singular; (c) IBVP for a three‐dimensional damped wave equation with spherically symmetric coefficients and both distributed and boundary damping; (d) IBVP for a system of two coupled hyperbolic equations constituting a Timoshenko beam model with variable coefficients and boundary damping; (e) IBVP for a coupled Euler‐Bernoulli and Timoshenko beam model with boundary energy dissipation (the model known in engineering literature as bending‐torsion vibration model); (f) IBVP for two coupled Timoshenko beams model, which is currently accepted as an appropriate model describing vibrational behavior of a longer double‐walled carbon nanotube. Problems (a)–(c) have been discussed in the first paper of the aforementioned set. Problems (d)–(f) are discussed in the present paper.

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Eigensystem of an L 2-perturbed harmonic oscillator is an unconditional basis

Cited by 26 publications

References 38 publications

On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators

On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators

Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator

On the completeness of root vectors generated by systems of coupled hyperbolic equations

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