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

Krejčiřı́k, David; Siegl, Petr; Železný, Jakub

doi:10.1007/s11785-013-0301-y

Cited by 26 publications

(34 citation statements)

References 39 publications

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“…This was demonstrated for a class of operators with non-Hermitian (not necessarily P T -symmetric) point interactions in Ref. [15], where, in addition, explicit formulas for the similarity transformation , metric operator Â, C operator, and similar self-adjoint operator h were presented in a closed form. Nevertheless, the non-Hermiticity and nonlocality are not always equivalent in the described sense [16,17].…”

Section: Introductionmentioning

confidence: 91%

On the metric operator for the imaginary cubic oscillator

2012

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We show that the eigenvectors of the PT-symmetric imaginary cubic oscillator are complete, but do not form a Riesz basis. This results in the existence of a bounded metric operator having intrinsic singularity reflected in the inevitable unboundedness of the inverse. Moreover, the existence of non-trivial pseudospectrum is observed. In other words, there is no quantum-mechanical Hamiltonian associated with it via bounded and boundedly invertible similarity transformations. These results open new directions in physical interpretation of PT-symmetric models with intrinsically singular metric, since their properties are essentially different with respect to self-adjoint Hamiltonians, for instance, due to spectral instabilities.Comment: 7 pages; completely rewritten, new result

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Section: Introductionmentioning

confidence: 91%

On the metric operator for the imaginary cubic oscillator

2012

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“…The spectral theory of H β is developed in [13,12,14]. Below we recall basic spectral properties of this operator.…”

Section: × 4 Examplementioning

confidence: 99%

The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics

2018

Self Cite

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We propose a unique way how to choose a new inner product in a Hilbert space with respect to which an originally non-self-adjoint operator similar to a self-adjoint operator becomes self-adjoint. Our construction is based on minimising a 'Hilbert-Schmidt distance' to the original inner product among the entire class of admissible inner products. We prove that either the minimiser exists and is unique, or it does not exist at all. In the former case we derive a system of Euler-Lagrange equations by which the optimal inner product is determined. A sufficient condition for the existence of the unique minimally anisotropic metric is obtained. The abstract results are supplied by examples in which the optimal inner product does not coincide with the most popular choice fixed through a charge-like symmetry.

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“…Similarity transformations, from non-selfadjoint to similar selfadjoint operators, have been recently studied in [18], where the authors focus on the particular case of 1D Schrödinger operators defined with non-selfadjoint Robin-type conditions occurring at the boundary of an interval. In the case of parity and time-reversal symmetry (PT -symmetry), the similarity of this model with a selfadjoint Hamiltonian is derived.…”

Section: Introductionmentioning

confidence: 99%

Wave operators, similarity and dynamics for a class of Schrödinger operators with generic non-mixed interface conditions in 1D

Mantile

2013

Journal of Mathematical Physics

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We consider a simple modification of the 1D-Laplacian where non-mixed interface conditions occur at the boundaries of a finite interval. It has recently been shown that Schrödinger operators having this form allow a new approach to the transverse quantum transport through resonant heterostructures. In this perspective, it is important to control the deformations effects introduced on the spectrum and on the time propagator by this class of non-selfadjont perturbations. In order to obtain uniform-in-time estimates of the perturbed semigroup, our strategy consists in constructing stationary waves operators allowing to intertwine the modified non-selfadjoint Schrödinger operator with a 'physical' Hamiltonian. For small values of a deformation parameter 'θ', this yields a dynamical comparison between the two models showing that the distance between the corresponding semigroups is dominated by |θ| uniformly in time in the L 2 -operator norm.

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

Cited by 26 publications

References 39 publications

On the metric operator for the imaginary cubic oscillator

On the metric operator for the imaginary cubic oscillator

The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics

Wave operators, similarity and dynamics for a class of Schrödinger operators with generic non-mixed interface conditions in 1D

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