Pseudospectra in non-Hermitian quantum mechanics

Abstract

We propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-Hermitian operators. We relate pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint operators, and basis properties of eigenfunctions. The abstract results are illustrated by unexpected wild properties of operators familiar from PT-symmetric quantum mechanics.

version accepted for publication in J. Math. Phys.: criterion excluding basis property (Proposition 6) added, unbounded time-evolution discussed, new references

References in corpus (22)

• Making Sense of Non-Hermitian Hamiltonians
• Complex Extension of Quantum Mechanics
• Pseudo-Hermitian Representation of Quantum Mechanics
• Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
• Pseudospectra of semi-classical (pseudo)differential operators
• On the reality of the eigenvalues for a class of PT-symmetric oscillators
• On the metric operator for the imaginary cubic oscillator
• Closed formula for the metric in the Hilbert space of a PT-symmetric model
• Examples of Pseudo-bosons in quantum mechanics
• PT-symmetric waveguides
• Root system of singular perturbations of the harmonic oscillator type operators
• Spectral projections and resolvent bounds for partially elliptic quadratic differential operators
• Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem
• Differential operators admitting various rates of spectral projection growth
• Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator
• PT-symmetric models in curved manifolds
• Spectral projections of the complex cubic oscillator
• On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators
• $J$-self-adjoint operators with $\mathcal{C}$-symmetries: extension theory approach
• Quadratic PT-symmetric operators with real spectrum and similarity to self-adjoint operators
• Non-Hermitian spectral effects in a PT-symmetric waveguide
• Estimation de résolvante et construction de quasimode près du bord du pseudospectre

Cited by in corpus (77)

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• Mending the broken PT-regime via an explicit time-dependent Dyson map
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• Unitarity corridors to exceptional points
• Appropriate Inner Product for PT-Symmetric Hamiltonians
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• Tridiagonality, supersymmetry and non self-adjoint Hamiltonians
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• Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via $\mathcal{P}\mathcal{T}-$symmetric and exactly solvable linear-Bose-Hubbard building blocks
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• Spectral instability of horizonless compact objects within astrophysical environments
• Hidden symmetries in non-self-adjoint graphs
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• Pseudospectra of the Schroedinger operator with a discontinuous complex potential
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• Interference of non-Hermiticity with Hermiticity at exceptional points
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• Intrinsic exceptional point -- a challenge in quantum theory
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• Computation and Verification of Spectra for Non-Hermitian Systems
• The shifted harmonic oscillator and the hypoelliptic Laplacian on the circle
• Three alternative model-building strategies using quasi-Hermitian time-dependent observables
• Comment on 'The operational foundations of PT-symmetric and quasi-Hermitian quantum theory'
• Some results on the rotated infinitely deep potential and its coherent states
• Cylindrical first order superintegrability with complex magnetic fields
• Critical time for the observability of Kolmogorov-type equations
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• Anisotropy and asymptotic degeneracy of the physical-Hilbert-space inner-product metrics in an exactly solvable crypto-unitary quantum model
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• The pseudospectrum of an operator with Bessel-type singularities
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• The virial theorem and the method of multipliers in spectral theory
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• Emergence and localization of exceptional points in an exactly solvable toy model