PT-symmetric models in curved manifolds

Abstract

We consider the Laplace-Beltrami operator in tubular neighbourhoods of curves on two-dimensional Riemannian manifolds, subject to non-Hermitian parity and time preserving boundary conditions. We are interested in the interplay between the geometry and spectrum. After introducing a suitable Hilbert space framework in the general situation, which enables us to realize the Laplace-Beltrami operator as an m-sectorial operator, we focus on solvable models defined on manifolds of constant curvature. In some situations, notably for non-Hermitian Robin-type boundary conditions, we are able to prove either the reality of the spectrum or the existence of complex conjugate pairs of eigenvalues, and establish similarity of the non-Hermitian m-sectorial operators to normal or self-adjoint operators. The study is illustrated by numerical computations.

37 pages, PDFLaTeX with 11 figures

References in corpus (13)

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Cited by in corpus (18)

• Mathematical and physical aspects of complex symmetric operators
• Pseudospectra in non-Hermitian quantum mechanics
• Breaking of PT-symmetry in bounded and unbounded scattering systems
• Perfect transmission scattering as a PT-symmetric spectral problem
• Non-self-adjoint graphs
• On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators
• Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding
• Quantum star-graph analogues of PT-symmetric square wells
• Resonant alteration of propagation in guiding structures with complex Robin parameter and its magnetic-field-induced restoration
• Solvable non-Hermitian discrete square well with closed-form physical inner product
• The Pauli equation with complex boundary conditions
• Metric operators for non-Hermitian quadratic su(2) Hamiltonians
• Quantum star-graph analogues of PT-symmetric square wells. II: Spectra
• Guiding structures with multiply connected cross-sections: evolution of propagation in external fields at complex Robin parameters
• Hidden symmetries in non-self-adjoint graphs
• Mathematical and physical meaning of the crossings of energy levels in $\mathcal{P}\mathcal{T}-$symmetric systems
• Contact interactions and Kronig-Penney Models in Hermitian and PT-symmetric Quantum Mechanics
• Spectral analysis of metamaterials in curved manifolds