Spectral singularities for Non-Hermitian one-dimensional Hamiltonians: puzzles with resolution of identity

Abstract

We examine the completeness of bi-orthogonal sets of eigenfunctions for non-Hermitian Hamiltonians possessing a spectral singularity. The correct resolutions of identity are constructed for delta like and smooth potentials. Their form and the contribution of a spectral singularity depend on the class of functions employed for physical states. With this specification there is no obstruction to completeness originating from a spectral singularity.

25 pages, more refs added

References in corpus (8)

• Making Sense of Non-Hermitian Hamiltonians
• Visualization of Branch Points in PT-Symmetric Waveguides
• Exponentially Fragile PT-Symmetry in Lattices with Localized Eigenmodes
• Spectral singularities and Bragg scattering in complex crystals
• Zero width resonance (spectral singularity) in a complex PT-symmetric potential
• Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: I. General properties
• Complex Lagrangians and phantom cosmology
• Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: II. Rigorous results

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