On the Dirac-Infeld-Plebanski delta function

The present work is a brief review of the progressive search of improper delta-functions which are of interest in Quantum Mechanics and in the problem of motion in General Relativity Theory.Comment: LaTeX file, 15 pages no figure

Bi-Orthogonal Approach to Non-Hermitian Hamiltonians with the Oscillator Spectrum: Generalized Coherent States for Nonlinear Algebras

A set of Hamiltonians that are not self-adjoint but have the spectrum of the harmonic oscillator is studied. The eigenvectors of these operators and those of their Hermitian conjugates form a bi-orthogonal system that provides a mathematical procedure to satisfy the superposition principle. In this form the non-Hermitian oscillators can be studied in much the same way as in the Hermitian approaches. Two different nonlinear algebras generated by properly constructed ladder operators are found and the corresponding generalized coherent states are obtained. The non-Hermitian oscillators can be steered to the conventional one by the appropriate selection of parameters. In such limit, the generators of the nonlinear algebras converge to generalized ladder operators that would represent either intensity-dependent interactions or multi-photon processes if the oscillator is associated with single mode photon fields in nonlinear media.Comment: this abridged version (37 pages, 11 figures) includes simplified formulae and correction of misprint

Rectangular Potentials in a Semi-Harmonic Background: Spectrum, Resonances and Dwell Time

We study the energy properties of a particle in one dimensional semi-harmonic rectangular wells and barriers. The integration of energies is obtained by solving a simple transcendental equation. Scattering states are shown to include cases in which the impinging particle is 'captured' by the semi-harmonic rectangular potentials. The 'time of capture' is connected with the dwell time of the scattered wave. Using the particle absorption method, it is shown that the dwell time $\tau^a_D$ coincides with the phase time $\tau_W$ of Eisenbud and Wigner, calculated as the energy derivative of the reflected wave phase shift. Analytical expressions are derived for the phase time $\tau_W$ of the semi-harmonic delta well and barrier potentials