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Completeness of Energy Eigenfunctions for the Reflectionless Potential in Quantum Mechanics
(Aip Publishing, 2024) Erman, Fatih; Turgut, O. Teoman
There are a few exactly solvable potentials in quantum mechanics for which the completeness relation of the energy eigenstates can be explicitly verified. In this article, we give an elementary proof that the set of bound (discrete) states together with the scattering (continuum) states of the reflectionless potential form a complete set. We also review a direct and elegant derivation of the energy eigenstates with proper normalization by introducing an analog of the creation and annihilation operators of the harmonic oscillator problem. We further show that, in the case of a single bound state, the corresponding wave function can be found from the knowledge of continuum eigenstates of the system. Finally, completeness is shown by using the even/odd parity eigenstates of the Hamiltonian, which provides another explicit demonstration of a fundamental property of quantum mechanical Hamiltonians.
Explicit Derivation of the Propagator for a Point Interaction in Three Dimensional Hyperbolic Space
(Springer/plenum Publishers, 2024) Erman, Fatih
The explicit expression for the propagator of the Dirac delta potential in three dimensional hyperbolic spaces is derived using the integral transform of the Krein's type of the resolvent formula, obtained after the renormalization procedure.

Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection