Inversion transformation in the Schrödinger equation

Sergio Scarlatti, Alessandro Teta

An explicit formula for the time-dependent propagator of the Schrodinger equation with one point interaction in three dimensions is given. The derivation is based on the inverse Laplace transformation applied to the corresponding resolvent.

John Morrison

JOHN C. MORRISON

Jean I. F. King

Quantum states of simple systems are shown to have composite root-pole structure in the complex transform plane. The Schrödinger condition becomes the inverse of a continuity equation expressing invariance of the Ψ-transform codon under discrete displacement. Four distinct quotient polynomial solutions model the Legendre, Hermite, Laguerre, and Jacobi polynomial families. Schrödinger coefficients are identified with pole strengths of quotient polynomials and are geometrically interpreted in terms of universal root and pole interactions.

Daniel Alonso, Juan Gonzalo Muga, R. Sala Mayato

Recently, Olavo has proposed several derivations of the Schr\"odinger equation from different sets of hypothesis [Phys. Rev. A 61, 052109 (2000)]. One of them is based on the infinitesimal inverse Weyl transform of a classically evolved phase-space density. We show, however, that the Schr\"odinger equation can only be obtained in that manner for linear or quadratic potential functions.