Schrödinger equation and quantum state codons in discrete transform space

Abstract

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.