Spectral expansion of Schwartz linear operators

TL;DR

This paper establishes a spectral expansion theorem for Schwartz linear operators with Schwartz eigenfamilies, paralleling finite-dimensional spectral theory and modeling quantum mechanical spectral expansions.

Contribution

It introduces a rigorous spectral expansion framework for Schwartz operators with eigenfamilies indexed by Euclidean spaces, bridging mathematical theory and quantum physics applications.

Findings

Proves a spectral expansion theorem for Schwartz linear operators.

Shows the spectral expansion resembles finite-dimensional cases with continuous superpositions.

Provides a mathematical model aligning with quantum mechanical spectral expansions.

Abstract

In this paper we prove and apply a theorem of spectral expansion for Schwartz linear operators which have an S-linearly independent Schwartz eigenfamily. This type of spectral expansion is the analogous of the spectral expansion for self-adjoint operators of separable Hilbert spaces, but in the case of eigenfamilies of vectors indexed by the real Euclidean spaces. The theorem appears formally identical to the spectral expansion in the finite dimensional case, but for the presence of continuous superpositions instead of finite sums. The Schwartz expansion we present is one possible rigorous and simply manageable mathematical model for the spectral expansions used frequently in Quantum Mechanics, since it appears in a form extremely similar to the current formulations in Physics.