Compact-form solution to the time-dependent Schr\"odinger equation with an arbitrary potential

TL;DR

This paper presents a method to derive exact, compact-form solutions for the time-dependent Schrödinger equation with arbitrary potentials, combining techniques from PDEs, combinatorics, and number theory.

Contribution

It introduces a novel approach to obtain explicit, finite-form solutions for complex quantum systems with arbitrary potentials, advancing analytical methods in quantum mechanics.

Findings

Exact solutions for arbitrary potentials are expressed in finite, explicit forms.

The approach unifies PDE, combinatorics, and number theory techniques.

Potential applications span physics, mathematics, and calculus.

Abstract

We obtain exact solutions to the class of parabolic partial differential equations of arbitrary dimensionality and with arbitrary potentials. The solutions are presented in a compact-form: as explicit mathematical expressions consisting of finite number of standard mathematical operations with finite (condition-independent) number of variables in a discrete or continous region of consideration. The general approach to obtain the compact-form solutions combines a number of methods in the fields of partial differential equations, enumerative combinatorics, and number theory, which we describe in detail. We discuss their advantages and perspectives for the various fields in physics, mathematics, and advanced calculus.