# Unitary time-evolution in stochastic time-dependent Hilbert spaces

## TL;DR

This paper investigates how quantum systems evolve unitarily within infinite-dimensional, stochastic, time-dependent Hilbert spaces, exploring two models and their physical implications.

## Contribution

It introduces a framework for unitary evolution in stochastic, time-dependent Hilbert spaces, extending quantum dynamics to more complex, realistic scenarios.

## Key findings

- Formulated unitary evolution in stochastic Hilbert spaces.
- Compared stochastic models with open quantum systems theory.
- Discussed potential physical applications.

## Abstract

In this work we study the unitary time-evolutions of quantum systems defined on infinite-dimensional separable time-dependent Hilbert spaces. Two possible cases are considered: a quantum system defined on a stochastic interval and another one defined on a Hilbert space with stochastic integration measure (stochastic time-dependent scalar product). The formulations of the two problems and a comparison with the general theory of open quantum systems are discussed. Possible physical applications of the situations considered are analyzed.