# Quantum time delay for unitary operators: general theory

## TL;DR

This paper develops a general, model-independent framework for defining and analyzing quantum time delay for unitary operators, establishing its equivalence with an operator-based approach using advanced functional analysis tools.

## Contribution

It introduces a new, general framework for quantum time delay in terms of sojourn times and proves its equivalence with an operator-based definition, applicable to a broad class of models.

## Key findings

- Time delay defined via sojourn times exists and matches the expectation of a unitary Eisenbud-Wigner operator.
- The framework is general and model-independent, relying on advanced functional analysis techniques.
- The approach unifies time-dependent and time-independent definitions of quantum time delay.

## Abstract

We present a suitable framework for the definition of quantum time delay in terms of sojourn times for unitary operators in a two-Hilbert spaces setting. We prove that this time delay defined in terms of sojourn times (time-dependent definition) exists and coincides with the expectation value of a unitary analogue of the Eisenbud-Wigner time delay operator (time-independent definition). Our proofs rely on a new summation formula relating localisation operators to time operators and on various tools from functional analysis such as Mackey's imprimititvity theorem, Trotter-Kato Formula and commutator methods for unitary operators. Our approach is general and model-independent.