General Adiabatic Evolution with a Gap Condition

TL;DR

This paper develops a generalized adiabatic theory for linear operators with a spectral gap, including non-diagonalizable cases, providing a method to approximate the evolution semigroup with controlled errors.

Contribution

It introduces a superadiabatic renormalization approach to construct projectors that approximate the evolution in complex spectral scenarios.

Findings

Constructed projectors closely follow instantaneous eigenprojectors in the adiabatic limit.

Provided a controlled approximation of the evolution semigroup with explicit error bounds.

Extended adiabatic theory to non-diagonalizable operators with eigenilpotents.

Abstract

We consider the adiabatic regime of two parameters evolution semigroups generated by linear operators that are analytic in time and satisfy the following gap condition for all times: the spectrum of the generator consists in finitely many isolated eigenvalues of finite algebraic multiplicity, away from the rest of the spectrum. The restriction of the generator to the spectral subspace corresponding to the distinguished eigenvalues is not assumed to be diagonalizable. The presence of eigenilpotents in the spectral decomposition of the generator forbids the evolution to follow the instantaneous eigenprojectors of the generator in the adiabatic limit. Making use of superadiabatic renormalization, we construct a different set of time-dependent projectors, close to the instantaneous eigeprojectors of the generator in the adiabatic limit, and an approximation of the evolution semigroup which intertwines exactly between the values of these projectors at the initial and final times. Hence, the evolution semigroup follows the constructed set of projectors in the adiabatic regime, modulo error terms we control.