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arXiv:1112.6338

Adiabatensätze mit und ohne Spektrallückenbedingung

Jochen Schmid

Published Dec 29, 2011• 0 citations

Abstract

In this work we generalize some of the previously known adiabatic theorems to situations with non-unitary evolutions in Banach spaces. We prove adiabatic theorems with uniform gap condition (generalizing a theorem of Abou Salem), adiabatic theorems with non-uniform gap condition (generalizing a theorem of Kato) and qualitative as well as quantitative adiabatic theorems without gap condition (generalizing theorems of Avron and Elgart, and Teufel). Additionally, we give a generalized version of an adiabatic theorem of higher order due to Nenciu. In all these adiabatic theorems the considered spectral values need not lie on the imaginary axis and in the adiabatic theorems with spectral gap condition and the adiabatic theorem of higher order compact subsets of the spectrum are sufficient (in particular, these subsets need not consist of eigenvalues). We explore the strength of the presented adiabatic theorems in numerous examples. In particular, we show that the theorems of the present work are more general than the previously known theorems. This work was finished in October 2010 and handed in as a diploma thesis at the mathematics department of the University of Stuttgart. The more recent results of Avron, Fraas, Graf und Grech which appeared in the meantime (in June 2011) are therefore not taken into consideration here. We point out, however, that the theorems of the present work are more general than the corresponding theorems of Avron, Fraas, Graf und Grech - with the exception of the adiabatic theorem of higher order which is in no logical relation to the adiabatic theorem of higher order of Avron, Fraas, Graf and Grech. We are about to gather the most important theorems of the present work in an article and will upload it to arXiv soon.

DOI: 10.48550/arxiv.1112.6338

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Adiabaticity occurs when, during its evolution, a physical system remains in the instantaneous eigenstate of the Hamiltonian. Unfortunately, existing results, such as the quantum adiabatic theorem based on a slow down evolution $[H(ϵt),ϵ\ensuremath{\rightarrow}0]$, are insufficient to describe an evolution driven by the Hamiltonian

H(t)

itself. Here we derive general criteria and exact bounds, for the state and its phase, ensuring an adiabatic evolution for any Hamiltonian

H(t)

. As a corollary, we demonstrate that the commonly used condition of a slow Hamiltonian variation rate, compared to the spectral gap, is indeed sufficient to ensure adiabaticity but only when the Hamiltonian is real and nonoscillating (for instance, containing exponential or polynomial but no sinusoidal functions).

Adiabaticity occurs when, during its evolution, a physical system remains in the instantaneous eigenstate of the hamiltonian. Unfortunately, existing results, such as the quantum adiabatic theorem based on a slow down evolution (H(epsilon t), epsilon ? 0), are insufficient to describe an evolution driven by the hamiltonian H(t) itself. Here we derive general criteria and exact bounds, for the state and its phase, ensuring an adiabatic evolution for any hamiltonian H(t). As a corollary we demonstrate that the commonly used condition of a slow hamiltonian variation rate, compared to the spectral gap, is indeed sufficient to ensure adiabaticity but only when the hamiltonian is real and non oscillating (for instance containing exponential or polynomial but no sinusoidal functions).