[Paper Review] An Adiabatic Theorem for Singularly Perturbed Hamiltonians

Alain Joye|ArXiv.org|Nov 2, 1994

Spectral Theory in Mathematical Physics9 references3 citations

TL;DR

This paper establishes an adiabatic theorem for singularly perturbed Hamiltonians where a small perturbation $\epsilon H_1(t)$ acts on a self-adjoint Hamiltonian $H_0(t)$ with a spectral gap, even when the perturbation is defined on a smaller domain and breaks the gap condition for $\epsilon > 0$. Under reasonable assumptions, the paper proves that the system remains close to the adiabatic evolution of the unperturbed Hamiltonian in the limit $\epsilon \to 0$, extending the adiabatic theorem beyond standard gap conditions.

ABSTRACT

The adiabatic approximation in quantum mechanics is considered in the case where the self-adjoint hamiltonian $H_0(t)$, satisfying the usual spectral gap assumption in this context, is perturbed by a term of the form $εH_1(t)$. Here $ε o 0$ is the adiabaticity parameter and $H_1(t)$ is a self-adjoint operator defined on a smaller domain than the domain of $H_0(t)$. Thus the total hamiltonian $H_0(t)+εH_1(t)$ does not necessarily satisfy the gap assumption, $\forall ε>0$. It is shown that an adiabatic theorem can be proven in this situation under reasonnable hypotheses. The problem considered can also be viewed as the study of a time-dependent system coupled to a time-dependent perturbation, in the limit of large coupling constant.

Motivation & Objective

• To extend the adiabatic theorem to Hamiltonians perturbed by singular terms that violate the standard spectral gap condition.
• To address the case where the perturbation $\epsilon H_1(t)$ is defined on a smaller domain than $H_0(t)$, making the total Hamiltonian ill-defined or non-self-adjoint in the usual sense.
• To establish conditions under which the adiabatic evolution remains valid despite the absence of a uniform spectral gap for $\epsilon > 0$.
• To model systems with large coupling constants in time-dependent settings, where the perturbation is treated as a small parameter $\epsilon \to 0$.

Proposed method

• The analysis uses functional analytic techniques in Hilbert spaces to handle the domain mismatch between $H_0(t)$ and $H_1(t)$, ensuring the total Hamiltonian remains well-defined for $\epsilon > 0$.
• The proof relies on the spectral gap assumption for $H_0(t)$ and control of the perturbation via operator norm estimates in the adiabatic limit.
• A key technical tool is the use of the time-ordered exponential and the adiabatic evolution operator, adapted to singular perturbations.
• The method involves constructing a unitary transformation that decouples the adiabatic subspace from the rest of the spectrum, even when the gap condition fails for the full Hamiltonian.
• The analysis considers the limit $\epsilon \to 0$ and derives bounds on the deviation of the true evolution from the adiabatic one.
• The framework allows for time-dependent Hamiltonians and incorporates the dynamics of the perturbation through a controlled asymptotic expansion.

Experimental results

Research questions

• RQ1 Can an adiabatic theorem be established when the perturbation breaks the spectral gap condition for the total Hamiltonian?
• RQ2 How can one define and analyze the adiabatic evolution when the perturbation operator $H_1(t)$ is defined on a smaller domain than $H_0(t)$?
• RQ3 What conditions ensure that the system remains in the adiabatic subspace despite the lack of a uniform spectral gap for $\epsilon > 0$?
• RQ4 Is it possible to extend the adiabatic theorem to singularly perturbed systems with large coupling constants in the time-dependent setting?

Key findings

• An adiabatic theorem holds for singularly perturbed Hamiltonians even when the total Hamiltonian $H_0(t) + \epsilon H_1(t)$ does not satisfy the standard spectral gap condition for $\epsilon > 0$.
• The adiabatic evolution remains valid under reasonable assumptions on the domain and norm behavior of $H_1(t)$, ensuring the system tracks the instantaneous eigenprojectors of $H_0(t)$ as $\epsilon \to 0$.
• The paper establishes uniform bounds on the deviation of the true time evolution from the adiabatic evolution, quantifying the error in terms of $\epsilon$ and the adiabatic parameter.
• The method applies to time-dependent systems with singular perturbations, generalizing the adiabatic theorem beyond the standard framework.
• The result is robust under weak assumptions on the perturbation, including cases where $H_1(t)$ is not globally defined on the domain of $H_0(t)$.
• The framework provides a rigorous foundation for studying systems with large coupling constants in quantum mechanics, where the perturbation is treated as a small parameter in the adiabatic limit.