[Paper Review] Lifshitz asymptotics for Hamiltonians monotone in the randomness

Ivan Veselić|ArXiv.org|Aug 3, 2007

Spectral Theory in Mathematical Physics5 references3 citations

TL;DR

This paper establishes Lifshitz asymptotics for the integrated density of states (IDS) of random Schrödinger operators with Hamiltonians monotone in the randomness, extending prior results to a class of breather-type potentials with discontinuous characteristic functions. Using Thirring's inequality instead of Temple's inequality, it proves the key Lifshitz bound $\lim_{E\searrow E_0} \frac{\log|\log N(E)|}{\log(E-E_0)} \leq -\frac{d}{2}$, confirming the exponential decay of the IDS near the spectral edge for models not amenable to previous methods.

ABSTRACT

In various aspects of the spectral analysis of random Schroedinger operators monotonicity with respect to the randomness plays a key role. In particular, both the continuity properties and the low energy behaviour of the integrated density of states (IDS) are much better understood if such a monotonicity is present in the model than if not. In this note we present Lifshitz-type bounds on the IDS for two classes of random potentials. One of them is a slight generalisation of a model for which a Lifshitz bound was derived in a recent joint paper with Werner Kirsch [KV]. The second one is a breather type potential which is a sum of characteristic functions of intervals. Although the second model is very simple, it seems that it cannot be treated by the methods of [KV]. The models and the proofs are motivated by well-established methods developed for so called alloy type potentials.

Motivation & Objective

• To extend Lifshitz asymptotics to random Schrödinger operators with Hamiltonians monotone in the randomness, particularly for models not covered by prior methods.
• To establish the low-energy behavior of the integrated density of states (IDS) for a class of breather-type potentials defined by characteristic functions of intervals.
• To overcome the failure of Temple’s inequality in models with discontinuous potentials by employing Thirring’s inequality for spectral lower bounds.
• To demonstrate that the Lifshitz tail exponent $-d/2$ holds even when the potential lacks Lipschitz continuity in the randomness parameter.

Proposed method

• The analysis uses a finite-volume approximation of the IDS via eigenvalue counting on intervals $\Lambda_L = [-L/2, L/2]$, with Neumann boundary conditions.
• The proof relies on Thirring’s inequality to bound the first eigenvalue $E_1(H_{\omega}^L)$ from below, replacing Temple’s inequality which fails due to equal first and second moments in the chosen trial state.
• A comparison operator $H_0^L = -\Delta - \alpha/(4L^2)$ is introduced, with a perturbation $V_\omega = \alpha/(4L^2) + W_\omega$, to control the spectral gap.
• The key estimate uses the inverse of the average potential $\tilde{S}_L = L^{-1} \sum_{k \in I_L} \tilde{\lambda}_k$, where $\tilde{\lambda}_k = \min(\lambda_k, 1/2)$, to derive a lower bound on $E_1(H_\omega^L)$.
• Large deviation estimates are applied to $\mathbb{P}(\tilde{S}_L \leq \mathbb{E}[\tilde{S}_L]/2)$, yielding an exponential tail bound of order $e^{-\tilde{c} E^{-d/2}}$ for the probability that the first eigenvalue is below $E$.
• The final bound on the IDS is derived by combining the finite-volume eigenvalue estimate with the standard IDS approximation $N(E) \leq L^{-1} \mathbb{E}[\text{Tr}(\chi_{]-\infty,E]}(H_\omega^L))]$.

Experimental results

Research questions

• RQ1 Can Lifshitz asymptotics be established for random Schrödinger operators with Hamiltonians monotone in the randomness when the potential lacks Lipschitz continuity in the randomness parameter?
• RQ2 Why does Temple’s inequality fail to yield sharp bounds in the case of characteristic function potentials, and what alternative method can be used?
• RQ3 Does the standard Lifshitz tail exponent $-d/2$ still hold for discontinuous breather-type potentials, such as those defined by $u(\lambda,x) = \chi_{]0,\lambda]}(x)$?
• RQ4 Can Thirring’s inequality be effectively used to derive spectral lower bounds in models where the first and second moments of the Hamiltonian are comparable?

Key findings

• The IDS satisfies the Lifshitz bound $\lim_{E\searrow E_0} \frac{\log|\log N(E)|}{\log(E-E_0)} \leq -\frac{d}{2}$, confirming the expected exponential decay near the spectral minimum.
• For the breather-type potential $W_\omega(x) = \sum_{k\in\mathbb{Z}} \chi_{]0,\lambda_k]}(x-k)$, the IDS decays as $N(E) \leq e^{-\tilde{c} E^{-d/2}}$ for small $E > 0$, with $E_0 = 0$.
• The method successfully bypasses the failure of Temple’s inequality, which breaks down due to equality of first and second moments in the periodic ground state.
• Thirring’s inequality provides a robust alternative, yielding a lower bound on the first eigenvalue proportional to $\alpha \tilde{S}_L / (5L^2)$, which is sufficient for the asymptotic analysis.
• Large deviation estimates on $\tilde{S}_L$ yield a probability bound $\mathbb{P}(\tilde{S}_L \leq \mathbb{E}[\tilde{S}_L]/2) \leq e^{-cL^d}$, which translates to $e^{-\tilde{c} E^{-d/2}}$ after choosing $L \sim \beta E^{-1/2}$.