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On the Integrated Density of States of Fractional Random Schrödinger Operators

This paper establishes the existence of the Integrated Density of States for fractional random Schrödinger operators with Gaussian potentials using isotropic $\alpha$ -stable Lévy processes, while also demonstrating Lifshitz tails and determining the asymptotic behavior at the right end of the spectrum.

Original authors: Peter Kern, Leonard Pleschberger

Published 2026-09-16

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CC BY 4.0

Original authors: Peter Kern, Leonard Pleschberger

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, particles do not move like tiny billiard balls rolling across a table. Instead, they behave like waves, spreading out and interfering with one another. To predict where these particles might be found, physicists use a mathematical tool called the Schrödinger operator. Think of this tool as a map that describes the energy landscape a particle travels through. In a perfect, empty universe, this landscape is smooth and predictable. But in the real world, matter is messy. Random impurities, fluctuating fields, and chaotic environments create a bumpy, unpredictable terrain. When scientists study how particles behave in these messy environments, they are looking at what is called a random Schrödinger operator. The goal is to count how many different energy states are available to the particles up to a certain level. This count, known as the integrated density of states, acts like a census of the quantum possibilities, telling us how crowded the energy levels are at different points.

For decades, researchers have understood how this census works when the particles move in the standard, continuous way we are used to seeing in everyday life. However, nature sometimes allows particles to move in stranger ways, making sudden, long jumps rather than smooth steps. This behavior is described by a concept called the fractional Laplacian, which governs particles that follow a specific type of random motion known as a Lévy process. These particles are self-similar, meaning their pattern of movement looks the same whether you zoom in or zoom out, but they are not continuous; they can teleport across gaps. Until now, it was unclear how to count the energy states for these jumping particles when they are also subjected to random, messy environments.

In a recent study, Peter Kern and Leonard Pleschberger tackled this problem by proving that the census of energy states, the integrated density of states, actually exists for these fractional random operators. They focused on two types of messy environments: one where the randomness comes from a smooth, bell-curve distribution known as a Gaussian potential, and another where the randomness comes from scattered points, like a Poissonian potential. The researchers showed that even with these strange, jumping particles and chaotic surroundings, the number of available energy states is well-defined and can be calculated. They did not just prove it exists; they determined exactly how the count behaves at the very lowest and very highest ends of the energy spectrum.

At the lowest energy levels, the researchers found that the number of available states drops off extremely rapidly, following a pattern known as Lifshitz tails. This means that as you look for very low energy states, they become vanishingly rare. Surprisingly, the speed at which these states disappear depends entirely on the nature of the random environment, not on the specific way the particles jump. Whether the particles are moving in a standard way or making long, fractional jumps, the low-energy behavior is dictated by the messiness of the landscape itself. This finding suggests that at the bottom of the energy scale, the specific rules of motion matter less than the random obstacles the particles encounter.

At the opposite end of the spectrum, looking at very high energy levels, the story changes. Here, the researchers discovered that the behavior of the energy states depends solely on the jumping nature of the particles, specifically a parameter that controls how far they tend to leap. The type of random environment, whether it is the smooth Gaussian kind or the scattered Poissonian kind, does not influence the high-energy count. The researchers were able to calculate the precise mathematical relationship for this growth, showing that the number of states increases in a predictable way determined by the dimension of the space and the jumpiness of the particles.

The study relied on a deep connection between the mathematics of these operators and the probability theory of random processes. By treating the particles as travelers moving through a random landscape and using a formula that links their movement to their energy, the authors could translate a difficult physics problem into a solvable probability problem. They proved that for the smooth Gaussian environment, the low-energy drop-off is governed by the variance of the random field, while the high-energy growth is governed by the jump parameter. For the scattered Poissonian environment, they confirmed that the high-energy behavior follows the same rule as the Gaussian case, depending only on the jump parameter.

This work fills a significant gap in our understanding of quantum systems in disordered media. It confirms that the concept of counting energy states remains valid even when the underlying motion of particles is non-local and discontinuous. The researchers demonstrated that the universe of quantum possibilities is orderly even in the most chaotic settings, with distinct rules governing the extremes of the energy spectrum. The low end is ruled by the chaos of the environment, while the high end is ruled by the nature of the motion itself. These results provide a solid foundation for future studies of complex quantum materials where particles might not move in the traditional, continuous way we expect.

Technical Summary: On the Integrated Density of States of Fractional Random Schrödinger Operators

Problem Statement
The paper investigates the spectral properties of fractional random Schrödinger operators defined on $\mathbb{R}^d$ by the equation:
$H_\omega := (-\Delta)^{\alpha/2} + V_\omega$
where $\alpha \in (0, 2]$ and $V_\omega$ represents a random potential (specifically Gaussian or Poissonian). The primary objective is to establish the existence of the Integrated Density of States (IDS), denoted as $IDS(\lambda)$ , which characterizes the average number of quantum states per unit volume up to an energy level $\lambda$ . Furthermore, the authors aim to determine the precise asymptotic behavior of the IDS at both the left end ( $\lambda \to -\infty$ ) and the right end ( $\lambda \to +\infty$ ) of the spectrum.

A key distinction of this work is the use of the fractional Laplacian $(-\Delta)^{\alpha/2}$ , a non-local operator, as opposed to the classical Laplacian ( $\alpha=2$ ). The authors note that while the existence and asymptotics for $\alpha=2$ (classical case) were established by Nakao [16] and generalized by $\bar{\text{Ok}}$ ura [18] for Poissonian potentials, the specific asymptotic behavior for fractional operators with Gaussian potentials required further rigorous analysis.

Methodology
The authors employ a probabilistic approach grounded in the theory of isotropic $\alpha$ -stable Lévy processes. The methodology relies on the following core components:

Stochastic Representation: The fractional Laplacian is treated as the generator of a strongly continuous semigroup associated with an isotropic $\alpha$ -stable Lévy process $X_t$ . The operator $H_\omega$ is analyzed via the Feynman-Kac formula, linking the semigroup $e^{-tH_\omega}$ to the expectation of functionals of the Lévy process:
$e^{-tH_\omega}[\psi](x) = \mathbb{E}_x \left[ e^{-\int_0^t V_\omega(X_s) ds} \psi(X_t) \right]$
For bounded domains $\Lambda$ , Dirichlet exterior conditions are imposed, corresponding to the first exit time $\tau_\Lambda$ of the process from the domain.
Existence Proof: The existence of the IDS is derived by applying a general theorem by $\bar{\text{Ok}}$ ura [18]. This involves verifying two conditions regarding the Lévy exponent $Q(\xi) = |\xi|^\alpha$ and the integrability of the exponential moments of the negative part of the potential $V_\omega^-$ .
Asymptotic Analysis via Laplace Transforms: To determine the asymptotics of the IDS, the authors analyze the Laplace-Stieltjes transform of the IDS, $L[IDS](t)$ .
- Left End ( $\lambda \to -\infty$ ): The behavior of $L[IDS](t)$ as $t \to +\infty$ is analyzed using bounds derived from $\bar{\text{Ok}}$ ura's work and properties of the Dirichlet form. A Tauberian theorem (Fukushima, Nagai, and Nakao [7]) is then applied to translate the asymptotic behavior of the transform at infinity to the behavior of the IDS at $-\infty$ .
- Right End ( $\lambda \to +\infty$ ): The behavior of the positive unilateral Laplace-Stieltjes transform as $t \to 0$ is derived. This relies on the scaling properties of the isotropic $\alpha$ -stable Lévy process and the fact that the potential term becomes negligible in the limit $t \to 0$ . A Hardy-Littlewood type Tauberian theorem is used to deduce the polynomial growth of the IDS at $+\infty$ .

Key Contributions and Results

Existence of IDS: The paper proves that the IDS exists for fractional random Schrödinger operators with stationary Gaussian potentials. This extends previous results which were primarily focused on the classical Laplacian ( $\alpha=2$ ) or Poissonian potentials.
Lifshitz Tails (Left End): The authors demonstrate that the IDS exhibits Lifshitz tails, characterized by an exponential decay as $\lambda \to -\infty$ . Specifically, for Gaussian potentials with covariance $c(x)$ , the asymptotics are given by:
$\lim_{\lambda \downarrow -\infty} \frac{\ln IDS(\lambda)}{\lambda^2} = -\frac{1}{2c(0)}$
Crucially, the authors show that this asymptotic behavior is independent of the fractional order $\alpha \in (0, 2)$ . The decay rate is determined entirely by the local properties of the random potential (specifically the covariance at the origin).
Right-End Asymptotics: The paper determines the asymptotic behavior of the IDS as $\lambda \to +\infty$ for both Gaussian and Poissonian potentials. In both cases, the IDS grows polynomially:
$\lim_{\lambda \to +\infty} \frac{IDS(\lambda)}{\lambda^{d/\alpha}} = \frac{p(1, 0)}{\Gamma(d/\alpha + 1)}$
where $p(1, 0)$ is the transition density of the isotropic $\alpha$ -stable Lévy process at time $t=1$ and position $0$. Unlike the left end, the asymptotics at the right end depend on the fractional parameter $\alpha$ .
Unified Treatment: The paper provides a unified framework showing that while the left-end behavior is potential-dominated, the right-end behavior is kinetic (operator) dominated.

Significance and Claims
The paper claims that self-similarity is a fundamental concept for these quantum operators due to the nature of isotropic $\alpha$ -stable Lévy processes, which are self-similar but not continuous. The primary significance lies in rigorously establishing the existence of the IDS for the fractional case with Gaussian potentials and clarifying the distinct roles of the random potential and the fractional kinetic term in determining spectral asymptotics.

The authors explicitly state that their stochastic proof differs from Nakao's technique [16] used for the classical case. They highlight that their results confirm that the "Lifshitz tail" phenomenon (exponential decay at the bottom of the spectrum) is robust against the introduction of non-locality (fractional Laplacian), as the decay rate remains unchanged from the classical case. Conversely, the high-energy behavior is shown to be sensitive to the fractional order $\alpha$ , reflecting the non-local nature of the operator.

The work does not propose new experimental applications or future physical models but serves as a mathematical foundation for understanding the spectral statistics of disordered quantum systems governed by non-local dynamics.

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