On the asymptotic number of low-lying states in the two-dimensional confined Stark effect

This paper establishes Weyl-type asymptotics and derives weak asymptotics for the spectral projector density of low-lying states for the Stark operator restricted to a bounded two-dimensional domain with Dirichlet boundary conditions, building upon known three-term semiclassical expansions for individual eigenvalues.

The Gist

The Big Picture: A Bouncing Ball in a Slanted Room

Imagine you have a tiny, super-fast ball (representing a quantum particle, like an electron) trapped inside a room. This room isn't just any room; it's a bounded shape (like a circle or an oval) with walls the ball cannot pass through.

Now, imagine the room is tilted. Gravity is pulling the ball down one side. In physics, this "tilt" is called the Stark Effect (usually caused by an electric field). The ball wants to roll to the lowest point of the room, but because it's a quantum particle, it can't just sit still at the bottom. It has to "jiggle" or vibrate.

This paper is about counting how many different ways this ball can vibrate (its "energy states") when the room is very small and the ball is moving very fast (a scenario physicists call the semiclassical limit).

The Key Characters

1. The Room ($\Omega$): A 2D shape with smooth walls.
2. The "Lowest" Spot ($X_0$): The point on the wall where the "gravity" (the electric field) is weakest. The ball loves to hang out here.
3. The Curvature ($\kappa_0$): How "curvy" the wall is at that lowest spot. Is it a sharp corner or a gentle curve? This matters a lot.
4. The "Airy Function": Think of this as a special mathematical recipe that describes how the ball behaves when it hits a wall that gets steeper and steeper. The paper uses the "zeros" of this recipe (the specific points where the recipe hits zero) to predict energy levels.

The Problem: Counting the "Low-Lying" States

In the past, scientists figured out the energy of the single lowest state, the second lowest, and so on. They found a pattern:

• The lowest energy is roughly the height of the floor ($x_0$).
• The next bit of energy depends on the "jiggle" size ($h^{2/3}$).
• The bit after that depends on how curvy the wall is ($h$).

The question this paper answers: Instead of looking at just one specific energy level, how many total energy states are there below a certain threshold?

Imagine you are counting how many steps you can take before you reach a certain height on a staircase. The authors are counting the "steps" (energy states) that are very close to the bottom of the room.

The Method: The "Box" and the "Tube"

To solve this, the author uses a clever trick called Dirichlet–Neumann Bracketing.

• The Analogy: Imagine you want to count how many fish are in a huge, irregularly shaped lake. It's hard to count them all at once. So, you build a grid of small, perfect square boxes over the lake.
  • In some boxes, you assume the fish can't escape the walls (Dirichlet).
  • In others, you assume the walls are slippery and the fish can slide out (Neumann).
  • By counting the fish in the "tight" boxes and the "loose" boxes, you get a range. As you make the boxes smaller and smaller, the two counts meet in the middle, giving you the exact number.

The author applies this to the quantum ball. They zoom in on the "lowest spot" ($X_0$) and stretch the coordinates to make the curved wall look like a flat, straight tube. This makes the math much easier to handle.

The Main Discoveries

The paper finds two main things, which are like two different ways of looking at the crowd of particles:

1. The Total Count (The "Headcount")

The authors derive a formula to count exactly how many energy states exist below a certain energy level.

• The Result: The number of states grows in a very specific way related to the curvature of the wall.
• The Metaphor: If the wall is very curved (like a sharp point), the "steps" of the staircase are packed tighter together, so you can fit more steps in a small space. If the wall is flat, the steps are spread out. The formula tells you exactly how the shape of the room changes the number of available states.

2. The Density Map (The "Heat Map")

Instead of just counting the total number, the authors also look at where these states are located.

• The Result: They create a "density map" showing where the particles are most likely to be found.
• The Metaphor: Imagine spraying paint on the floor to show where the ball spends its time. The paper shows that the paint isn't spread evenly. It piles up heavily near the lowest spot ($X_0$) and spreads out in a specific pattern along the wall (like a wave) and away from the wall (like a ripple).
• The paper proves that as the room gets smaller, this "paint pattern" settles into a predictable shape determined by the Airy function (the wall-hitting recipe) and the curvature of the wall.

Why Does This Matter?

You might ask, "Who cares about counting quantum steps in a tilted room?"

• Real World: This math applies to semiconductors and nanotechnology. When we build tiny electronic devices (like transistors in your phone), electrons get trapped in small spaces and are pushed by electric fields.
• The Insight: Understanding exactly how many energy states exist and where they are located helps engineers predict how these tiny devices will conduct electricity. If the shape of the device changes slightly (changing the curvature), the number of available states changes, which changes how the device works.

Summary in One Sentence

This paper uses advanced math to count and map the "vibrations" of a quantum particle trapped in a curved, tilted room, revealing that the shape of the room's wall dictates exactly how many energy states are available and where the particle likes to hang out.

Technical Summary

1. Problem Statement

The paper investigates the spectral properties of the Stark operator restricted to a bounded, open, connected domain $\Omega \subset \mathbb{R}^2$ with Dirichlet boundary conditions. The operator is defined as:
$$L_h = -h^2 \Delta + x_1$$
where $h > 0$ is a semiclassical parameter tending to zero.

Key Physical Context:

• The potential $V(x) = x_1$ is linear (repulsive in the $x_1$ direction).
• The domain $\Omega$ has a unique point $X_0 = (x_0, y_0)$ on its boundary $\partial \Omega$ that minimizes the first coordinate $x_1$.
• The boundary is smooth near $X_0$ with positive curvature $\kappa_0 > 0$.
• Phenomenon: As $h \to 0$, the low-lying eigenstates (bound states) concentrate near $X_0$. The energy levels split due to the interplay between the linear potential (acting in the normal direction) and the boundary curvature (acting as an effective harmonic oscillator in the tangential direction).

Previous Work:
Cornean et al. [1] established a three-term asymptotic expansion for the $k$-th eigenvalue $\lambda_k(L_h)$:
$$\lambda_k(L_h) = x_0 + z_1 h^{2/3} + (2k-1)\sqrt{\frac{\kappa_0}{2}} h + O(h^{4/3})$$
where $-z_1 \approx -2.338$ is the first zero of the Airy function.

The Gap:
While individual eigenvalue asymptotics were known, the paper addresses the accumulation of eigenvalues beneath these leading-order terms. Specifically, it seeks to determine the asymptotic number of eigenvalues below a threshold $\Lambda(h)$ and the asymptotic density of the corresponding spectral projector.

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2. Methodology

The author employs a combination of semiclassical analysis, geometric transformations, and spectral theory techniques:

A. Tubular Coordinates and Rescaling

The analysis is localized near the point $X_0$ using tubular coordinates $(s, t)$, where $s$ is the arc-length along the boundary and $t$ is the distance into the domain (normal direction).

• The mapping $\tau(s, t) = \gamma(s) - t \mathbf{n}(s)$ transforms the domain near $X_0$ into a strip.
• The Jacobian of this transformation is $1 - \kappa(s)t$.
• The author performs specific rescalings of coordinates depending on the energy regime:
  • $s \sim h^{1/3}$ and $t \sim h^{2/3}$ for the leading order regime.
  • $s \sim h^{\alpha/2}$ and $t \sim h^{2/3}$ for higher-order regimes ($\alpha \in (2/3, 1)$).

B. Dirichlet–Neumann Bracketing

To bound the counting function $N(L_h, \Lambda)$, the author restricts the operator to a shrinking region $W_h$ around $X_0$ and applies Dirichlet–Neumann bracketing.

• The operator is bounded from above and below by operators with Dirichlet and Neumann boundary conditions on sub-intervals of the strip.
• This reduces the problem to analyzing operators on flat strips with effective potentials derived from the Taylor expansion of the boundary.

C. Separation of Variables and Quasi-States

The core of the analysis involves separating the variables in the rescaled operator:

1. Normal Component ($t$): The operator approximates the Airy operator $-\frac{d^2}{dt^2} + t$ (with Dirichlet BCs). Its eigenvalues are related to the zeros of the Airy function ($z_k$).
2. Tangential Component ($s$): The curvature $\kappa_0$ induces an effective harmonic oscillator potential $\frac{\kappa_0}{2}s^2$.
3. Quasi-States: The author constructs trial functions (quasi-states) using the product of Airy eigenfunctions and harmonic oscillator eigenfunctions to approximate the true eigenfunctions.

D. Riesz Means and Weak Asymptotics

Instead of just counting eigenvalues (which corresponds to the 0-th Riesz mean), the paper analyzes the $\gamma$-Riesz means:
$$\text{Tr}(L_h - \Lambda)_-^\gamma$$
This allows for a more robust analysis of the spectral density. The author also derives weak asymptotics for the density of the spectral projector $\rho_h$, showing how the mass of the eigenfunctions distributes in the rescaled coordinates.

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3. Key Contributions and Results

Result 1: Weyl-Type Asymptotics for Counting Functions (Theorem 1.1)

The paper establishes the asymptotic number of eigenvalues below specific energy thresholds. Two regimes are analyzed:

1. Regime 1 (Leading Order): Threshold $\Lambda = x_0 + \mu h^{2/3}$.
   $$\lim_{h \to 0} h^{(1-2\gamma)/3} \text{Tr}(L_h - \mu h^{2/3})_-^\gamma = 4\pi L_{\gamma, 2}^{cl} \frac{1}{\sqrt{2\kappa_0}} \sum_{k=1}^\infty (\mu - z_k)_+^{\gamma+1}$$
   Here, the sum runs over Airy zeros $z_k$. The term $(\cdot)_+$ denotes the positive part. This result generalizes Weyl's law to the confined Stark effect, showing the eigenvalue count depends on the curvature $\kappa_0$ and the Airy zeros.

2. Regime 2 (Next Order): Threshold $\Lambda = x_0 + z_1 h^{2/3} + \mu h^\alpha$ (with $\alpha \in (2/3, 1)$).
   $$\lim_{h \to 0} h^{1-\alpha(1+\gamma)} \text{Tr}(L_h - z_1 h^{2/3} - \mu h^\alpha)_-^\gamma = 4\pi L_{\gamma, 2}^{cl} \frac{1}{\sqrt{2\kappa_0}} \mu^{\gamma+1}$$
   In this regime, only the first Airy mode ($k=1$) contributes, and the asymptotics depend on the curvature and the parameter $\mu$.

Result 2: Weak Asymptotics of Spectral Density (Theorem 1.2)

The paper derives the weak limit of the density of states $\rho_h$ in the rescaled coordinates $(s, t) \in \mathbb{R}^2_+$.

• For the first regime:
  $$h^{4/3} \rho_h(\tau(h^{1/3}s, h^{2/3}t); \mu h^{2/3}) \rightharpoonup \frac{1}{\pi} \sum_{k=1}^\infty \left( \mu - \frac{\kappa_0}{2}s^2 - z_k \right)_+^{1/2} |a_k(t)|^2$$
• For the second regime:
  $$h^{5/3-\alpha/2} \rho_h(\tau(h^{\alpha/2}s, h^{2/3}t); z_1 h^{2/3} + \mu h^\alpha) \rightharpoonup \frac{1}{\pi} \left( \mu - \frac{\kappa_0}{2}s^2 \right)_+^{1/2} |a_1(t)|^2$$
  Here, $a_k(t)$ are the normalized eigenfunctions of the Airy operator. This result describes the spatial distribution of the low-lying states, confirming they concentrate near $X_0$ with a profile determined by the Airy function in the normal direction and a semi-circle profile (harmonic oscillator) in the tangential direction.

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4. Significance and Impact

1. Extension of Weyl's Law: The paper successfully adapts the classical Weyl law (which usually applies to bulk potentials) to a boundary-dominated spectral problem where the potential is linear and the domain is bounded. It explicitly quantifies how boundary curvature modifies the density of states.
2. Multi-Scale Analysis: It rigorously handles the different scaling regimes ($h^{2/3}$ vs $h^\alpha$) required to resolve the splitting of energy levels in the confined Stark effect.
3. Spectral Density Distribution: Unlike previous works that focused only on eigenvalue counting, this paper provides the spatial density of the eigenfunctions. This is crucial for understanding physical observables like charge distribution in quantum systems subject to strong electric fields near boundaries.
4. Methodological Rigor: The use of Dirichlet-Neumann bracketing combined with precise quasi-state constructions and perturbation theory provides a robust framework that can likely be extended to other singular perturbation problems involving boundary layers and curvature effects.

In summary, Larry Read provides a comprehensive mathematical description of the "confined Stark effect," proving that the accumulation of low-energy states is governed by a universal law involving Airy zeros and the local curvature of the boundary, and explicitly characterizing the spatial shape of these states.