On the discrete spectrum of non-selfadjoint operators with applications to Schrödinger operators with complex potentials

This paper establishes a novel upper bound on the number of discrete eigenvalues for relatively form-compact perturbations of non-negative selfadjoint operators in terms of a partial trace of the Birman--Schwinger operator, thereby generalizing classical Cwikel--Lieb--Rozenblum and Lieb--Thirring inequalities to the non-selfadjoint setting of Schrödinger operators with complex potentials.

The Gist

Imagine you are a detective trying to solve a mystery in a vast, invisible city called Quantum Mechanics. In this city, there are "buildings" (mathematical objects called operators) that determine how particles like electrons move.

Usually, these buildings are very predictable. If you push a button (add energy), the lights turn on in a specific, orderly way. Mathematicians call these self-adjoint operators. They are like well-behaved citizens who always follow the rules. For these, we have a very famous rulebook (the CLR inequality) that tells us exactly how many "ghosts" (discrete eigenvalues) can hide in the building based on how strong the "wind" (the potential) is blowing outside.

The Problem: The Chaotic City
But what happens when the city gets chaotic? What if the buildings are non-self-adjoint?
In this chaotic version, the "wind" (the potential) isn't just blowing; it's swirling in complex, imaginary directions. The ghosts (eigenvalues) don't just hide in the basement (negative numbers); they can hide anywhere in the complex sky, swirling around and potentially piling up infinitely close to the city limits.

For a long time, mathematicians had no rulebook for this chaotic city. They knew the old rules didn't work, but they didn't know how to count the ghosts or predict where they would gather.

The New Detective Work: The "Birman-Schwinger" Mirror
The authors of this paper, Sabine and Sukrid, have built a new tool to solve this mystery. They use a clever trick called the Birman-Schwinger principle.

Think of the original chaotic building as a giant, confusing maze. It's hard to count the ghosts inside. But the Birman-Schwinger principle is like a magic mirror.

1. You look at the maze through the mirror.
2. The mirror transforms the confusing maze into a much simpler, smaller object (a compact operator).
3. The ghosts in the maze correspond to specific "shadows" in the mirror.

In the old, orderly world, counting the ghosts was easy: you just counted how many shadows were darker than a certain threshold. But in the chaotic world, the shadows are tilted and distorted. The authors' breakthrough is figuring out how to count these tilted shadows even when they are messy.

The "Anti-Symmetric" Team
How did they do it? They used a technique involving antisymmetric tensor products.
Imagine you have a team of detectives. In the old method, you just looked at them one by one. But in the chaotic world, the detectives get confused if they stand next to each other.
So, the authors created a special rule: No two detectives can stand in the same spot. If they try, they cancel each other out (like a "do not enter" sign).
By forcing the detectives to stand in unique, non-overlapping positions, the authors could organize the chaos. This "teamwork" allowed them to prove that even in the messiest, most complex building, the number of ghosts is still limited by the strength of the wind, provided you look in the right direction.

The Results: New Rules for the Chaos
With this new method, they derived two major findings:

1. The Generalized Counting Rule (The New CLR):
   They proved that even with complex, swirling winds, you can still put a cap on how many ghosts hide in specific "half-planes" (areas of the sky). The number of ghosts depends on how strong the "negative part" of the wind is. It's like saying: "No matter how crazy the wind swirls, if it's not strong enough in a specific direction, you can't have an infinite number of ghosts hiding there."

2. The "Lieb-Thirring" Energy Sum:
   They also created a new way to calculate the total "energy" of these ghosts. In the orderly world, you just add up their energies. In the chaotic world, they found a way to add them up while ignoring the ones that are too close to the city limits (the essential spectrum). This helps physicists understand how stable matter is, even when the forces acting on it are weird and complex.

Why Does This Matter?
Think of this as upgrading the map of a city. Before, we only had a map for the sunny, orderly suburbs. Now, thanks to this paper, we have a map for the stormy, chaotic downtown.

• For Physicists: It helps them understand quantum systems with complex potentials (like those found in optics or certain materials) where energy isn't conserved in the usual way.
• For Mathematicians: It fills a huge gap in theory, showing that even when things get messy and non-real, there are still strict mathematical laws governing how many "solutions" can exist.

In a Nutshell:
The authors took a chaotic, confusing mathematical problem where the usual rules broke down. They built a new "magic mirror" (using a clever team-based counting method) to translate the chaos into something countable. They proved that even in the wildest, most complex quantum environments, the number of hidden states is still controlled by the strength of the forces acting on them. They didn't just find a few ghosts; they wrote the rulebook for counting them all.

Technical Summary

1. Problem Statement

The paper addresses a fundamental gap in the spectral theory of non-selfadjoint operators, specifically Schrödinger operators with complex potentials ($H = -\Delta + V$ where $V$ is complex-valued).

• The Challenge: In the selfadjoint setting (real potentials), the Cwikel–Lieb–Rozenblum (CLR) inequality provides a quantitative bound on the number of negative eigenvalues based on the $L^{d/2}$ norm of the potential. Similarly, Lieb–Thirring (LT) inequalities bound sums of powers of eigenvalues.
• The Obstruction: For non-selfadjoint operators, discrete eigenvalues can be complex and may accumulate at any point on the essential spectrum $[0, \infty)$. Classical CLR and LT inequalities fail in this setting; specifically, the total number of discrete eigenvalues is not controlled by the $L^{p}$ norm of the potential alone.
• The Specific Gap: While previous work (e.g., Frank et al.) established LT-type bounds for eigenvalues in specific regions (like the left half-plane) for $\gamma \ge 1$, the case for $0 \le \gamma < 1$ remained open. The standard proof techniques relying on the convexity of the function $x \mapsto x^\gamma$ (used for $\gamma \ge 1$) fail for $\gamma < 1$.

2. Methodology

The authors develop a novel abstract framework to derive eigenvalue counting estimates for non-selfadjoint operators, which they then apply to Schrödinger operators.

A. Abstract Framework: Birman–Schwinger Principle in Antisymmetric Spaces

The core innovation is a non-selfadjoint analogue of the classical variational arguments used in the selfadjoint case.

• Birman–Schwinger (BS) Operator: The authors utilize the BS principle, which relates eigenvalues of $H$ to eigenvalues of a compact operator $S$. For a perturbation $V$, $S = (H_0 + \varepsilon)^{-1/2} V (H_0 + \varepsilon)^{-1/2}$.
• Antisymmetric Tensor Products: Instead of using the variational principle directly on the operator $H$ (which is unavailable for non-selfadjoint operators), the authors work in antisymmetric tensor product spaces ($\wedge^N \mathcal{H}$).
  • They lift the operator to the $N$-fold tensor product space.
  • They utilize properties of the antisymmetrization operator to relate the sum of eigenvalues of the BS operator to the trace of the operator acting on the antisymmetric subspace.
• Handling Algebraic Multiplicities: A critical technical hurdle is that non-selfadjoint operators can have non-trivial Jordan chains (algebraic multiplicity > geometric multiplicity). The authors employ a perturbation technique (Lemma 7) using a finite-rank operator $K_0$ to break Jordan chains into simple eigenvalues, allowing them to apply geometric multiplicity estimates and then pass to the limit.

B. Application to Schrödinger Operators

The abstract result is applied to $H = -\Delta + V$ in $L^2(\mathbb{R}^d)$.

• Weak Schatten Class Estimates: To obtain concrete bounds, the authors combine their abstract counting estimate with Cwikel's estimates for weak Schatten class operators. This allows them to bound the singular values of the BS operator in terms of the $L^p$ norms of the potential.
• Complex Potentials: They specifically analyze the real and imaginary parts of the potential ($V = \text{Re} V + i \text{Im} V$) to define appropriate half-planes in the complex spectrum.

3. Key Contributions and Results

A. Abstract Eigenvalue Counting Estimate (Theorem 2)

The paper establishes a general bound for the number of discrete eigenvalues $N$ of a non-negative selfadjoint operator $H_0$ perturbed by a relatively form-compact operator, located in a half-plane defined by $\text{Re} \lambda + \alpha \text{Im} \lambda < -\varepsilon$.
$$N \le \sum_{j=1}^N E_j(-\text{Re} S - \alpha \text{Im} S)$$
where $S$ is the Birman–Schwinger operator and $E_j$ denotes the $j$-th eigenvalue ordered non-increasingly. This is the first connection of its kind between the number of discrete eigenvalues and the BS operator in the non-selfadjoint theory.

B. Generalized Cwikel–Lieb–Rozenblum (CLR) Inequality (Theorem 9)

For Schrödinger operators with complex potentials, the authors prove a quantitative bound on the number of eigenvalues in a half-plane:
$$N(\text{Re} \lambda + \alpha \text{Im} \lambda < -\varepsilon; -\Delta + V) \le C_{d,p} \varepsilon^{-\gamma} \int_{\mathbb{R}^d} (\text{Re} V(x) + \alpha \text{Im} V(x))_-^p \, dx$$

• Significance: This generalizes the classical CLR inequality (recovered when $d \ge 3, \gamma=0, \alpha=0$, and $V$ is real) to the non-selfadjoint case. It proves that while eigenvalues can accumulate, their density in regions separated from the essential spectrum is controlled by the potential's norm.
• Sharpness: The constant is proven to be sharp in the 1D case ($d=1, p=1$).

C. Lieb–Thirring (LT) Type Inequalities for $\gamma < 1$ (Theorem 14 & 17)

The paper resolves the open problem for $0 \le \gamma < 1$:

1. Half-Plane Bounds (Theorem 14): They derive LT inequalities for sums of powers of eigenvalues in half-planes:
   $$\sum_{\lambda_j} (\text{Re} \lambda_j + \alpha \text{Im} \lambda_j)_-^\gamma \le \tilde{C}_{d,p} \int (\text{Re} V + \alpha \text{Im} V)_-^p$$
   This holds for $\gamma > 1/2$ ($d=1$), $\gamma > 0$ ($d=2$), and $\gamma \ge 0$ ($d \ge 3$).
2. Sector-Free Bounds (Theorem 17): By integrating over the angle of the half-plane, they derive an inequality for all discrete eigenvalues, weighted by their distance to the essential spectrum:
   $$\sum_{\lambda \in \sigma_d} \frac{\text{dist}(\lambda, [0, \infty))^p}{|\lambda|^{d/2}} f\left(-\log \frac{\text{dist}(\lambda, [0, \infty))}{|\lambda|}\right) \le C \int |V|^p$$
   This provides quantitative information on the accumulation rate of eigenvalues near the essential spectrum, showing that they must approach the real axis sufficiently fast.

4. Significance

• Theoretical Breakthrough: The paper successfully extends the powerful machinery of the Birman–Schwinger principle and antisymmetric tensor spaces to the non-selfadjoint regime, overcoming the lack of a variational principle.
• Resolution of Open Problems: It closes the gap for Lieb–Thirring inequalities with $\gamma < 1$ in the non-selfadjoint setting, a regime where previous convexity-based methods failed.
• Physical Relevance: Complex potentials arise in various physical contexts (e.g., PT-symmetric quantum mechanics, open quantum systems, optical lattices). These results provide rigorous bounds on the stability and spectral properties of such systems, ensuring that the "instability" caused by complex potentials is quantifiable and controlled.
• Optimality: The authors discuss the sharpness of constants and the optimality of the bounds, noting that while the constants may differ from the selfadjoint case, the structural form of the inequalities remains robust.

In summary, this work provides a comprehensive and rigorous extension of spectral bounds from selfadjoint to non-selfadjoint Schrödinger operators, offering new tools to analyze the discrete spectrum of systems with complex interactions.