[Paper Review] Blaschke-type conditions in unbounded domains, generalized convexity and applications in perturbation theory

S. Favorov, Leonid Golinskiĭ|arXiv (Cornell University)|Apr 19, 2012

Spectral Theory in Mathematical Physics17 references3 citations

TL;DR

This paper introduces $r$-convexity, a generalized convexity concept for compact sets in the complex plane, to establish Blaschke-type conditions for subharmonic functions on unbounded domains with boundary-governed growth. It applies these results to Schatten–von Neumann perturbation theory, proving discrete spectrum convergence estimates with explicit decay rates depending on the spectral set's geometry and perturbation strength.

ABSTRACT

We introduce a new geometric characteristic of compact sets on the plane called $r$-convexity, which fits nicely into the concept of generalized convexity and extends essentially the conventional convexity. For a class of subharmonic functions on unbounded domains with $r$-convex compact complement, with the growth governed by the distance to the boundary, we obtain the Blaschke--type condition for their Riesz measures. The result is applied to the study of the convergence of the discrete spectrum for the Schatten--von Neumann perturbations of bounded linear operators in the Hilbert space.

Motivation & Objective

• To develop a geometric framework for unbounded domains with compact complements that generalizes convexity and enables analysis of subharmonic functions with boundary-controlled growth.
• To establish a Blaschke-type condition for the Riesz measures of subharmonic functions on unbounded domains with $r$-convex compact complements.
• To apply the resulting estimates to Schatten–von Neumann perturbation theory, particularly for the discrete spectrum of bounded linear operators under trace-class or Schatten-p perturbations.
• To provide quantitative decay estimates for eigenvalues away from the essential spectrum, depending on the geometry of the spectrum and the Schatten norm of the perturbation.

Proposed method

• Introduce $r$-convexity as the intersection of all closed exteriors of disks of radius $r$ that contain the set, generalizing standard convexity.
• Define the outer neighborhood $\Omega_t = \{z : \text{dist}(z,E) > t\}$ and use Green's functions on $\Omega_t$ to analyze subharmonic functions with growth controlled by $\psi(\text{dist}(z,E))$.
• Apply the Riesz decomposition theorem to represent subharmonic functions as the sum of a harmonic function and a potential with Riesz measure $\mu = \frac{1}{2\pi}\Delta v$.
• Derive a key integral estimate involving the Green function and the Riesz measure, leading to a Blaschke-type condition: $\sum \Phi(\text{dist}(\lambda,E)) \leq C \|B\|_{{\mathcal{S}}_q}^q$.
• Use conformal mapping and distortion estimates to transfer results from the unit disk to unbounded domains, particularly when the spectrum is real or on the unit circle.
• Establish bounds on the support of the Riesz measure in terms of the operator norm $\|B\|$ and the function $\Psi$, ensuring the measure is supported in a controlled region.

Experimental results

Research questions

• RQ1 Can Blaschke-type conditions for Riesz measures be established in unbounded domains with non-convex, but $r$-convex, compact complements?
• RQ2 How does the geometry of the spectrum—specifically $r$-convexity—affect the decay rate of discrete eigenvalues under Schatten–von Neumann perturbations?
• RQ3 Can the standard conformal mapping approach to perturbation theory be generalized beyond finite unions of intervals or single segments using potential-theoretic methods?
• RQ4 What is the precise dependence of the eigenvalue decay rate on the Schatten norm $\|B\|_{{\mathcal{S}}_q}$ and the spectral set's Minkowski dimension?
• RQ5 Under what geometric conditions on the spectrum $\sigma(A_0)$ can one obtain uniform bounds on the discrete spectrum of $A = A_0 + B$ independent of the spectral transform?

Key findings

• For $r$-convex compact sets $E$, the Blaschke-type condition $\sum_{n}(1-|z_n|)\text{dist}^p(z_n,E) \leq C K_f$ holds for subharmonic functions with growth $|\log|f(z)|| \leq K_f / \text{dist}^q(z,E)$, where $p = \max(q + \kappa(E) - 1 + \varepsilon, 0)$ and $\kappa(E)$ is the upper Minkowski dimension of $E$.
• The Riesz measure $\mu$ of a subharmonic function $v$ on an unbounded domain $\Omega = \overline{\mathbb{C}} \setminus E$ with $v(\infty) = 0$ and $v(z) \leq K_v \psi(\text{dist}(z,E))$ satisfies a weighted summability condition: $\sum \Phi(\text{dist}(\lambda,E)) \leq C \|B\|_{{\mathcal{S}}_q}^q$ for suitable weight functions $\Phi$.
• For a bounded operator $A_0$ with real or unitary spectrum, and a Schatten–$q$ perturbation $B$, the discrete spectrum $\sigma_d(A)$ satisfies $\sum_{\lambda \in \sigma_d(A)} d^{pq+1+\varepsilon}(\lambda) < \infty$ when $\psi(t) = t^{-p}$, with $p > 0$, and $q > 0$.
• In the case of Hilbert–Schmidt perturbations ($q=2$) of a normal operator with a $C^2$-smooth or BC-arc spectrum $\gamma$, the bound $\sum \Phi(d(\lambda)) \leq C \|K\|_{{\mathcal{S}}_2}^2$ holds with $\Phi(x) = x^{3+\varepsilon}$ for $x \leq 1$ and $x^{2-\varepsilon}$ for $x > 1$.
• The support of the Riesz measure is uniformly bounded in terms of $\|B\|$, ensuring that the discrete spectrum lies within a controlled region depending on the perturbation norm and the function $\Psi$.