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The Birman--Solomyak theorem revisited: a novel elementary proof, generalisation, and applications

V. Bach, A. F. M. ter Elst, J. Rehberg

TL;DR

The work revisits the Birman–Solomyak theorem by delivering an elementary proof of the Hilbert–Schmidt case (p=2) for Lipschitz functions, then extends the result to arbitrary unbounded self-adjoint A,B with Hilbert–Schmidt perturbations. It also derives a resolvent–based HS bound, linking $f(A)-f(B)$ to the HS difference of resolvents, and provides a constructive contractive iteration framework for solving strongly monotone operator equations. A central application is to a Schrödinger–Poisson system within density functional theory, where the particle density map is shown to be locally Lipschitz and the SP solution depends Lipschitz-continuously on data. The results yield both theoretical insights and practical tools for density matrices and Kohn–Sham–type problems, with a focus on elementary proofs and accessibility.

Abstract

We provide a new short proof for the Birman--Solomyak theorem for Hilbert--Schmidt operators and give an application to a Schrödinger--Poisson system.

The Birman--Solomyak theorem revisited: a novel elementary proof, generalisation, and applications

TL;DR

The work revisits the Birman–Solomyak theorem by delivering an elementary proof of the Hilbert–Schmidt case (p=2) for Lipschitz functions, then extends the result to arbitrary unbounded self-adjoint A,B with Hilbert–Schmidt perturbations. It also derives a resolvent–based HS bound, linking to the HS difference of resolvents, and provides a constructive contractive iteration framework for solving strongly monotone operator equations. A central application is to a Schrödinger–Poisson system within density functional theory, where the particle density map is shown to be locally Lipschitz and the SP solution depends Lipschitz-continuously on data. The results yield both theoretical insights and practical tools for density matrices and Kohn–Sham–type problems, with a focus on elementary proofs and accessibility.

Abstract

We provide a new short proof for the Birman--Solomyak theorem for Hilbert--Schmidt operators and give an application to a Schrödinger--Poisson system.

Paper Structure

This paper contains 5 sections, 26 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. The elementary proof
  3. Resolvent estimates
  4. A contractive iteration scheme
  5. An application in density functional theory

Key Result

Theorem 1.1

Let $A,B$ be two self-adjoint operators in a Hilbert space $H$. Let $C \in \mathcal{L}_{\rm HS}(H)$ and suppose that $A = B + C$. Let $f \colon \mathds{R} \to \mathds{R}$ be a Lipschitz continuous function with Lipschitz constant $L$. Then $D(A) = D(B) \subset D(f(A)) = D(f(B))$ and the operator $f(

Theorems & Definitions (51)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 41 more