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Scattering matrices for perturbations of Laplace operator by infinite sums of zero-range potentials

Adamyan Vadym

TL;DR

This work addresses scattering for the three-dimensional Laplacian perturbed by infinite sums of zero-range potentials. It employs Krein's resolvent formula in an abstract singular-perturbation framework to relate the perturbed resolvent $R_L(z)$ to the unperturbed one via $R_L(z)=R(z)-G(z)[Q(z)+L]^{-1}G(\bar z)^*$, establishing conditions under which the resolvent difference is trace class and hence the perturbation is 'close'. When closeness holds, wave operators exist and the scattering operator $S(A_L,A)$ acts by multiplication by an operator-valued function $S(\lambda)$ in the spectral representation, with an explicit expression involving $\Gamma(\lambda+i0)=[J_L+\tilde Q(\lambda+i0)]^{-1}$. The theory is then specialized to $A=-\Delta$ in $L_2(\mathbb{R}^3)$ with both finite and infinite point interactions: for a lattice of zero-range potentials, the resolvent and scattering matrix are given by Krein's formula with explicit kernels on the sphere $\mathbf{S}_2$, and the kernel is continuous in $\lambda>0$ under the prescribed conditions, enabling concrete computations of scattering data in these singular perturbation models.

Abstract

This paper analyzes the scattering matrix for two unbounded self-adjoint operators: the standard Laplace operator in three-dimensional space and a second operator that differs from the first by an infinite sum of zero-range potentials.

Scattering matrices for perturbations of Laplace operator by infinite sums of zero-range potentials

TL;DR

This work addresses scattering for the three-dimensional Laplacian perturbed by infinite sums of zero-range potentials. It employs Krein's resolvent formula in an abstract singular-perturbation framework to relate the perturbed resolvent to the unperturbed one via , establishing conditions under which the resolvent difference is trace class and hence the perturbation is 'close'. When closeness holds, wave operators exist and the scattering operator acts by multiplication by an operator-valued function in the spectral representation, with an explicit expression involving . The theory is then specialized to in with both finite and infinite point interactions: for a lattice of zero-range potentials, the resolvent and scattering matrix are given by Krein's formula with explicit kernels on the sphere , and the kernel is continuous in under the prescribed conditions, enabling concrete computations of scattering data in these singular perturbation models.

Abstract

This paper analyzes the scattering matrix for two unbounded self-adjoint operators: the standard Laplace operator in three-dimensional space and a second operator that differs from the first by an infinite sum of zero-range potentials.

Paper Structure

This paper contains 4 sections, 7 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Resolvents of singular selfadjoint perturbations of Laplace operator
  3. Scatttering matrices
  4. Clarification for perturbations of the Laplace operator by infinite sums of zero-range potentials

Key Result

Theorem 1.1

Let $\mathcal{H}$ and $\mathcal{K}$ be Hilbert spaces, $A$ be an unbounded selfadjoint operator in $\mathcal{H}$ and $R(z),\; \mathrm{Im}z\neq 0,$ is the resolvent of $A$. Let $G(z)$ be a bounded holomorphic in the open upper and lower half-planes operator function from $\mathcal{K}$ to $\mathcal{H} is the resolvent of some singular selfadjoint perturbation $A_{L}$ of $A$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 2.1: GHM
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof
  • Proposition 3.2: AdP1
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • ...and 1 more