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Determinant Formulas for Scattering Matrices of Schrödinger Operators with Finitely Many Concentric $δ$-Shells

Masahiro Kaminaga

Abstract

We study stationary scattering for Schrödinger operators in $\mathbb R^3$ with finitely many concentric $δ$--shell interactions of constant real strengths. Starting from the self--adjoint realization and the boundary resolvent formula for this model, we show that, after partial--wave reduction, the same finite-dimensional boundary matrices that arise in the resolvent formula also determine the channel scattering coefficients. More precisely, for each angular momentum $\ell$, the channel coefficient $S_\ell(k)$ satisfies $S_\ell(k)=\det K_\ell(k^2-i0)/\det K_\ell(k^2+i0)$ for almost every $k>0$, where $K_\ell(z)=I_N+m_\ell(z)Θ$ is the $\ell$--th reduced boundary matrix. Thus, in each channel, the positive--energy scattering problem is reduced to a finite-dimensional matrix problem, and the scattering phase is recovered from $\det K_\ell(k^2+i0)$. We then study the first nontrivial case of two concentric shells in the $s$--wave channel, where the interaction between the shells produces nontrivial threshold effects. We derive an explicit formula for $S_0(k)$ and analyze its behavior as $k\downarrow0$. In the regular threshold regime, we obtain an explicit scattering length. We further identify a threshold--critical configuration characterized by the existence of a nontrivial zero--energy radial solution, regular at the origin, whose exterior constant term vanishes. In the corresponding nondegenerate exceptional case, the usual finite scattering length breaks down, and instead $S_0(k)\to -1$ as $k\downarrow0$.

Determinant Formulas for Scattering Matrices of Schrödinger Operators with Finitely Many Concentric $δ$-Shells

Abstract

We study stationary scattering for Schrödinger operators in with finitely many concentric --shell interactions of constant real strengths. Starting from the self--adjoint realization and the boundary resolvent formula for this model, we show that, after partial--wave reduction, the same finite-dimensional boundary matrices that arise in the resolvent formula also determine the channel scattering coefficients. More precisely, for each angular momentum , the channel coefficient satisfies for almost every , where is the --th reduced boundary matrix. Thus, in each channel, the positive--energy scattering problem is reduced to a finite-dimensional matrix problem, and the scattering phase is recovered from . We then study the first nontrivial case of two concentric shells in the --wave channel, where the interaction between the shells produces nontrivial threshold effects. We derive an explicit formula for and analyze its behavior as . In the regular threshold regime, we obtain an explicit scattering length. We further identify a threshold--critical configuration characterized by the existence of a nontrivial zero--energy radial solution, regular at the origin, whose exterior constant term vanishes. In the corresponding nondegenerate exceptional case, the usual finite scattering length breaks down, and instead as .
Paper Structure (12 sections, 16 theorems, 293 equations)

This paper contains 12 sections, 16 theorems, 293 equations.

Table of Contents

  1. Introduction
  2. The model and the resolvent formula
  3. The operator and its quadratic form
  4. Boundary operators and the resolvent formula
  5. Partial--wave reduction
  6. Scattering matrices for finitely many concentric shells
  7. Free spectral representation and partial--wave decomposition
  8. Outgoing solutions in a fixed partial wave
  9. Determination of the channel scattering coefficient
  10. The double $\delta$--shell case
  11. The $s$--wave channel
  12. Low--energy behavior in the double $\delta$--shell case

Key Result

Theorem 2.1

The quadratic form $h$ in eq:form is closed and lower semibounded on $H^1(\mathbb{R}^3)$, and therefore defines a self--adjoint operator $H$ in $L^2(\mathbb{R}^3)$. Moreover, a function $u\in L^2(\mathbb{R}^3)$ belongs to the operator domain $D(H)$ if and only if $u$ is piecewise $H^2$ on the region Here $\partial_r u(R_j\pm0,\omega)$ denotes the radial derivative taken from the exterior and inter

Theorems & Definitions (35)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 25 more