Scattering theory for difference equations with operator coefficients (2501.11194v1)

Abstract: We consider a second order difference equation with operator-valued coefficients. More precisely, we study either compact or trace class perturbations of the discrete Laplacian in the Hilbert space of bi-infinite square-summable sequence with entries in a fixed Hilbert space. We discuss its continuous and discrete spectrum, as well as properties of the associated scattering matrix.

• The paper extends classical scattering theory to discrete difference equations involving operator-valued coefficients, analyzing perturbations of the discrete Laplacian.
• It constructs Jost solutions and the scattering matrix for these operators, providing insights into both discrete and continuous spectra and establishing conditions for spectral stability.
• The study highlights the significance of moment conditions for understanding eigenvalue accumulation and suggests applications in quantum mechanics and discrete systems with infinite-dimensional components.

Overview of "Scattering Theory for Difference Equations with Operator Coefficients"

The paper authored by David Sher, Luis Silva, Boris Vertman, and Monika Winklmeier addresses the scattering theory for a class of difference equations characterized by operator-valued coefficients. The primary focus is on examining perturbations of the discrete Laplacian within the framework of bi-infinite square-summable sequences in a separable Hilbert space. This exploration extends classical results about Schrödinger operators to this novel discrete setting. The research explores various aspects, including the spectral analysis of the perturbed operator, the construction of Jost solutions, and the characterization of the scattering matrix, offering insights into both the discrete and continuous spectra of the operators.

Perturbed Discrete Laplacian and Spectrum Analysis

The paper considers a Jacobi operator generated by second-order difference equations with uniformly bounded, self-adjoint operator coefficients in a Hilbert space, defined as:

$(Tu)_n = A_{n-1}u_{n-1} + B_nu_n + A_nu_{n+1}$

This work extends the problem to operator coefficients $A_n$ and $B_n$, establishing self-adjointness of the Jacobi operator $J$. By considering k-th and exponential moment conditions, the paper ensures the perturbation's boundedness relative to the discrete free Laplacian $J_0$ and explores its spectral properties. The paper identifies conditions under which the spectrum remains stable or is perturbed, examining the implications of compact and trace class perturbations. The essential spectrum $o_{\text{ess}}(J)$ is shown to align with $[-2, 2]$ under specified conditions, allowing discrete eigenvalues to potentially accumulate only within this range.

Jost Solutions and Scattering Matrix

A significant contribution is the introduction of Jost solutions for the perturbed operator. By utilizing moment conditions, the authors construct solutions that asymptotically behave like plane waves, which facilitates the analysis of scattering phenomena. The Jost solutions are proved to form a fundamental system of solutions to the difference equations, critical for understanding wave propagation in discrete systems.

The paper constructs the scattering matrix, S(z), for the studied operator, which represents the relationship between incoming and outgoing wave components. Under certain conditions, such as the second moment condition and assuming specific properties of the Wronskian have closed range, continuity of the scattering matrix is established across the unit circle, excluding points ±1.

Discrete Spectrum Accumulation

Attention is given to the conditions under which eigenvalues of the Jacobi operator do not accumulate, particularly near the endpoints of the essential spectrum. The third moment condition and closed range properties of the Wronskian are critical for showing that the discrete spectrum's accumulation points are only within the essential spectrum range. The authors further provide results limiting eigenvalue accumulation through an analysis involving the perturbation determinant for trace-class perturbations.

Implications and Future Directions

This paper's implications reverberate through mathematical physics and operator theory, enhancing understanding of operator-valued difference equations and their spectral properties. The results extend the applicability of scattering theory to discrete systems with infinite-dimensional constituents, suggesting potential applications in quantum mechanics and dynamical systems with nontrivial boundary conditions.

Future research could focus on extending these methods to more general non-linear perturbations or multidimensional lattice operations, possibly exploring the computational aspects to assess numerical approximations' viability using the derived theoretical framework. The introduction of operator-valued coefficients sets the stage for further exploration into complex systems where traditional scalar methods are insufficient. This approach opens potential avenues for analyzing complex networks and discrete models in physical sciences more broadly.