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An analysis of non-selfadjoint first-order differential operators with non-local point interactions

Christoph Fischbacher, Danie Paraiso, Chloe Povey-Rowe, Brady Zimmerman

TL;DR

This paper analyzes non-selfadjoint, first-order differential operators on $(0,2\pi)$ with a non-local point interaction $f(2\pi)k$ and a boundary condition $f(0)=\rho f(2\pi)$. It derives an explicit resolvent and an eigenvalue equation via a similarity to a simpler model, showing eigenvalues are zeros of an entire function $\Phi(\lambda)$ and that the root vectors form a Riesz basis when the interaction parameter is nonzero. In the dissipative setting, the authors completely characterize maximally dissipative extensions $A_{\rho,k}$ and prove that at most one real eigenvalue can occur, providing a constructive way to realize any prescribed real spectral value; they also establish the long-time behavior of the generated semigroups. Collectively, these results yield precise spectral localization, a robust basis property for root vectors, and detailed dynamical behavior, advancing the theory of non-selfadjoint differential operators with non-local interactions and informing potential applications to dissipative systems and time evolution analyses.

Abstract

We study the spectra of non-selfadjoint first-order operators on the interval with non-local point interactions, formally given by ${i\partial_x+V+k\langle δ,\cdot\rangle}$. We give precise estimates on the location of the eigenvalues on the complex plane and prove that the root vectors of these operators form Riesz bases of $L^2(0,2π)$. Under the additional assumption that the operator is maximally dissipative, we prove that it can have at most one real eigenvalue, and given any $λ\in\mathbb{R}$, we explicitly construct the unique operator realization such that $λ$ is in its spectrum. We also investigate the time-evolution generated by these maximally dissipative operators.

An analysis of non-selfadjoint first-order differential operators with non-local point interactions

TL;DR

This paper analyzes non-selfadjoint, first-order differential operators on with a non-local point interaction and a boundary condition . It derives an explicit resolvent and an eigenvalue equation via a similarity to a simpler model, showing eigenvalues are zeros of an entire function and that the root vectors form a Riesz basis when the interaction parameter is nonzero. In the dissipative setting, the authors completely characterize maximally dissipative extensions and prove that at most one real eigenvalue can occur, providing a constructive way to realize any prescribed real spectral value; they also establish the long-time behavior of the generated semigroups. Collectively, these results yield precise spectral localization, a robust basis property for root vectors, and detailed dynamical behavior, advancing the theory of non-selfadjoint differential operators with non-local interactions and informing potential applications to dissipative systems and time evolution analyses.

Abstract

We study the spectra of non-selfadjoint first-order operators on the interval with non-local point interactions, formally given by . We give precise estimates on the location of the eigenvalues on the complex plane and prove that the root vectors of these operators form Riesz bases of . Under the additional assumption that the operator is maximally dissipative, we prove that it can have at most one real eigenvalue, and given any , we explicitly construct the unique operator realization such that is in its spectrum. We also investigate the time-evolution generated by these maximally dissipative operators.
Paper Structure (9 sections, 22 theorems, 107 equations, 1 figure)

This paper contains 9 sections, 22 theorems, 107 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and some preliminaries
  3. First-order differential operators with non-local point-interactions
  4. The spectrum of $A_{\rho,k}$
  5. Riesz basis property of root functions
  6. Dissipative extensions of $A_{min}$
  7. Basic definitions and characterization of all dissipative extensions of $A_{min}$
  8. Extensions of $A_{min}$ with a real eigenvalue
  9. Long-time behavior of the semigroup generated by $iA_{\rho,k}$

Key Result

Theorem 3.1

The spectrum of $A_{\rho,k}$ consists only of eigenvalues and is given by where the geometric multiplicity of each eigenvalue is $\dim(\ker(A_{\rho,k}-\lambda))=1$. The corresponding eigenfunctions are given by Moreover, for $\lambda\in \varrho(A_{\rho,k})=\mathbb{C}\setminus\sigma(A_{\rho,k})$, the resolvent operator $(A_{\rho,k}-\lambda)^{-1}$ is given by where

Figures (1)

  • Figure 1: Eigenvalues of $P_{1,\widetilde{K}}$ for $\widetilde{K}(t)=\frac{1-i}{2}(t-\pi)$. The recatngle and circles illustrate the result stated in Theorem \ref{['thm:3.14']}.

Theorems & Definitions (40)

  • Definition 2.1
  • Theorem 3.1
  • proof
  • Corollary 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Theorem 3.6
  • proof
  • Corollary 3.7
  • Proposition 3.8
  • ...and 30 more