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Fractional Laplacian in bended strip

Fedor Bakharev, Sergey Matveenko

TL;DR

This work analyzes the restricted fractional Laplacian in a smoothly bent waveguide, proving the existence of discrete eigenvalues below the essential spectrum threshold $λ_1$ under curvature magnitude and distribution constraints. It reframes the nonlocal problem via the Caffarelli–Silvestre extension, translating to a local weighted energy in a curved tubular domain and deriving curvature-induced attractions that enable bound states. A carefully constructed quasimode, together with IMS-type estimates and a detailed parameter tuning (lengths, cutoffs, and curvature scales), yields a Rayleigh quotient strictly below $λ_1$ for sufficiently long bends. The results extend classic bent-waveguide spectral phenomena to nonlocal operators, highlighting how curvature distribution, not merely magnitude, governs bound-state existence with potential implications for nonlocal quantum and waveguide models.

Abstract

The spectral properties of the restricted fractional Laplacian with Dirichlet boundary conditions in a smoothly bent waveguide is investigated. The existence of eigenvalues below the threshold of the continuous spectrum is proved, generalizing classical results known for the local Laplace operator. Our approach utilizes the Caffarelli--Silvestre extension, addressing the specific geometric difficulties arising from the operator non-locality. The sufficient conditions on the curvature magnitude and distribution to ensure the existence of these trapped modes is established.

Fractional Laplacian in bended strip

TL;DR

This work analyzes the restricted fractional Laplacian in a smoothly bent waveguide, proving the existence of discrete eigenvalues below the essential spectrum threshold under curvature magnitude and distribution constraints. It reframes the nonlocal problem via the Caffarelli–Silvestre extension, translating to a local weighted energy in a curved tubular domain and deriving curvature-induced attractions that enable bound states. A carefully constructed quasimode, together with IMS-type estimates and a detailed parameter tuning (lengths, cutoffs, and curvature scales), yields a Rayleigh quotient strictly below for sufficiently long bends. The results extend classic bent-waveguide spectral phenomena to nonlocal operators, highlighting how curvature distribution, not merely magnitude, governs bound-state existence with potential implications for nonlocal quantum and waveguide models.

Abstract

The spectral properties of the restricted fractional Laplacian with Dirichlet boundary conditions in a smoothly bent waveguide is investigated. The existence of eigenvalues below the threshold of the continuous spectrum is proved, generalizing classical results known for the local Laplace operator. Our approach utilizes the Caffarelli--Silvestre extension, addressing the specific geometric difficulties arising from the operator non-locality. The sufficient conditions on the curvature magnitude and distribution to ensure the existence of these trapped modes is established.

Paper Structure

This paper contains 15 sections, 2 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Caffarelli -- Silvestre extension
  3. Geometry of the bent strip and tubular coordinates
  4. CS energy in the embedded superstrip and unitary flattening
  5. Existence of an eigenvalue below $\lambda_1$
  6. (i) Upper bound for the $s$–term.
  7. (ii) Localized CS identity in $(n,y)$.
  8. (iii) Curvature gain.
  9. (iv) Rayleigh quotient and choice of parameters.
  10. (v) Estimates in case $\alpha\in(0, 1/2)$.
  11. (vi) Estimates in case $\alpha=1/2$.
  12. Step 1 (large admissible superstrip and amplitude).
  13. Step 2 (bending scale to ensure $C_0'>B_0$).
  14. Step 3 (control the $s$–cutoff error).
  15. Step 4 (suppress the IMS remainder).

Key Result

Proposition 4.1

Let $\widetilde{W}\in C_c^\infty(\Pi_\rho \times[0,\infty))$ with $\widetilde{W}(\cdot,\pm\rho,\cdot)=0$. Then with

Theorems & Definitions (5)

  • Proposition 4.1
  • Remark 4.2
  • proof
  • Theorem 5.1
  • proof