Unitary dimension reduction for a class of self-adjoint extensions with applications to graph-like structures

Abstract

We consider a class of self-adjoint extensions using the boundary triple technique. Assuming that the associated Weyl function has the special form $M(z)=\big(m(z)\Id-T\big) n(z)^{-1}$ with a bounded self-adjoint operator $T$ and scalar functions $m,n$ we show that there exists a class of boundary conditions such that the spectral problem for the associated self-adjoint extensions in gaps of a certain reference operator admits a unitary reduction to the spectral problem for $T$. As a motivating example we consider differential operators on equilateral metric graphs, and we describe a class of boundary conditions that admit a unitary reduction to generalized discrete laplacians.

19 pages

References in corpus (7)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• On the spectra of carbon nano-structures
• Large gaps in point-coupled periodic systems of manifolds
• Localization effects in a periodic quantum graph with magnetic field and spin-orbit interaction
• Equilateral quantum graphs and boundary triples
• Localization on quantum graphs with random edge lengths
• Localization on quantum graphs with random vertex couplings

Cited by in corpus (17)

• Spectral Theory of Infinite Quantum Graphs
• An example of unitary equivalence between self-adjoint extensions and their parameters
• Scattering on periodic metric graphs
• Cantor spectrum of graphene in magnetic fields
• Spectral Estimates for Infinite Quantum Graphs
• Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-posedness and p-independence of spectra
• Continuum limit of the lattice quantum graph Hamiltonian
• New relations between discrete and continuous transition operators on (metric) graphs
• Absence of absolutely continuous spectrum for the Kirchhoff Laplacian on radial trees
• Zero Measure and Singular Continuous Spectra for Quantum Graphs
• Cantor spectra of magnetic chain graphs
• Non-compact quantum graphs with summable matrix potentials
• Laplacians on infinite graphs: discrete vs continuous
• Embedded trace operator for infinite metric trees
• Magnetic square lattice with vertex coupling of a preferred orientation
• A remark on the discriminant of Hill's equation and Herglotz functions
• Topologically induced spectral behavior: the example of quantum graphs