Dirac Operators with Delta-Interactions on Smooth Hypersurfaces in Rn

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Abstract

We consider the Dirac operators with singular potentials DA,Φ,m,ΓδΣ=DA,Φ,m+ΓδΣwhere DA,Φ,m=∑j=1nαj(-i∂xj+Aj)+αn+1m+ΦINis a Dirac operator on Rn with variable magnetic and electrostatic potentials A=(A1, … , An) ∈ L(Rn, Cn) and Φ ∈ L(Rn) , and the variable mass m∈ L(Rn). In formula (2) αj are the N× N Dirac matrices, that is αjαk+ αkαj= 2 δjkIN, IN is the unit N× N matrix, N= 2 [(n+1)/2]. In formula (1) Γ δΣ is a singular delta-type potential supported by a C2-hypersurface Σ ⊂ Rn which is the common boundary of the open sets Ω ±. Let H1±, CN) be the Sobolev spaces of N-dimensional vector-valued distributions u on Ω ±, and H1(Rn╲Σ,CN)=H1(Ω+,CN)⊕H1(Ω-,CN).We associate with the formal Dirac operator DA,Φ,m,ΓδΣ an unbounded in L2(Rn, CN) operator DA,Φ,m,BΣ defined by the Dirac operator DA,Φ,m with domain domDA,Φ,m,BΣ⊂H1(Rn╲Σ,CN) defined by an interaction conditions. The main aims of the paper are the study of self-adjointmess of the operators DA,Φ,m,BΣ for uniformly regular C2-hypersurfaces Σ ⊂ Rn and the essential spectra of DA,Φ,m,BΣ for closed C2-hypersurfaces Σ ⊂ Rn.

Original languageEnglish
Article number20
JournalJournal of Fourier Analysis and Applications
Volume28
Issue number2
DOIs
StatePublished - Apr 2022

Keywords

  • Delta-interactions
  • Dirac operators
  • Essential spectrum
  • Self-adjointness
  • Singular potentials

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