Dirac-Krein Systems on Star Graphs

• We study the spectrum of a self-adjoint Dirac–Krein operator with potential on a compact star graph G with a finite number n of edges. This operator is defined by a Dirac–Krein differential expression with summable matrix potentials on each edge, by self-adjoint boundary conditions at the outer vertices, and by a self-adjoint matching condition at the common central vertex of G. Special attention is paid to Robin matching conditions with parameter τ∈R∪{∞}. Choosing the decoupled operator with Dirichlet condition at the central vertex as a reference operator, we derive Krein’s resolvent formula, introduce corresponding Weyl–Titchmarsh functions, study the multiplicities, dependence on τ, and interlacing properties of the eigenvalues, and prove a trace formula. Moreover, we show that, asymptotically for R→∞, the difference of the number of eigenvalues in the intervals [0,R) and [−R,0) deviates from some integer κ0, which we call dislocation index, at most by n+2.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Nyckelord

Dirac operator

Dirac-Krein system

star graph

Krein's resolvent formula

trace formula

dislocation index

Publikations- och innehållstyp

ref (ämneskategori)

art (ämneskategori)