We study the spectrum of a self-adjoint Dirac-Krein operator with potential on a compact star graph $mathcal G$ 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 $mathcal G$. Special attention is paid to Robin matching conditions with parameter $tau inmathbb Rcup{infty}$. Choosing the decoupled operator with Dirichlet condition at the central vertex as a reference operator, we derive Kreins resolvent formula, introduce corresponding Weyl-Titchmarsh functions, study the multiplicities, dependence on $tau$, and interlacing properties of the eigenvalues, and prove a trace formula. Moreover, we show that, asymptotically for $Rto infty$, the difference of the number of eigenvalues in the intervals $[0,R)$ and $[-R,0)$ deviates from some integer $kappa_0$, which we call dislocation index, at most by $n+2$.
تحميل البحث