The trace formula for quantum graphs with general self adjoint boundary conditions(9)

We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbi

Employingthisdiagonalisationintherepresentation(3.4)oftheS-matrixthenleadstotheexpression

S(A,B;k)=W λ1 ik

SincetheunitaryWisindependent λkr 1s W.(3.5)d+i1 ofkthe rststatementofthe lemmaisobvious.

TheunitarityofS(A,B;k)forrealkalsofollowsimmediatelybyobservingthatthediagonalentriesin(3.5)areallofunitabsolutevalue.

Thethirdstatementfollowsinacompletelyanalogousfashionfromtherepresentation(3.5).

KnowingthattheS-matrixisanalyticink,onewouldliketocalculateitsderivative.Thisinfactisrequiredintheproofofthetraceformulabelow.ItisevenpossibletorelatethederivativeofS(k)totheS-matrixitself.

Lemma3.2.UnderthesameassumptionsasinLemma3.1oneobtainsfork∈C\

[±iσ(L)\{0}],

d

2k S(A,B;k) S(A,B,k) 1 S(A,B;k).(3.6)

WeremarkthatforrealktheunitarityoftheS-matrixcanbeinvokedtoobtainfrom(3.6)thatitisindependentofk,i itisselfadjoint.

Proof.Letus rstassumethatk∈R\{0}andabbreviateS(A,B;k)asS(k).Wealsode-notederivativesw.r.t.kbyadashandusetherelationd

2 S(k) 1 andC(k)B= 1

2k S(k)+1S(k) 1

= 1

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