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On the Weyl law for quantum graphs. (English) Zbl 1432.11131
Summary: Roughly speaking, the Weyl law describes the asymptotic distribution of eigenvalues of the Laplacian that can be attached to different objects and can be analyzed in different settings. The form of the remainder term in the Weyl law is very significant in applications, and a power-saving exponent in the remainder term is appreciated. We are dealing with Laplacian defined on compact metric graph with general self-adjoint boundary conditions. The main purpose of this paper is to present application of the special form of the Tauberian theorem for the Laplace transform to the suitably transformed trace formula in the above-mentioned quantum graphs setting. The key feature of our method is that it produces a power-saving form of the reminder term and hence represents improvement in classical methods, which may be applied in other settings as well. The obtained form of the Weyl law is with the power saving of $$1 / 3$$ in the remainder term.

##### MSC:
 11M45 Tauberian theorems 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
##### Keywords:
Weyl law; quantum graphs; Laplacian; Tauberian theorems
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