On a Model Graph with a Loop and Small Edges. Holomorphy Property of Resolvent
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Journal of Mathematical Sciences, Vol. 251, No. 5, December, 2020
ON A MODEL GRAPH WITH A LOOP AND SMALL EDGES. HOLOMORPHY PROPERTY OF RESOLVENT D. I. Borisov ∗ Institute of Mathematics, UFRC RAS 112, Chernyshevskii St., Ufa 450008, Russia Bashkir State University 32, Zaki Validi St., Ufa 450076, Russia University of Hradec Kr´ alov´e 62, Rokitansk´eho, Hradec Kr´ alov´e 50003, Czech Republic [email protected]
A. I. Mukhametrakhimova Bashkir State Pedagogical University 3a, October Revolution St., Ufa 450000, Russia [email protected]
UDC 517.956+517.958
We consider the Schr¨ odinger operator on a graph consisting of two infinite edges, a loop, and a glued (at the start and end points of the loop) graph obtained by ε−1 times contraction of some fixed graph. The Kirchhoff conditions are imposed at interior vertices and the Dirichlet or Neumann conditions are imposed at boundary vertices of the graph. We show that the resolvent of the Schr¨ odinger operator is holomorphic with respect to the small parameter ε and write out the first three terms of the asymptotic expansion of the resolvent. Bibliography: 10 titles. Illustrations: 2 figures.
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Introduction
One of the new directions in the theory of quantum graphs (cf. for example, [1, 2]) is connected with the theory of perturbation for quantum graphs with small edges. In the first publications in this direction, for a graph with small edges the boundary condition at a vertex was approximated in the sense of the uniform resolvent convergence (cf. [3]; we also mention the known woven membrane model [1, 4]). General results for a graph of an arbitrary structure with arbitrary scale invariant boundary conditions were recently obtained in [5], where the limit operators were written and the uniform resolvent convergence, as well as the convergence of spectra were investigated. The results of [5] motivated the further study of the asymptotic behavior of resolvents and spectral characteristics ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 106, 2020, pp. 17-41. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2515-0573
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of elliptic operators on graphs. In particular, the simplest model example was studied in [6, 7] for an elementary star-shaped graph consisting of two finite or infinite edges and one small edge. On such a graph, the Schr¨ odinger operator was given with potential depending singularly on the small edge length, and the δ- or δ -interaction condition was imposed at the vertex. For this problem the first terms of the asymptotic expansion of the resolvent and the corresponding estimate for the remainder were obtained in [6, 7]. An unexpected fact was revealed when studying the behavior of eigenvalues. Namely, the eigenvalues turned out to be holomorphic functions of the small parameter. This fact is extraordinary because small edges present an example of singular perturbations. Under singular perturbations, only asymptotic expansions of eigenvalues were studied in the literature, while the
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