Non-compact Quantum Graphs with Summable Matrix Potentials
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Annales Henri Poincar´ e
Non-compact Quantum Graphs with Summable Matrix Potentials Yaroslav Granovskyi, Mark Malamud and Hagen Neidhardt
Abstract. Let G be a metric non-compact connected graph with finitely many edges. The main object of the paper is the Hamiltonian Hα associated in L2 (G; Cm ) with a matrix Sturm–Liouville expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying the technique of boundary triplets and the corresponding Weyl functions, we show that the singular continuous spectrum of the Hamiltonian Hα as well as any other self-adjoint realization of the Sturm–Liouville expression is empty. We also indicate conditions on the graph ensuring pure absolute continuity of the positive part of Hα . Under an additional condition on the potential matrix, a Bargmann-type estimate for the number of negative eigenvalues of Hα is obtained. Additionally, for a star graph G a formula is found for the scattering matrix of the pair {Hα , HD }, where HD is the Dirichlet operator on G. Mathematics Subject Classification. Primary 34B45; Secondary 81Q10, 35Pxx, 58J50.
Contents 1. 2.
Introduction Preliminaries 2.1. Boundary Triplets and Weyl Functions 2.2. Weyl Function and Spectrum 2.3. Weyl Function and Scattering Matrix 2.3.1. Direct Integral and Spectral Representation 2.3.2. Scattering Matrix
Hagen Neidhardt deceased on 23 March 2019. Research supported by the “RUDN University Program 5-100” (M.M.) and by the European Research Council (ERC) under Grant No. AdG 267802 “AnaMultiScale” (H.N.).
Y. Granovskyi et al.
Ann. Henri Poincar´e
3. 4.
Vector-Valued Sturm–Liouville Operators General Non-compact Quantum Graphs with Finitely Many Edges 4.1. Framework 4.2. Absence of Singular Continuous Spectrum for any Realization 4.3. Hamiltonian with Delta Interactions 4.4. Hamiltonians with δ-Interactions on the Line 5. Bargmann-Type Estimates 5.1. Quadratic Form 5.2. Classical Bargmann Estimate 5.3. Bargmann-Type Estimate for Graphs 6. Non-compact Star Graphs with Finitely Many Edges 6.1. Boundary Triplets and Weyl Functions 6.2. Negative Spectrum 6.3. Scattering Matrix for the Star Graph 6.4. Perturbation Determinants Acknowledgements References
1. Introduction The spectral theory of quantum graphs with a finite or infinite number of edges has been actively developed over the last three decades (see [11,12,21,25,28, 29,50,51] and the references therein). In particular, Schr¨ odinger and Laplace operators on the lattices, carbon nanostructures and periodic metric graphs have attracted a lot of attention (see, for example, [35,36,39,40]). In this paper, we consider a non-compact connected quantum graph G = (V, E) with finitely many edges E and vertices V assuming that at least one of its edges is infinite. We assume that G has no “loops” (“tadpoles”) and multiple edges, i.e. no edge starts and ends at the same vertex, and no edges connecting two same vertices; this can always be achieved by introducing additional vertices, if necessary. The main object of the
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