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Limits of quantum graph operators with shrinking edges. (English) Zbl 1422.81101

Summary: We address the question of convergence of Schrödinger operators on metric graphs with general self-adjoint vertex conditions as lengths of some of graph’s edges shrink to zero. We determine the limiting operator and study convergence in a suitable norm resolvent sense. It is noteworthy that, as edge lengths tend to zero, standard Sobolev-type estimates break down, making convergence fail for some graphs. We use a combination of functional-analytic bounds on the edges of the graph and Lagrangian geometry considerations for the vertex conditions to establish a sufficient condition for convergence. This condition encodes an intricate balance between the topology of the graph and its vertex data. In particular, it does not depend on the potential, on the differences in the rates of convergence of the shrinking edges, or on the lengths of the unaffected edges.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
47A10 Spectrum, resolvent
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
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[1] Alonso, A.; Simon, B., The Birman-Krein-Vishik theory of selfadjoint extensions of semibounded operators, J. Operator Theory. J. Operator Theory, J. Operator Theory, 6, 407-270 (1981), Addenda · Zbl 0476.47019
[2] Alon, L.; Band, R.; Berkolaiko, G., Nodal statistics on quantum graphs, preprint · Zbl 1406.81037
[3] Ariturk, S., Eigenvalue estimates on quantum graphs, preprint
[4] Band, R.; Levy, G., Quantum graphs which optimize the spectral gap, Ann. Henri Poincaré, 18, 3269-3323 (2017) · Zbl 1431.81055
[5] Barra, F.; Gaspard, P., On the level spacing distribution in quantum graphs, J. Stat. Phys., 101, 283-319 (2000) · Zbl 1088.81507
[6] Berkolaiko, G.; Kennedy, J.; Kurasov, P.; Delio, M., Edge connectivity and the spectral gap of combinatorial and quantum graphs, J. Phys. A, 50, Article 365201 pp. (2017) · Zbl 1456.81193
[7] Berkolaiko, G., An elementary introduction to quantum graphs, (Geom. and Comp. Spec. Theory. Geom. and Comp. Spec. Theory, Contemp. Math., vol. 700 (2017), CRM Proc., AMS: CRM Proc., AMS Providence, RI), 41-72 · Zbl 1388.34020
[8] Berkolaiko, B.; Kuchment, P., Introduction to Quantum Graphs, Math. Surv. Monog., vol. i186 (2013), AMS · Zbl 1318.81005
[9] Berkolaiko, B.; Kuchment, P., Dependence of the spectrum of a quantum graph on vertex conditions and edge lengths, (Spectral Geometry. Spectral Geometry, Proc. Sympos. Pure Math., vol. 84 (2012), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 117-137 · Zbl 1317.81127
[10] Booss-Bavnbek, B.; Furutani, K., The Maslov index: a functional analytical definition and the spectral flow formula, Tokyo J. Math., 21, 1-34 (1998) · Zbl 0932.37063
[11] Burenkov, V. I., Sobolev Spaces on Domains (1998), B.G. Teubner Stuttgart-Leipzig · Zbl 0893.46024
[12] Buttazzo, G.; Ruffini, B.; Velichkov, B., Shape optimization problems for metric graphs, ESAIM Control Optim. Calc. Var., 20, 1-22 (2014) · Zbl 1286.49050
[13] Cacciapuoti, C., Scale invariant effective Hamiltonians for a graph with a small compact core, Symmetry, 11, 3, 359 (2019) · Zbl 1423.81082
[14] Cheon, T.; Exner, P.; Turek, O., Approximation of a general singular vertex coupling in quantum graphs, Ann. Phys., 325, 548-578 (2010) · Zbl 1192.81159
[15] Del Pezzo, L. M.; Rossi, J. D., The first eigenvalue of the p-Laplacian on quantum graphs, Anal. Math. Phys., 6, 365-391 (2016) · Zbl 1353.81060
[16] Exner, P.; Jex, M., On the ground state of quantum graphs with attractive \(δ\)-coupling, Phys. Lett. A, 376, 713-717 (2012) · Zbl 1255.81160
[17] Exner, P.; Post, O., Convergence of spectra of graph-like thin manifolds, J. Geom. Phys., 54, 77-115 (2005) · Zbl 1095.58007
[18] Friedlander, L., Extremal properties of eigenvalues for a metric graph, Ann. Inst. Fourier (Grenoble), 55, 199-211 (2005) · Zbl 1074.34078
[19] Howard, P.; Sukhtayev, A., The Maslov and Morse indices for Schrödinger operators on \([0, 1]\), J. Differential Equations, 260, 4499-4549 (2016) · Zbl 1337.47064
[20] Harmer, M., Hermitian symplectic geometry and extension theory, J. Phys. A: Math. Gen., 33, 9193-9203 (2000) · Zbl 0983.53051
[21] Kostrykin, V.; Schrader, R., Kirchhoff’s rule for quantum wires, J. Phys. A: Math. Gen., 32, 595-630 (1999) · Zbl 0928.34066
[22] Kostrykin, V.; Schrader, R., Laplacians on metric graphs: eigenvalues, resolvents and semigroups, (Quantum Graphs and Their Applications. Quantum Graphs and Their Applications, Contemp. Math., vol. 415 (2006), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 201-225 · Zbl 1122.34066
[23] Kennedy, J. B.; Kurasov, P.; Malenová, G.; Mugnolo, D., On the spectral gap of a quantum graph, Ann. Henri Poincaré, 17, 1-35 (2016)
[24] Kurasov, P.; Malenová, G.; Naboko, S., Spectral gap for quantum graphs and their edge connectivity, J. Phys. A, 46, Article 275309 pp. (2013) · Zbl 1270.81100
[25] Latushkin, Y.; Sukhtaiev, S., The Maslov index and the spectra of second order differential operators, Adv. Math., 329, 422-486 (2018) · Zbl 06863449
[26] Latushkin, Y.; Sukhtaiev, S.; Sukhtayev, A., The Morse and Maslov indices for Schrödinger operators, J. Anal. Math., 135, 1, 345-387 (June 2018) · Zbl 1407.35073
[27] Libermann, P.; Marle, Ch-M., Symplectic Geometry and Analytical Mechanics (1987), D. Reidel, Dordrecht: D. Reidel, Dordrecht Holland · Zbl 0643.53002
[28] McDuff, D.; Salamon, D., Introduction to Symplectic Topology (1998), Clarendon Press: Clarendon Press Oxford · Zbl 1066.53137
[29] Mugnolo, D., Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems (2014), Springer · Zbl 1306.47001
[30] Pavlov, B. S., The theory of extensions and explicitly-soluble models, Russian Math. Surveys, 42, 6, 127-168 (1987) · Zbl 0665.47004
[31] Post, O., Spectral convergence of quasi-one-dimensional spaces, Ann. Henri Poincaré, 5, 933-973 (2006) · Zbl 1187.81124
[32] Post, O., Convergence result for thick graphs, (Mathematical Results in Quantum Physics, QMath 11 Proceedings (2011), World Sci. Publ.: World Sci. Publ. Hackensack, NJ), 60-78 · Zbl 1238.81122
[33] Post, O., Spectral Analysis on Graph-like Spaces, Lecture Notes in Mathematics, vol. 2039 (2012), Springer: Springer Heidelberg · Zbl 1247.58001
[34] Reed, M.; Simon, B., Methods of Modern Mathematical Physics, vol. 1 (1980), Academic Press · Zbl 0459.46001
[35] Rohleder, J., Eigenvalue estimates for the Laplacian on a metric tree, Proc. Amer. Math. Soc., 145, 2119-2129 (2017) · Zbl 1367.34029
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