Inverse problems for locally perturbed lattices – Discrete Hamiltonian and quantum graph
[Problèmes inverses pour treillis localement perturbés – Hamiltonien discret et graphe quantique]
Blåsten, Emilia1 ; Exner, Pavel2 ; Isozaki, Hiroshi3 ; Lassas, Matti4 ; Lu, Jinpeng4
1Computational Engineering, LUT University, 15210 Lahti (Finland)
2Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University in Prague, Břehová 7, 115 19 Prague, Czechia, and Nuclear Physics Institute, Czech Academy of Science, 250 68 Řež (Czechia)
3Graduate School of Pure and Applied Sciences, Professor Emeritus, University of Tsukuba, Tsukuba, 305-8571 (Japan)
4Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki (Finland)
Annales Henri Lebesgue, Tome 7 (2024), pp. 267-305

We consider the inverse scattering problems for two types of Schrödinger operators on locally perturbed periodic lattices. For the discrete Hamiltonian, the knowledge of the S-matrix for all energies determines the graph structure and the coefficients of the Hamiltonian. For locally perturbed equilateral metric graphs, the knowledge of the S-matrix for all energies determines the graph structure.

Nous considérons les problèmes de scattering inverse pour deux types d’opérateurs de Schrödinger sur des réseaux périodiques localement perturbés. Pour le hamiltonien discret, la connaissance de la S-matrice pour toutes les énergies détermine la structure du graphe et les coefficients du hamiltonien. Pour les graphes métriques équilatéraux localement perturbés, la connaissance de la S-matrice pour toutes les énergies détermine la structure du graphe.

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DOI : 10.5802/ahl.201
Classification : 81Q10, 81Q35, 81U40
Keywords: lattice, metric graph, discrete Hamiltonian, S-matrix, inverse porblem

Blåsten, Emilia  1   ; Exner, Pavel  2   ; Isozaki, Hiroshi  3   ; Lassas, Matti  4   ; Lu, Jinpeng  4

1 Computational Engineering, LUT University, 15210 Lahti (Finland)
2 Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University in Prague, Břehová 7, 115 19 Prague, Czechia, and Nuclear Physics Institute, Czech Academy of Science, 250 68 Řež (Czechia)
3 Graduate School of Pure and Applied Sciences, Professor Emeritus, University of Tsukuba, Tsukuba, 305-8571 (Japan)
4 Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki (Finland)
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {Inverse problems for locally perturbed lattices {\textendash} {Discrete} {Hamiltonian} and quantum graph},
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Blåsten, Emilia; Exner, Pavel; Isozaki, Hiroshi; Lassas, Matti; Lu, Jinpeng. Inverse problems for locally perturbed lattices – Discrete Hamiltonian and quantum graph. Annales Henri Lebesgue, Tome 7 (2024), pp. 267-305. doi: 10.5802/ahl.201

[AIM16] Ando, Kazunori; Isozaki, Hiroshi; Morioka, Hisashi Spectral properties for Schrödinger operators on perturbed lattices, Ann. Henri Poincaré, Volume 17 (2016), pp. 2103-2171 | MR | DOI | Zbl

[AIM18] Ando, Kazunori; Isozaki, Hiroshi; Morioka, Hisashi Inverse scattering for Schrödinger operators on perturbed periodic lattices, Ann. Henri Poincaré, Volume 19 (2018), pp. 3397-3455 | MR | DOI | Zbl

[And13] Ando, Kazunori Inverse scattering theory for discrete Schrödinger operators on the hexagonal lattice, Ann. Henri Poincaré, Volume 14 (2013), pp. 347-383 | MR | DOI | Zbl

[BER15] Bolte, Jens; Egger, Sebastian; Rueckriemen, Ralf Heat-kernel and resolvent asymptotics for Schrödinger operators on metric graphs, AMRX, Appl. Math. Res. Express, Volume 2015 (2015) no. 1, pp. 129-165 | MR | DOI | Zbl

[BILL23a] Blåsten, Emilia; Isozaki, Hiroshi; Lassas, Matti; Lu, Jinpeng Gelfand’s inverse problem for the graph Laplacian, J. Spectr. Theory, Volume 13 (2023) no. 1, pp. 1-45 | MR | Zbl | DOI

[BILL23b] Blåsten, Emilia; Isozaki, Hiroshi; Lassas, Matti; Lu, Jinpeng Inverse problems for discrete heat equations and random walks for a class of graphs, SIAM J. Discrete Math., Volume 37 (2023), pp. 831-863 | MR | DOI | Zbl

[BK13] Berkolaiko, Gregory; Kuchment, Peter Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186, American Mathematical Society, 2013 | MR | Zbl

[Cat97] Cattaneo, Carla The spectrum of the continuous Laplacian on a graph, Monatsh. Math., Volume 124 (1997) no. 3, pp. 215-235 | MR | DOI | Zbl

[CET10] Cheon, Taksu; Exner, Pavel; Turek, Ondřej Approximation of a general singular vertex coupling in quantum graphs, Ann. Phys., Volume 325 (2010), pp. 548-578 | MR | DOI | Zbl

[Exn96] Exner, Pavel Weakly coupled states on branching graphs, Lett. Math. Phys., Volume 38 (1996), pp. 313-320 | MR | DOI | Zbl

[Exn97] Exner, Pavel A duality between Schrödinger operators on graphs and certain Jacobi matrices, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 66 (1997), pp. 359-371 | MR | Zbl | Numdam

[GR22] Gernandt, Hannes; Rohleder, Jonathan A Calderéron type inverse problem for the tree graphs, Linear Algebra Appl., Volume 646 (2022), pp. 29-42 | MR | DOI | Zbl

[IK12] Isozaki, Hiroshi; Korotyaev, Evgeny Inverse problems, trace formulae for discrete Schrödinger operators, Ann. Henri Poincaré, Volume 13 (2012), pp. 751-788 | MR | DOI | Zbl

[IM15] Isozaki, Hiroshi; Morioka, Hisashi Inverse scattering at a fixed energy for discrete Schrödinger operators on the square lattice, Ann. Inst. Fourier, Volume 65 (2015), pp. 1153-1200 | MR | DOI | Zbl | Numdam

[IN95] Isakov, Victor; Nachman, Adrian I. Global uniqueness for a two-dimensional semilinear elliptic inverse problem, Trans. Am. Math. Soc., Volume 347 (1995), pp. 3375-3390 | MR | DOI | Zbl

[KKL01] Katchalov, Alexander; Kurylev, Yaroslav; Lassas, Matti Inverse boundary spectral problems, Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman & Hall / CRC Press, 2001 | MR | Zbl | DOI

[KKLM04] Katchalov, Alexander; Kurylev, Yaroslav; Lassas, Matti; Mandache, Niculae Equivalence of time-domain inverse problems and boundary spectral problems, Inverse Probl., Volume 20 (2004), pp. 419-436 | MR | DOI | Zbl

[KS99] Kostrykin, Vadim V.; Schrader, Robert Kirchhoff’s rule for quantum wires, J. Phys. A. Math. Gen., Volume 32 (1999), pp. 595-630 | MR | DOI | Zbl

[Pan06] Pankrashkin, Konstantin Spectra of Schrödinger operators on equilateral quantum graphs, Lett. Math. Phys., Volume 77 (2006), pp. 139-154 | MR | DOI | Zbl

[Pan13] Pankrashkin, Konstantin An example of unitary equivalence between self-adjoint extensions and their parameters, J. Funct. Anal., Volume 265 (2013), pp. 2910-2936 | MR | DOI | Zbl

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