Can one distinguish quantum trees from the boundary?

Kurasov, Pavel

doi:10.1090/S0002-9939-2011-11077-3

Can one distinguish quantum trees from the boundary?
HTML articles powered by AMS MathViewer

by Pavel Kurasov

Proc. Amer. Math. Soc. 140 (2012), 2347-2356

DOI: https://doi.org/10.1090/S0002-9939-2011-11077-3

Published electronically: October 26, 2011

PDF | Request permission

Abstract:

Schrödinger operators on metric trees are considered. It is proven that for certain matching conditions the Titchmarsh-Weyl matrix function does not determine the underlying metric tree; i.e. there exist quantum trees with equal Titchmarsh-Weyl functions. The constructed trees form one-parameter families of isospectral and isoscattering graphs.

References

Sergei Avdonin and Pavel Kurasov, Inverse problems for quantum trees, Inverse Probl. Imaging 2 (2008), no. 1, 1–21. MR 2375320, DOI 10.3934/ipi.2008.2.1
Sergei Avdonin, Pavel Kurasov, and Marlena Nowaczyk, Inverse problems for quantum trees II: recovering matching conditions for star graphs, Inverse Probl. Imaging 4 (2010), no. 4, 579–598. MR 2726415, DOI 10.3934/ipi.2010.4.579
Ram Band, Ori Parzanchevski, and Gilad Ben-Shach, The isospectral fruits of representation theory: quantum graphs and drums, J. Phys. A 42 (2009), no. 17, 175202, 42. MR 2539297, DOI 10.1088/1751-8113/42/17/175202
R. Band, A. Sawicki, and U. Smilansky, Scattering from isospectral quantum graphs, J. Phys. A 43 (2010), no. 41, 415201, 17. MR 2726689, DOI 10.1088/1751-8113/43/41/415201
M. I. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems 20 (2004), no. 3, 647–672. MR 2067494, DOI 10.1088/0266-5611/20/3/002
M. I. Belishev and A. F. Vakulenko, Inverse problems on graphs: recovering the tree of strings by the BC-method, J. Inverse Ill-Posed Probl. 14 (2006), no. 1, 29–46. MR 2218385, DOI 10.1163/156939406776237474
Jan Boman and Pavel Kurasov, Symmetries of quantum graphs and the inverse scattering problem, Adv. in Appl. Math. 35 (2005), no. 1, 58–70. MR 2141505, DOI 10.1016/j.aam.2004.10.002
B. M. Brown and R. Weikard, A Borg-Levinson theorem for trees, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2062, 3231–3243. MR 2172226, DOI 10.1098/rspa.2005.1513
B. M. Brown and R. Weikard, On inverse problems for finite trees, Methods of spectral analysis in mathematical physics, Oper. Theory Adv. Appl., vol. 186, Birkhäuser Verlag, Basel, 2009, pp. 31–48. MR 2732071, DOI 10.1007/978-3-7643-8755-6_{2}
Robert Carlson, Inverse eigenvalue problems on directed graphs, Trans. Amer. Math. Soc. 351 (1999), no. 10, 4069–4088. MR 1473434, DOI 10.1090/S0002-9947-99-02175-3
G. Freiling and V. Yurko, Inverse problems for differential operators on trees with general matching conditions, Appl. Anal. 86 (2007), no. 6, 653–667. MR 2345888, DOI 10.1080/00036810701303976
N. I. Gerasimenko and B. S. Pavlov, A scattering problem on noncompact graphs, Teoret. Mat. Fiz. 74 (1988), no. 3, 345–359 (Russian, with English summary); English transl., Theoret. and Math. Phys. 74 (1988), no. 3, 230–240. MR 953298, DOI 10.1007/BF01016616
N. I. Gerasimenko, The inverse scattering problem on a noncompact graph, Teoret. Mat. Fiz. 75 (1988), no. 2, 187–200 (Russian, with English summary); English transl., Theoret. and Math. Phys. 75 (1988), no. 2, 460–470. MR 959124, DOI 10.1007/BF01017484
Boris Gutkin and Uzy Smilansky, Can one hear the shape of a graph?, J. Phys. A 34 (2001), no. 31, 6061–6068. MR 1862642, DOI 10.1088/0305-4470/34/31/301
M. Harmer, Inverse scattering on matrices with boundary conditions, J. Phys. A 38 (2005), no. 22, 4875–4885. MR 2148630, DOI 10.1088/0305-4470/38/22/012
V. Kostrykin and R. Schrader, Kirchhoff’s rule for quantum wires. II. The inverse problem with possible applications to quantum computers, Fortschr. Phys. 48 (2000), no. 8, 703–716. MR 1778728, DOI 10.1002/1521-3978(200008)48:8<703::AID-PROP703>3.0.CO;2-O
P. Kurasov, Inverse problems for Aharonov-Bohm rings, Math. Proc. Cambridge Philos. Soc. 148 (2010), no. 2, 331–362. MR 2600145, DOI 10.1017/S030500410999034X
P. Kurasov On the inverse problem for quantum graphs with one cycle, Acta Physica Polonica A, 116 (2009), 765-771.
Pavel Kurasov and Marlena Nowaczyk, Geometric properties of quantum graphs and vertex scattering matrices, Opuscula Math. 30 (2010), no. 3, 295–309. MR 2669120, DOI 10.7494/OpMath.2010.30.3.295
P. Kurasov and F. Stenberg, On the inverse scattering problem on branching graphs, J. Phys. A 35 (2002), no. 1, 101–121. MR 1891815, DOI 10.1088/0305-4470/35/1/309
M. G. Gasymov and B. M. Levitan, Sturm-Liouville differential operators with discrete spectrum, Mat. Sb. (N.S.) 63 (105) (1964), 445–458 (Russian). MR 0160962
Ori Parzanchevski and Ram Band, Linear representations and isospectrality with boundary conditions, J. Geom. Anal. 20 (2010), no. 2, 439–471. MR 2579517, DOI 10.1007/s12220-009-9115-6
T. Shapira and U. Smilansky, Quantum graphs which sound the same, in: Non-Linear Dynamics and Fundamental Interactions NATO Science Series, 2006, 213, Part 1, 17-29.
V. Yurko, Inverse spectral problems for Sturm-Liouville operators on graphs, Inverse Problems 21 (2005), no. 3, 1075–1086. MR 2146822, DOI 10.1088/0266-5611/21/3/017
V. A. Yurko, On the reconstruction of Sturm-Liouville operators on graphs, Mat. Zametki 79 (2006), no. 4, 619–630 (Russian, with Russian summary); English transl., Math. Notes 79 (2006), no. 3-4, 572–582. MR 2251311, DOI 10.1007/s11006-006-0064-0