An Ambarzumian type theorem on graphs with odd cycles

M.  Kiss

doi:10.37863/umzh.v74i12.6734

M. Kiss Inst. Math. Budapest Univ. Technology and Economics, Hungary

DOI: https://doi.org/10.37863/umzh.v74i12.6734

Анотація

УДК 517.9

Теорема типу Амбарцумяна для графів з непарними циклами

Розглянуто обернену задачу для операторів Шредінгера на зв’язному рівносторонньому графі $G$ зі стандартними умовами узгодження. Граф $G$ складається принаймні з двох непарних циклів, що склеєні в спільній вершині. Доведено результат типу Амбарцумяна, тобто якщо певна частина спектра така ж сама, як і у випадку нульового потенціалу, то потенціал повинен бути нульовим.

Посилання

E. Akkermans, A. Comtet, J. Desbois, G. Montambaux, Ch. Texier, Spectral determinant on quantum graphs, Annal. Phys., 284, № 1, 10–51 (2000). DOI: https://doi.org/10.1006/aphy.2000.6056

V.~Ambarzumian, Über eine Frage der Eigenwerttheorie, Z. Phys., 53, 690–695 (1929). DOI: https://doi.org/10.1007/BF01330827

G.~Berkolaiko, P.~Kuchment, Introduction to quantum graphs, Math. Surveys and Monogr., Amer. Math. Soc. (2013). DOI: https://doi.org/10.1090/surv/186

B. Bollobás, Modern graph theory, Springer, New York (1998). DOI: https://doi.org/10.1007/978-1-4612-0619-4

J. Bolte, S. Egger, R. Rueckriemen, Heat-kernel and resolvent asymptotics for Schrödinger operators on metric graphs, Appl. Math. Res. Express, 2015, № 1, 129–165 (2014). DOI: https://doi.org/10.1093/amrx/abu009

J.~Boman, P.~Kurasov, R.~Suhr, Schrödinger operators on graphs and geometry II. Spectral estimates for $L_{1}$-potentials and an Ambartsumian theorem, Integral Equat. and Operator Theory, 90, № 3 (2018). DOI: https://doi.org/10.1007/s00020-018-2467-1

G.~Borg, Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe, Acta Math., 78, 1–96 (1946). DOI: https://doi.org/10.1007/BF02421600

R. Carlson, V. Pivovarchik, Ambarzumian's theorem for trees, Electron. J. Different. Equat., 2007, № 142, 1–9 (2007).

R. Carlson, V. Pivovarchik, Spectral asymptotics for quantum graphs with equal edge lengths, J. Phys. A: Math. and Theor., 41, № 14, Article 145202 (2008). DOI: https://doi.org/10.1088/1751-8113/41/14/145202

Y. H.~Cheng, Tui-En Wang, Chun-Jen Wu, A note on eigenvalue asymptotics for Hill's equation, Appl. Math. Lett., 23, № 9, 1013–1015 (2010). DOI: https://doi.org/10.1016/j.aml.2010.04.028

S. Currie, B. A. Watson, Eigenvalue asymptotics for differential operators on graphs, J. Comput. and Appl. Math., 182, № 1, 13–31 (2005). DOI: https://doi.org/10.1016/j.cam.2004.11.038

E. B. Davies, An inverse spectral theorem, J. Operator Theory, 69, № 1, 195–208 (2013). DOI: https://doi.org/10.7900/jot.2010sep14.1881

Yves~Colin De~Verdière, Semi-classical measures on quantum graphs and the Gauss map of the determinant manifold, Ann. Henri Poincaré, 16, 347–364 (2015). DOI: https://doi.org/10.1007/s00023-014-0326-4

J. Desbois, Spectral determinant of Schrödinger operators on graphs, J. Phys. A: Math. and Gen., 33, № 7 (2000). DOI: https://doi.org/10.1088/0305-4470/33/7/103

L. Friedlander, Determinant of the Schrödinger operator on a metric graph, Contemp. Math., 415, 151–160 (2006). DOI: https://doi.org/10.1090/conm/415/07866

J.~M. Harrison, K.~Kirsten, C.~Texier, Spectral determinants and zeta functions of Schrödinger operators on metric graphs, J. Phys. A: Math. and Theor., 45, № 12, Article 125206 (2012). DOI: https://doi.org/10.1088/1751-8113/45/12/125206

M.~Horváth, Inverse spectral problems and closed exponential systems, Ann. Math., 162, № 2, 885–918 (2005). DOI: https://doi.org/10.4007/annals.2005.162.885

M.~Horváth, On the stability in Ambarzumian theorems, Inverse Problems, 31, № 2, Article 025008 (2015). DOI: https://doi.org/10.1088/0266-5611/31/2/025008

I.~Kac, V.~Pivovarchik, On multiplicity of a quantum graph spectrum, J. Phys. A: Math. and Theor., 44, № 10, Article~105301 (2011). DOI: https://doi.org/10.1088/1751-8113/44/10/105301

M. Kiss, Spectral determinants and an Ambarzumian type theorem on graphs, Integral Equat. and Operator Theory, 92, 1–11 (2020). DOI: https://doi.org/10.1007/s00020-020-02579-4

P. Kuchment, Quantum graphs: an introduction and a brief survey, Analysis on Graphs and its Applications, 291–314 (2008). DOI: https://doi.org/10.1090/pspum/077/2459876

Chun-Kong Law, Eiji Yanagida, {A solution to an Ambarzumyan problem on trees}, Kodai Math. J., 35, № 2, 358–373 (2012). DOI: https://doi.org/10.2996/kmj/1341401056

M. Möller, V. Pivovarchik, Spectral theory of operator pencils, Hermite–Biehler functions, and their applications, Oper. Theory: Adv. and Appl., Springer (2015). DOI: https://doi.org/10.1007/978-3-319-17070-1

K. Pankrashkin, Spectra of Schrödinger operators on equilateral quantum graphs, Lett. Math. Phys., 77, № 2, 139–154 (2006). DOI: https://doi.org/10.1007/s11005-006-0088-0

V. N.~Pivovarchik, Ambarzumian's theorem for a Sturm–Liouville boundary value problem on a star-shaped graph, Funct. Anal. and Appl., 39, № 2, 148–151 (2005). DOI: https://doi.org/10.1007/s10688-005-0029-1

Yu.~V. Pokornyi, A. V.~Borovskikh, Differential equations on networks (geometric graphs), J. Math. Sci., 119, № 6, 691–718 (2004). DOI: https://doi.org/10.1023/B:JOTH.0000012752.77290.fa

Ch. Texier, $zeta$-Regularized spectral determinants on metric graphs, J. Phys. A: Math. and Theor., 43, № 42, Article~425203 (2010). DOI: https://doi.org/10.1088/1751-8113/43/42/425203

Chuan-Fu Yang, Xiao-Chuan Xu, Ambarzumyan-type theorems on graphs with loops and double edges, J. Math. Anal. and Appl., 444, № 2, 1348–1358 (2016). DOI: https://doi.org/10.1016/j.jmaa.2016.07.030