On the number of the negative eigenvalues on a finite compact metric tree

El Aïdi, Mohammed

doi:10.1090/proc/16822

On the number of the negative eigenvalues on a finite compact metric tree
HTML articles powered by AMS MathViewer

by Mohammed El Aïdi;

Proc. Amer. Math. Soc.

DOI: https://doi.org/10.1090/proc/16822

Published electronically: April 23, 2024

HTML | PDF | Request permission

Abstract:

The purpose of the present article is to provide an upper bound of the number of the negative eigenvalues of a generalized Schrödinger operator defined on a finite compact metric tree.

References

Gregory Berkolaiko and Peter Kuchment, Introduction to quantum graphs, Mathematical Surveys and Monographs, vol. 186, American Mathematical Society, Providence, RI, 2013. MR 3013208, DOI 10.1090/surv/186
M. Sh. Birman and M. Z. Solomyak, Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations, Estimates and asymptotics for discrete spectra of integral and differential equations (Leningrad, 1989–90) Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 1–55. MR 1306507
Mohammed El Aïdi, On the eigenvalues for a weighted $p$-Laplacian operator on metric graphs, Complex Var. Elliptic Equ. 64 (2019), no. 4, 541–547. MR 3902953, DOI 10.1080/17476933.2018.1434630
Mohammed El Aïdi, Upper bounds of the eigenvalues related to a weighted fractional $p$-Laplacian on metric graphs, Graphs Combin. 34 (2018), no. 3, 501–508. MR 3784931, DOI 10.1007/s00373-018-1889-5
Mohammed El Aïdi, On the Riesz-means of negative eigenvalues for a fractional Schrödinger operator, Integral Transforms Spec. Funct. 27 (2016), no. 12, 974–980. MR 3569735, DOI 10.1080/10652469.2016.1245301
Mohammed El Aïdi, CLR-type inequality on a suitable smooth manifold, Ric. Mat. 65 (2016), no. 1, 155–161. MR 3513905, DOI 10.1007/s11587-016-0257-0
Mohammed El Aïdi, Sur le nombre des valeurs propres négatives d’un opérateur elliptique, Bull. Sci. Math. 137 (2013), no. 4, 434–456 (French, with English and French summaries). MR 3054269, DOI 10.1016/j.bulsci.2012.10.005
Mohammed El Aïdi, An explicit upper bound of the number of negative eigenvalues associated to an elliptic operator, J. Pseudo-Differ. Oper. Appl. 6 (2015), no. 3, 391–405. MR 3385203, DOI 10.1007/s11868-015-0119-7
Mohammed El Aïdi, On a new embedding theorem and the CLR-type inequality for Euclidean and hyperbolic spaces, Bull. Sci. Math. 138 (2014), no. 3, 335–342. MR 3206471, DOI 10.1016/j.bulsci.2013.05.003
Yuri Egorov and Vladimir Kondratiev, On spectral theory of elliptic operators, Operator Theory: Advances and Applications, vol. 89, Birkhäuser Verlag, Basel, 1996. MR 1409364, DOI 10.1007/978-3-0348-9029-8
Yu. V. Egorov and M. El Aidi, Spectre négatif d’un opérateur elliptique avec des conditions au bord de Robin, Publ. Mat. 45 (2001), no. 1, 125–148 (French, with English summary). MR 1829580, DOI 10.5565/PUBLMAT_{4}5101_{0}5
Yu. V. Egorov and V. A. Kondrat′ev, The negative spectrum of an elliptic operator, Mat. Sb. 181 (1990), no. 2, 147–166 (Russian); English transl., Math. USSR-Sb. 69 (1991), no. 1, 155–177. MR 1046596, DOI 10.1070/SM1991v069n01ABEH001234
Yu. V. Egorov and V. A. Kondratiev, On moments of negative eigenvalues of an elliptic operator, Math. Nachr. 174 (1995), 73–79. MR 1349038, DOI 10.1002/mana.19951740107
I. M. Glazman, Pryamye metody kachestvennogo spektral′nogo analiza, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR 185471
A. Laptev and Yu. Netrusov, On the negative eigenvalues of a class of Schrödinger operators, Differential operators and spectral theory, Amer. Math. Soc. Transl. Ser. 2, vol. 189, Amer. Math. Soc., Providence, RI, 1999, pp. 173–186. MR 1730512, DOI 10.1090/trans2/189/14
Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
G. Rozenblyum and M. Solomyak, The Cwikel-Lieb-Rozenblyum estimator for generators of positive semigroups and semigroups dominated by positive semigroups, Algebra i Analiz 9 (1997), no. 6, 214–236 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 9 (1998), no. 6, 1195–1211. MR 1610184
G. Rozenblum and M. Solomyak, On the number of negative eigenvalues for the two-dimensional magnetic Schrödinger operator, Differential operators and spectral theory, Amer. Math. Soc. Transl. Ser. 2, vol. 189, Amer. Math. Soc., Providence, RI, 1999, pp. 205–217. MR 1730514, DOI 10.1090/trans2/189/16
Grigori Rozenblum and Alexander V. Sobolev, Discrete spectrum distribution of the Landau operator perturbed by an expanding electric potential, Spectral theory of differential operators, Amer. Math. Soc. Transl. Ser. 2, vol. 225, Amer. Math. Soc., Providence, RI, 2008, pp. 169–190. MR 2509783, DOI 10.1090/trans2/225/12
Heinz Siedentop and Rudi Weikard, Asymptotically correct lower bound for the sum of negative eigenvalues of Schrödinger operators through a decomposition of unity given by Macke, Asymptotic Anal. 8 (1994), no. 1, 65–72. MR 1265125, DOI 10.3233/ASY-1994-8104