On spectral estimates for the Schr\"odinger operators in global dimension~2

G. Rozenblum ; M. Solomyak

Algebra i analiz, Tome 25 (2013) no. 3, pp. 185-199.

Résumé

The problem of finding eigenvalue estimates for the Schrödinger operator turns out to be most complicated for the dimension 2. Some important results for this case have been obtained recently. In the paper, these results are discussed, and their counterparts are established for the operator on the combinatorial and metric graphs corresponding to the lattice $\mathbb Z^2$.

Détail

Keywords: eigenvalue estimates, Schrödinger operator, metric graphs, local dimension.

Bibliographie

[1] Grigoryan A., Nadirashvili N., Negative eigenvalues of two-dimensional Schrödinger operators, arXiv: 1112.4986

[2] Krasnosel'skii M. A., Rutickii Ya. B., Convex functions and Orlicz spaces, P. Noordhof Ltd., Groningen, 1961 | MR

[3] “Quantum graphs. I. Some basic structures. Special section on quantum graphs”, Kuchment P., 14:1 (2004), S107–S128 | MR | Zbl

[4] Kuchment P., “Quantum Graphs. II. Some spectral properties of quantum and combinatorial graphs”, J. Phys. A, 38:22 (2005), 4887–4900 | DOI | MR | Zbl

[5] Laptev A., Netrusov Yu., “On the negative eigenvalues of a class of Schrödinger operators”, Differential operators and spectral theory, Amer. Math. Soc. Transl., Ser. 2, 189, Amer. Math. Soc., Providence, RI, 1999, 173–186 | MR | Zbl

[6] Laptev A., Solomyak M., “On the negative spectrum of the two-dimensional Schrödinger operator with radial potential”, Comm. Math. Phys., 314:1 (2012), 229–241 http://www.ams.org.proxy.lib.chalmers.se/mathscinet/search/journaldoc.html?cn=Comm_Math_Phys | DOI | MR | Zbl

[7] Laptev A., Solomyak M., On the spectral estimates for the two-dimensional Schrödinger operator, arXiv: 1201.3074

[8] Levin D., Solomyak M., “Rozenblum–Lieb–Cwikel inequality for Markov generators”, J. Anal. Math., 71 (1997), 173–193 | DOI | MR | Zbl

[9] Molchanov S., Vainberg B., On general Cwikel–Lieb–Rozenblum and Lieb–Thirring inequalities, arXiv: ; See also: Around the research of Vladimir Maz'ya, v. III, Intern. Math. Ser., 13, Springer, New York; Tamara Rozhkovskaya, Novosibirsk, 2010, 201–246 0812.2968 | DOI | MR | Zbl

[10] Molchanov S., Vainberg B., Bargmann type estimates of the counting function for general Schrödinger operators, arXiv: 1201.3135 | MR

[11] Rao M. M., Ren Z. D., Theory of Orlicz spaces, Marcel Dekker, NY, 1991 | MR | Zbl

[12] Rozenblum G., Solomyak M., “Counting Schrödinger boundstates: semiclassics and beyond Sobolev Spaces in Mathematics. II”, Applications in Analysis and Partial Differential Equations, Intern. Math. ser., 8, Springer, NY; Tamara Rozhkovskaya Publ., Novosibirsk, 2008, 329–354 | MR

[13] Rozenblyum G., Solomyak M., “O spektralnykh otsenkakh dlya operatorov tipa Shrëdingera na $\mathbb Z^d$, $d\ge3$”, Probl. mat. anal., 41, Tamara Rozhkovskaya, Novosibirsk, 2009, 107–120 | MR

[14] Rozenblyum G., Solomyak M., “O spektralnykh otsenkakh dlya operatorov tipa Shrëdingera: sluchai maloi lokalnoi razmernosti”, Funkts. anal. i ego pril., 44:4 (2010), 21–33 | DOI | MR

[15] Rozenblyum G., Solomyak M., “Spektralnye otsenki dlya operatorov Shrëdingera s razrezhennymi potentsialami na grafakh”, Probl. mat. anal., 57, Tamara Rozhkovskaya, Novosibirsk, 2011, 151–164 | MR

[16] Shargorodsky E., On negative eigenvalues of two-dimensional Schrödinger operators, arXiv: 1203.4833

[17] Solomyak M., “Piecewise-polynomial approximation of functions from $H^l((0,1)^d)$, $2l=d$, and applications to the spectral theory of the Schrödinger operator”, Israel J. Math., 86:1–3 (1994), 253–275 | DOI | MR | Zbl

[18] Solomyak M., “On a class of spectral problems on the half-line and their applications to multi-dimensional problems”, J. Spectral Theory, 3 (2013), 215–235 ; arXiv: 1203.1156 | DOI | MR | Zbl