| Title:
|
On spectral problems of discrete Schrödinger operators (English) |
| Author:
|
Chan, Chi-Hua |
| Author:
|
Huang, Po-Chun |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
66 |
| Issue:
|
3 |
| Year:
|
2021 |
| Pages:
|
325-344 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A special type of Jacobi matrices, discrete Schrödinger operators, is found to play an important role in quantum physics. In this paper, we show that given the spectrum of a discrete Schrödinger operator and the spectrum of the operator obtained by deleting the first row and the first column of it can determine the discrete Schrödinger operator uniquely, even though one eigenvalue of the latter is missing. Moreover, we find the forms of the discrete Schrödinger operators when their smallest and largest eigenvalues attain the extrema under certain constraints by use of the notion of generalized directional derivative and the method of Lagrange multiplier. (English) |
| Keyword:
|
discrete Schrödinger operator |
| MSC:
|
34B09 |
| idZBL:
|
07361058 |
| idMR:
|
MR4263154 |
| DOI:
|
10.21136/AM.2021.0203-19 |
| . |
| Date available:
|
2021-05-20T13:32:32Z |
| Last updated:
|
2023-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148897 |
| . |
| Reference:
|
[1] Agranovich, Z. S., Marchenko, V. A.: The Inverse Problem of Scattering Theory.Gordon and Breach, New York (1963). Zbl 0117.06003, MR 0162497 |
| Reference:
|
[2] Anderson, P. W.: Absence of diffusion in certain random lattices.Phys. Rev. 109 (1958), 1492-1505. 10.1103/PhysRev.109.1492 |
| Reference:
|
[3] Astrauskas, A.: Extremal theory for spectrum of random discrete Schrödinger operator. III. Localization properties.J. Stat. Phys. 150 (2013), 889-907. Zbl 1266.82033, MR 3028390, 10.1007/s10955-012-0669-5 |
| Reference:
|
[4] Borg, G.: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe: Bestimmung der Differentialgleichung durch die Eigenwerte.Acta Math. 78 (1946), 1-96 German. Zbl 0063.00523, MR 0015185, 10.1007/BF02421600 |
| Reference:
|
[5] Damanik, D., Hundertmark, D., Killip, R., Simon, B.: Variational estimates for discrete Schrödinger operators with potentials of indefinite sign.Commun. Math. Phys. 238 (2003), 545-562. Zbl 1052.47027, MR 1993385, 10.1007/s00220-003-0868-7 |
| Reference:
|
[6] Gel'fand, I. M., Levitan, B. M.: On the determination of a differential equation from its spectral function.Am. Math. Soc., Transl., II. Ser. 1 (1955), 253-304. Zbl 0066.33603, MR 0073805, 10.1090/trans2/001 |
| Reference:
|
[7] Gesztesy, F., Simon, B.: $m$-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices.J. Anal. Math. 73 (1997), 267-297. Zbl 0924.15005, MR 1616422, 10.1007/BF02788147 |
| Reference:
|
[8] Gladwell, G. M. L.: Inverse Problems in Vibrations.Mechanics: Dynamical Systems 9. Martinus Nijhoff Publishers, Dordrecht (1986). Zbl 0646.73013, MR 0874749, 10.1007/978-94-015-1178-0 |
| Reference:
|
[9] Hardy, G. H., Littlewood, J. E., Pólya, G.: Inequalities.Cambridge University Press, Cambridge (1952). Zbl 0047.05302, MR 0046395 |
| Reference:
|
[10] Hochstadt, H.: On some inverse problems in matrix theory.Arch. Math. 18 (1967), 201-207. Zbl 0147.27701, MR 0213379, 10.1007/BF01899647 |
| Reference:
|
[11] Hochstadt, H.: The inverse Sturm-Liouville problem.Commun. Pure Appl. Math. 26 (1973), 715-729. Zbl 0281.34015, MR 0330607, 10.1002/cpa.3160260514 |
| Reference:
|
[12] Levinson, N.: The inverse Sturm-Liouville problem.Mat. Tidsskr. B 1949 (1949), 25-30. Zbl 0041.42310, MR 0032067 |
| Reference:
|
[13] Levitan, B. M.: Inverse Sturm-Liouville Problem.VNU Science Press, Utrecht (1987). Zbl 0749.34001, MR 0933088 |
| Reference:
|
[14] Levitan, B. M., Gasymov, M. G.: Determination of a differential equation by two of its spectra.Russ. Math. Surv. 19 (1964), 1-63 translation from Usp. Mat. Nauk 19 1964 1-63. Zbl 0145.10903, MR 0162996, 10.1070/RM1964v019n02ABEH001145 |
| Reference:
|
[15] Pelinovsky, D. E., Stefanov, A.: On the spectral theory and dispersive estimates for a discrete Schrödinger equation in one dimension.J. Math. Phys. 49 (2008), Article ID 113501, 17 pages. Zbl 1159.81336, MR 2468536, 10.1063/1.3005597 |
| Reference:
|
[16] Pöschel, J., Trubowitz, E.: Inverse Spectral Theory.Pure and Applied Mathematics 130. Academic Press, Boston (1987). Zbl 0623.34001, MR 0894477, 10.1016/s0079-8169(08)x6138-0 |
| . |
