| Title:
|
Spectrum of the weighted Laplace operator in unbounded domains (English) |
| Author:
|
Filinovskiy, Alexey |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
136 |
| Issue:
|
4 |
| Year:
|
2011 |
| Pages:
|
415-427 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We investigate the spectral properties of the differential operator $-r^s \Delta $, $s\ge 0$ with the Dirichlet boundary condition in unbounded domains whose boundaries satisfy some geometrical condition. Considering this operator as a self-adjoint operator in the space with the norm $\|u\|^2_{L_{2, s} (\Omega )}= \int _{\Omega } r^{-s} |u|^2 {\rm d} x $, we study the structure of the spectrum with respect to the parameter $s$. Further we give an estimate of the rate of condensation of discrete spectra when it changes to continuous. (English) |
| Keyword:
|
Laplace operator |
| Keyword:
|
multiplicative perturbation |
| Keyword:
|
Dirichlet problem |
| Keyword:
|
Friedrichs extension |
| Keyword:
|
purely discrete spectra |
| Keyword:
|
purely continuous spectra |
| MSC:
|
35J15 |
| MSC:
|
35J20 |
| MSC:
|
35J25 |
| MSC:
|
35P05 |
| MSC:
|
35P15 |
| idZBL:
|
Zbl 1249.35076 |
| idMR:
|
MR2985551 |
| DOI:
|
10.21136/MB.2011.141701 |
| . |
| Date available:
|
2011-11-10T15:53:48Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141701 |
| . |
| Reference:
|
[1] T., Lewis R.: Singular elliptic operators of second order with purely discrete spectra.Trans. Am. Math. Soc. 271 (1982), 653-666. Zbl 0507.35069, MR 0654855, 10.1090/S0002-9947-1982-0654855-X |
| Reference:
|
[2] M., Eidus D.: The perturbed Laplace operator in a weighted $L\sp 2$-space.J. Funct. Anal. 100 (1991), 400-410. Zbl 0762.35020, MR 1125232, 10.1016/0022-1236(91)90117-N |
| Reference:
|
[3] A., Ladyzhenskaya O., N., Uraltseva N.: Linear and Quasilinear Equations of Elliptic Type.Second edition, revised. Nauka, Moskva (1973), 576 Russian. MR 0509265 |
| Reference:
|
[4] M., Glazman I.: Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators.Oldbourne Press, London (1965), 234. Zbl 0143.36505, MR 0190800 |
| Reference:
|
[5] A., Berezin F., A., Shubin M.: The Schrodinger Equation.Moskov. Gos. Univ., Moskva (1983), 392 Russian. MR 0739327 |
| Reference:
|
[6] M., Abramowitz, I.A., Stegun: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables.Dover Publications (1964), 1058. Zbl 0171.38503, MR 1225604 |
| Reference:
|
[7] M., Landis E.: On some properties of solutions of elliptic equations.Dokl. Akad. Nauk SSSR 107 (1956), 640-643 Russian. Zbl 0075.28201, MR 0078557 |
| . |
