Spectrum of the weighted Laplace operator in unbounded domains

Alexey Filinovskiy

Mathematica Bohemica (2011)

  • Volume: 136, Issue: 4, page 415-427
  • ISSN: 0862-7959

Abstract

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We investigate the spectral properties of the differential operator - r s Δ , s 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 L 2 , s ( Ω ) 2 = Ω r - s | u | 2 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.

How to cite

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Filinovskiy, Alexey. "Spectrum of the weighted Laplace operator in unbounded domains." Mathematica Bohemica 136.4 (2011): 415-427. <http://eudml.org/doc/196401>.

@article{Filinovskiy2011,
abstract = {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 $\Vert u\Vert ^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.},
author = {Filinovskiy, Alexey},
journal = {Mathematica Bohemica},
keywords = {Laplace operator; multiplicative perturbation; Dirichlet problem; Friedrichs extension; purely discrete spectra; purely continuous spectra; weighted Laplace operator; Friedrichs extension; purely discrete spectra; purely continuous spectra},
language = {eng},
number = {4},
pages = {415-427},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Spectrum of the weighted Laplace operator in unbounded domains},
url = {http://eudml.org/doc/196401},
volume = {136},
year = {2011},
}

TY - JOUR
AU - Filinovskiy, Alexey
TI - Spectrum of the weighted Laplace operator in unbounded domains
JO - Mathematica Bohemica
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 136
IS - 4
SP - 415
EP - 427
AB - 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 $\Vert u\Vert ^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.
LA - eng
KW - Laplace operator; multiplicative perturbation; Dirichlet problem; Friedrichs extension; purely discrete spectra; purely continuous spectra; weighted Laplace operator; Friedrichs extension; purely discrete spectra; purely continuous spectra
UR - http://eudml.org/doc/196401
ER -

References

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  2. M., Eidus D., 10.1016/0022-1236(91)90117-N, J. Funct. Anal. 100 (1991), 400-410. (1991) Zbl0762.35020MR1125232DOI10.1016/0022-1236(91)90117-N
  3. A., Ladyzhenskaya O., N., Uraltseva N., Linear and Quasilinear Equations of Elliptic Type, Second edition, revised. Nauka, Moskva (1973), 576 Russian. (1973) MR0509265
  4. M., Glazman I., Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Oldbourne Press, London (1965), 234. (1965) Zbl0143.36505MR0190800
  5. A., Berezin F., A., Shubin M., The Schrodinger Equation, Moskov. Gos. Univ., Moskva (1983), 392 Russian. (1983) MR0739327
  6. M., Abramowitz, I.A., Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publications (1964), 1058. (1964) Zbl0171.38503MR1225604
  7. M., Landis E., On some properties of solutions of elliptic equations, Dokl. Akad. Nauk SSSR 107 (1956), 640-643 Russian. (1956) Zbl0075.28201MR0078557

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