Discreteness of the spectrum for some differential operators with unbounded coefficients in R n

Giorgio Metafune; Diego Pallara

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2000)

  • Volume: 11, Issue: 1, page 9-19
  • ISSN: 1120-6330

Abstract

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We give sufficient conditions for the discreteness of the spectrum of differential operators of the form A ⁢ u = - △ ⁢ u + ∇ ⁡ F , ∇ ⁡ u in L μ 2 ⁢ R n where d ⁢ μ ⁢ x = e - F ⁢ x ⁢ d ⁢ x and for Schrödinger operators in L 2 ⁢ R n . Our conditions are also necessary in the case of polynomial coefficients.

How to cite

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Metafune, Giorgio, and Pallara, Diego. "Discreteness of the spectrum for some differential operators with unbounded coefficients in \( \mathbb{R}^{n} \)." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 11.1 (2000): 9-19. <http://eudml.org/doc/252372>.

@article{Metafune2000,
abstract = {We give sufficient conditions for the discreteness of the spectrum of differential operators of the form \( A u = - \triangle u + ( \nabla F,\nabla u) \) in \( L^\{2\}\_\{\mu\}(\mathbb\{R\}^\{n\}) \) where \( d \mu(x) = e^\{-F(x)\} dx \) and for Schrödinger operators in \( L^\{2\}(\mathbb\{R\}^\{n\}) \). Our conditions are also necessary in the case of polynomial coefficients.},
author = {Metafune, Giorgio, Pallara, Diego},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Singular differential operators; Discrete spectrum; Schrödinger operators; Sobolev embeddings},
language = {eng},
month = {3},
number = {1},
pages = {9-19},
publisher = {Accademia Nazionale dei Lincei},
title = {Discreteness of the spectrum for some differential operators with unbounded coefficients in \( \mathbb\{R\}^\{n\} \)},
url = {http://eudml.org/doc/252372},
volume = {11},
year = {2000},
}

TY - JOUR
AU - Metafune, Giorgio
AU - Pallara, Diego
TI - Discreteness of the spectrum for some differential operators with unbounded coefficients in \( \mathbb{R}^{n} \)
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2000/3//
PB - Accademia Nazionale dei Lincei
VL - 11
IS - 1
SP - 9
EP - 19
AB - We give sufficient conditions for the discreteness of the spectrum of differential operators of the form \( A u = - \triangle u + ( \nabla F,\nabla u) \) in \( L^{2}_{\mu}(\mathbb{R}^{n}) \) where \( d \mu(x) = e^{-F(x)} dx \) and for Schrödinger operators in \( L^{2}(\mathbb{R}^{n}) \). Our conditions are also necessary in the case of polynomial coefficients.
LA - eng
KW - Singular differential operators; Discrete spectrum; Schrödinger operators; Sobolev embeddings
UR - http://eudml.org/doc/252372
ER -

References

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