Two separation criteria for second order ordinary or partial differential operators

Richard C. Brown; Don B. Hinton

Mathematica Bohemica (1999)

  • Volume: 124, Issue: 2-3, page 273-292
  • ISSN: 0862-7959

Abstract

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We generalize a well-known separation condition of Everitt and Giertz to a class of weighted symmetric partial differential operators defined on domains in . Also, for symmetric second-order ordinary differential operators we show that where is a singular point guarantees separation of on its minimal domain and extend this criterion to the partial differential setting. As a particular example it is shown that is separated on its minimal domain if is superharmonic. For the criterion is used to give examples of a separation inequality holding on the domain of the minimal operator in the limit-circle case.

How to cite

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Brown, Richard C., and Hinton, Don B.. "Two separation criteria for second order ordinary or partial differential operators." Mathematica Bohemica 124.2-3 (1999): 273-292. <http://eudml.org/doc/248473>.

@article{Brown1999,
abstract = {We generalize a well-known separation condition of Everitt and Giertz to a class of weighted symmetric partial differential operators defined on domains in $\mathbb \{R\}^n$. Also, for symmetric second-order ordinary differential operators we show that $\limsup _\{t\rightarrow c\} (pq^\{\prime \})^\{\prime \}/q^2=\theta <2$ where $c$ is a singular point guarantees separation of $-(py^\{\prime \})^\{\prime \}+qy$ on its minimal domain and extend this criterion to the partial differential setting. As a particular example it is shown that $-\Delta y+qy$ is separated on its minimal domain if $q$ is superharmonic. For $n=1$ the criterion is used to give examples of a separation inequality holding on the domain of the minimal operator in the limit-circle case.},
author = {Brown, Richard C., Hinton, Don B.},
journal = {Mathematica Bohemica},
keywords = {separation; ordinary or partial differential operator; limit-point; essentially selfadjoint; separation; ordinary or partial differential operator; limit-point; essentially selfadjoint},
language = {eng},
number = {2-3},
pages = {273-292},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Two separation criteria for second order ordinary or partial differential operators},
url = {http://eudml.org/doc/248473},
volume = {124},
year = {1999},
}

TY - JOUR
AU - Brown, Richard C.
AU - Hinton, Don B.
TI - Two separation criteria for second order ordinary or partial differential operators
JO - Mathematica Bohemica
PY - 1999
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 124
IS - 2-3
SP - 273
EP - 292
AB - We generalize a well-known separation condition of Everitt and Giertz to a class of weighted symmetric partial differential operators defined on domains in $\mathbb {R}^n$. Also, for symmetric second-order ordinary differential operators we show that $\limsup _{t\rightarrow c} (pq^{\prime })^{\prime }/q^2=\theta <2$ where $c$ is a singular point guarantees separation of $-(py^{\prime })^{\prime }+qy$ on its minimal domain and extend this criterion to the partial differential setting. As a particular example it is shown that $-\Delta y+qy$ is separated on its minimal domain if $q$ is superharmonic. For $n=1$ the criterion is used to give examples of a separation inequality holding on the domain of the minimal operator in the limit-circle case.
LA - eng
KW - separation; ordinary or partial differential operator; limit-point; essentially selfadjoint; separation; ordinary or partial differential operator; limit-point; essentially selfadjoint
UR - http://eudml.org/doc/248473
ER -

References

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