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 ℝ n . Also, for symmetric second-order ordinary differential operators we show that lim sup t → c ( p q ' ) ' / q 2 = θ < 2 where c is a singular point guarantees separation of - ( p y ' ) ' + q y on its minimal domain and extend this criterion to the partial differential setting. As a particular example it is shown that - Δ y + q y 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.

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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  1. Atkinson F. V., On some results of Everitt and Giertz, Proc. Royal Soc. Edinburg 71A (1972/3), 151-58. (1972) MR0326045
  2. Brown R. C, Hinton D. B., 10.1016/0022-247X(85)90263-X, J. Math. Anal. Appl. 112 (1985), 563-578. (1985) Zbl0587.26011MR0813620DOI10.1016/0022-247X(85)90263-X
  3. Brown R. C, Hinton D. B., Shaw M. F., Some separation criteria and inequalities associated with linear second order differential operators, Preprint. Zbl0982.34032MR1974789
  4. Coddington E. A., Levinson N., Theory of ordinary differential equations, McGraw-Hill Book Company, New York, 1955. (1955) Zbl0064.33002MR0069338
  5. Davies E. B., Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, U.K., 1989. (1989) Zbl0699.35006MR0990239
  6. Dunford N., Schwartz J. T, Linear operators. Part II: Spectral theory, Interscience, New York, 1963. (1963) Zbl0128.34803MR1009163
  7. Eastham M. S. P., Evans W. D., McLeod J. B., 10.1007/BF00250679, Arch. Rational Mech. Anal. 60 (1976), 185-204. (1976) Zbl0326.35018MR0417564DOI10.1007/BF00250679
  8. Evans W. D., On the essential self-adjointness of powers of Schrödinger-type operators, Proc. Royal Soc. Edinburgh 79A (1977), 61-77. (1977) Zbl0374.35014MR0481617
  9. Evans W. D., Zettl A., Dirichlet and separation results for Schrödinger-type operators, Proc. Royal Soc. Edinburgh 80A (1978), 151-162. (1978) Zbl0397.47022MR0529574
  10. Everitt W. N., On the strong limit-point condition of second-order differential expressions, International Conference of Differential Equatioпs. (H. A. Antosiewicz, ed.), Proceedings of an international conference held at the University of Southern California, September 3-7, 1974, Academic Press, New York, 1974, pp. 287-306. (1974) MR0435497
  11. Everitt W.N., 10.4153/CJM-1976-033-3, Can. J. Math. 28 (2) (1976), 312-320. (1976) MR0430391DOI10.4153/CJM-1976-033-3
  12. Everitt W. N., Giertz M., Some properties of the domains of certain differential operators, Proc. London Math. Soc. (3) 23 (1971), 301-24. (1971) Zbl0224.34018MR0289840
  13. Everitt W. N., Giertz M., 10.1007/BF01110336, Math. Z. 126 (1972), 308-328. (1972) MR0303001DOI10.1007/BF01110336
  14. Everitt W.N., Giertz M., On limit-point and separation criteria for linear differential expressions, Proceedings of the 1972 Equadiff Conference, Brno, 1972, pp. 31-41. (1972) MR0367354
  15. Everitt W.N., Giertz M., Inequalities and separation for certain ordinary differential operators, Proc. London Math. Soc 28 (3) (1974), 352-372. (1974) Zbl0278.34009MR0342758
  16. Everitt W. N., Giertz M., Inequalities and separation for Schrödinger type operators in L 2 R n , Proc. Royal Soc. Edinburgh 79A (1977), 257-265. (1977) Zbl0423.35043MR0491381
  17. Everitt W. N., Giertz M., Weidmann J., 10.1007/BF01428264, Math. Ann. 200 (1973), 335-346. (1973) MR0326047DOI10.1007/BF01428264
  18. Jörgens K. T., 10.7146/math.scand.a-10722, Math. Scand 15 (1964), 5-17. (1964) MR0180755DOI10.7146/math.scand.a-10722
  19. Kalf H., 10.1007/BF01238041, Math. Z. 133 (1973), 249-255. (1973) Zbl0266.35018MR0328308DOI10.1007/BF01238041
  20. Kalf H., Schmincke U. W., Walter J., Wüst R., On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, Proceedings of the 1974 Dundee Symposium. Lecture Notes in Mathematics. vol. 448, Springer-Verlag, Berlin, 1975, pp. 182-226. (1974) MR0397192
  21. Kato T., Schrödinger operators with singular potentials, Israel J. Math., 135-148. Zbl0246.35025MR0333833
  22. Kato T., Read, T, Zettl A., The deficiency index problem for powers of differential operators, Lecture Notes in Mathematics, vol. 621, Springer-Verlag, New York, 1977. (1977) 
  23. Knowles I., 10.1016/0022-247X(78)90254-8, J. Math. Anal. Appl. 66 (1978), 574-585. (1978) MR0517747DOI10.1016/0022-247X(78)90254-8
  24. Naimark M. A., Linear Differential Operators. Part II, Frederick Ungar, New York, 1968. (1968) Zbl0227.34020MR0262880
  25. Opic B., Kufner A., Hardy-type inequalities, Longman Scientific and Technical, Harlow, Essex, UK, 1990. (1990) Zbl0698.26007MR1069756
  26. Read T. T., A limit-point criterion for expressions with intermittently positive coefficients, J. London Math. Soc. (2) 15, 271-276. Zbl0406.34037MR0437844
  27. Simon B., 10.1007/BF01427943, Math. Ann. 201 (1973), 211-220. (1973) Zbl0234.47027MR0337215DOI10.1007/BF01427943

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