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The essential self-adjointness of differential operators

Published online by Cambridge University Press:  14 November 2011

R. G. Keller
R. G. Keller
Affiliation:
Mathematics Institute, University of Oxford
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Synopsis

We consider the formally self-adjoint 2mth-order elliptic differential operator in ℝn given by where lt is an operator of order t, and establish conditions under which the operator on is essentially self-adjoint in L2. A feature is that the major conditions have to be imposed only in an increasing sequence of annular regions surrounding the origin.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 82 , Issue 3-4 , 1979 , pp. 305 - 344
Copyright
Copyright © Royal Society of Edinburgh 1979

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References

1Brusencev, A. G. and Rofe-Beketov, F. S.. The self-adjointness of higher order elliptic operators. Funkcional. Anal, i Prilozen. 7/4 (1973), 78–79. English transl. Functional Anal. Appl. 7 (1973), 322–323 (1974).Google Scholar
2Brusencev, A. G. and Rofe-Beketov, F. S.. Conditions for the self-adjointness of strongly elliptic systems of arbitrary order. Mat. Sb. 137 (1974), 108–129, 160.Google Scholar
3Eastham, M. S. P., Evans, W. D. and McLeod, J. B.. The essential self-adjointness of Schrödinger-type operators. Arch. Rational Mech. Anal. 60 (1976), 184–204.CrossRefGoogle Scholar
4Evans, W. D.. On non-integrable square solutions of a fourth-order differential equation and the limit-2 classification. J. London Math. Soc. 7 (1973), 343–354.CrossRefGoogle Scholar
5Evans, W. D. and Zettl, A.. On the deficiency indices of powers of real 2nth.order symmetric differential expressions. J. London Math. Soc. 13 (1976), 543–556.CrossRefGoogle Scholar
6Evans, W. D. and Zettl, A.. Interval limit-point criteria for differential expressions and their powers. J. London Math. Soc. 15 (1977), 119–133.CrossRefGoogle Scholar
7Everitt, W. N.. On the limit-point classification of fourth-order differential equations. J. London Math. Soc. 44 (1969), 273–281.CrossRefGoogle Scholar
8Hinton, D. B.. Limit-point criteria for differential equations. Canad. J. Math. 24 (1972), 293–305.CrossRefGoogle Scholar
9Ikebe, T. and Kato, T.. Uniqueness of self-adjoint extensions of singular elliptic differential operators. Arch. Rational Mech. Anal. 9 (1962), 77–92.CrossRefGoogle Scholar
10Kato, T.. Schrödinger operators with singular potentials. Israel J. Math. 13 (1972), 135–148.CrossRefGoogle Scholar
11Keller, R. G.. Some Problems in Differential Operators (Essential Self-Adjointness) (Oxford Univ. D. Phil. Thesis, 1977).Google Scholar
12Keller, R. G.. The essential self-adjointness of differential operators with positive coefficients. Proc. Roy. Soc. Bdinb. Sect. A 82 (1979), 345–360.CrossRefGoogle Scholar
13Knowles, I.. On essential self-adjointness for singular elliptic differential operators. Math. Ann. 227 (1977), 155–172.CrossRefGoogle Scholar
14Schechter, M.. Spectra of Partial Differential Operators (Amsterdam; North-Holland, 1971).Google Scholar