On the essential self-adjointness of the general second order elliptic operators

Author:
I. M. Oleinik

Journal:
Proc. Amer. Math. Soc. **127** (1999), 889-900

MSC (1991):
Primary 58G03; Secondary 35J10

DOI:
https://doi.org/10.1090/S0002-9939-99-04551-7

MathSciNet review:
1468201

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we give sufficient conditions for the essential self-adjointness of second order elliptic operators. It turns out that these conditions coincide with those for the Schrödinger operator on a manifold whose metric essentially depends on the principal coefficients of a given operator.

**1.**R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, tensor analysis, and applications. Addison-Wesley, Reading, Mass., 1983. MR**84h:58001****2.**Yu. M. Berezanskii, Self-adjoint operators in spaces of functions of infinitely many variables. Translations of mathematical monographs, vol. 63. Amer. Math. Soc., Providence, RI, 1986. MR**87i:47023****3.**F. A. Berezin and M. A. Shubin, The Schrödinger equation. Kluwer Academic Publishers Group, Dordrecht, 1991. MR**93i:81001****4.**M. Braverman, On self-adjointness of a Schrödinger operator on differential forms,*Proc. Amer. Math. Soc.***126**(1998), 617-623. CMP**98:03****5.**P. R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations,*J. Func. Anal.***12**(1973), 401-414. MR**51:6119****6.**P. R. Chernoff, Schrödinger and Dirac operators with singular potentials and hyperbolic equations,*Pacific J. Math.***72**(1977), 361-382. MR**58:23150****7.**A. A. Chumak, Self-adjointness of the Beltrami-Laplace operator on a complete paracompact manifold without boundary,*Ukrainian Math. J.***25**(1973), 649-655, in Russian.**8.**A. Devinatz, Essential self-adjointness of Schrödinger type operators,*Func. Anal.***25**(1977), 58-69. MR**56:884****9.**M. Gaffney, A special Stoke's theorem for complete Riemannian manifolds,*Ann. of Math.***60**(1954), 140-145. MR**15:986d****10.**P. Hartman, The number of -solutions of*Amer. J. Math.***43**(1951), 635-645. MR**13:462a****11.**B. Hellwig, A criterion for self-adjointness of singular elliptic operators,*J. Math. Anal. Appl.***26**(1969), 279-291. MR**38:6254****12.**T. Ikebe and T. Kato, Uniqueness of the self-adjoint extension of singular elliptic differential operators,*Arch. Rational Mech. Anal.***9**(1962),77-99. MR**26:461****13.**R. S. Ismagilov, Conditions for self-adjointness of differential operators of higher order,*Soviet Math. Dokl.***3**(1962), 279-283. MR**24:A1443****14.**S. A. Laptev, Closure in the metric of the generized Dirichlet integral,*J. Differential Equations***7**(1971), 727-736. MR**44:2030****15.**I. M. Oleinik, On a connection between classical and quantum mechanical completeness of the potential at infinity on a complete Riemannian manifold,*Mat. Zametki***55**(1994), no. 4, 65-73. MR**95h:35051****16.**Yu. B. Orochko, The hyperbolic equation method in the theory of operators of Schrödinger type with locally integrable potential,*Russian Math Surveys*,**43**(1988), no. 2, 51-102. MR**89k:35065****17.**M. Reed and B. Simon, Methods of modern mathematical physics II: Fourier analysis, self-adjointness. Academic Press, New York, 1975. MR**58:12429b****18.**M. Riesz, L'intégrale de Riemann-Liouville et le problème de Cauchy,*Acta Math.***81**(1949), 1-223. MR**10:713c****19.**F. S. Rofe- Beketov, Conditions for the self-adjointness of the Schrödinger operator,*Math. Notes*,**8**(1970), 741-751. MR**43:743****20.**F. S. Rofe- Beketov, Necessary and sufficient conditions for a finite rate of propagation for elliptic operators,*Ukrain. Mat. Zh.***37**(1985), 668-670. MR**87c:35022**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
58G03,
35J10

Retrieve articles in all journals with MSC (1991): 58G03, 35J10

Additional Information

**I. M. Oleinik**

Affiliation:
Department of Mathematics, Northeastern University, Boston, Massachusetts 02115

Address at time of publication:
PCI Services, Inc., 30 Winter Street, 12th Floor, Boston, Massachusetts 02108

Email:
oleinik@neu.edu, igoro@pciwiz.com

DOI:
https://doi.org/10.1090/S0002-9939-99-04551-7

Received by editor(s):
May 20, 1996

Received by editor(s) in revised form:
June 4, 1997

Communicated by:
Christopher D. Sogge

Article copyright:
© Copyright 1999
American Mathematical Society