Proceedings of the American Mathematical Society

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

Author(s): I. M. Oleinik
Journal: Proc. Amer. Math. Soc. 127 (1999), 889-900.
MSC (1991): Primary 58G03; Secondary 35J10
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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 $x''\,+\,q(t)x\,=\, 0,$ 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 Equations7(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

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