The form sum and the Friedrichs extension of Schrödingertype operators on Riemannian manifolds
Author:
Ognjen Milatovic
Journal:
Proc. Amer. Math. Soc. 132 (2004), 147156
MSC (2000):
Primary 35P05, 58G25; Secondary 47B25, 81Q10
Published electronically:
April 24, 2003
MathSciNet review:
2021257
Fulltext PDF Free Access
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Abstract: We consider , where is a Riemannian manifold (not necessarily complete), and is the scalar Laplacian on . We assume that , where and ( is a constant) are realvalued, and is semibounded below on . Let be the Friedrichs extension of . We prove that the form sum coincides with the selfadjoint operator associated to the closure of the restriction to of the sum of two closed quadratic forms of and . This is an extension of a result of Cycon. The proof adopts the scheme of Cycon, but requires the use of a more general version of Kato's inequality for operators on Riemannian manifolds.
 1.
M. Braverman, O. Milatovic, M. A. Shubin, Essential selfadjointness of Schrödinger type operators on manifolds, Russian Math. Surveys, 57 (4) (2002), 641692.
 2.
Hans
L. Cycon, On the form sum and the Friedrichs extension of
Schrödinger operators with singular potentials, J. Operator
Theory 6 (1981), no. 1, 75–86. MR 637002
(82k:47068)
 3.
William
G. Faris, Selfadjoint operators, Lecture Notes in
Mathematics, Vol. 433, SpringerVerlag, BerlinNew York, 1975. MR 0467348
(57 #7207)
 4.
David
Gilbarg and Neil
S. Trudinger, Elliptic partial differential equations of second
order, SpringerVerlag, BerlinNew York, 1977. Grundlehren der
Mathematischen Wissenschaften, Vol. 224. MR 0473443
(57 #13109)
 5.
HeinzWilli
Goelden, On nondegeneracy of the ground state of Schrödinger
operators, Math. Z. 155 (1977), no. 3,
239–247. MR 0609535
(58 #29426)
 6.
H.
Hess, R.
Schrader, and D.
A. Uhlenbrock, Domination of semigroups and generalization of
Kato’s inequality, Duke Math. J. 44 (1977),
no. 4, 893–904. MR 0458243
(56 #16446)
 7.
H.
Hess, R.
Schrader, and D.
A. Uhlenbrock, Kato’s inequality and the spectral
distribution of Laplacians on compact Riemannian manifolds, J.
Differential Geom. 15 (1980), no. 1, 27–37
(1981). MR
602436 (82g:58090)
 8.
Tosio
Kato, Perturbation theory for linear operators, Classics in
Mathematics, SpringerVerlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
(96a:47025)
 9.
Tosio
Kato, A second look at the essential selfadjointness of the
Schrödinger operators, Physical reality and mathematical
description, Reidel, Dordrecht, 1974, pp. 193–201. MR 0477431
(57 #16958)
 10.
Michael
Reed and Barry
Simon, Methods of modern mathematical physics. I. Functional
analysis, Academic Press, New YorkLondon, 1972. MR 0493419
(58 #12429a)
Michael
Reed and Barry
Simon, Methods of modern mathematical physics. II. Fourier
analysis, selfadjointness, Academic Press [Harcourt Brace Jovanovich,
Publishers], New YorkLondon, 1975. MR 0493420
(58 #12429b)
 11.
Barry
Simon, An abstract Kato’s inequality for generators of
positivity preserving semigroups, Indiana Univ. Math. J.
26 (1977), no. 6, 1067–1073. MR 0461209
(57 #1194)
 12.
Barry
Simon, Kato’s inequality and the comparison of
semigroups, J. Funct. Anal. 32 (1979), no. 1,
97–101. MR
533221 (80e:47036), http://dx.doi.org/10.1016/00221236(79)90079X
 13.
Barry
Simon, Maximal and minimal Schrödinger forms, J. Operator
Theory 1 (1979), no. 1, 37–47. MR 526289
(81m:35104)
 14.
Michael
E. Taylor, Partial differential equations. II, Applied
Mathematical Sciences, vol. 116, SpringerVerlag, New York, 1996.
Qualitative studies of linear equations. MR 1395149
(98b:35003)
 1.
 M. Braverman, O. Milatovic, M. A. Shubin, Essential selfadjointness of Schrödinger type operators on manifolds, Russian Math. Surveys, 57 (4) (2002), 641692.
 2.
 H. L. Cycon, On the form sum and the Friedrichs extension of Schrödinger operators with singular potentials, J. Operator Theory, 6 (1981), 7586. MR 82k:47068
 3.
 W. G. Faris, Selfadjoint Operators, Lecture Notes in Mathematics No. 433, SpringerVerlag, Berlin e.a., 1975. MR 57:7207
 4.
 D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Springer, New York, 1977. MR 57:13109
 5.
 H.W. Goelden, On nondegeneracy of the ground state of Schrödinger operators, Math. Z., 155 (1977), 239247. MR 58:29426
 6.
 H. Hess, R. Schrader, D.A. Uhlenbrock, Domination of semigroups and generalization of Kato's inequality, Duke Math. J., 44 (1977), 893904. MR 56:16446
 7.
 H. Hess, R. Schrader, D. A. Uhlenbrock, Kato's inequality and the spectral distribution of Laplacians on compact Riemannian manifold, J. Differential Geom., 15 (1980), 2737. MR 82g:58090
 8.
 T. Kato, Perturbation theory for linear operators, SpringerVerlag, New York, 1980. reprint MR 96a:47025
 9.
 T. Kato, A second look at the essential selfadjointness of the Schrödinger operators, Physical reality and mathematical description, Reidel, Dordrecht, 1974, 193201. MR 57:16958
 10.
 M. Reed, B. Simon, Methods of Modern Mathematical Physics I, II: Functional analysis. Fourier analysis, selfadjointness, Academic Press, New York e.a., 1975. MR 58:12429a; MR 58:12429b
 11.
 B. Simon, An abstract Kato's inequality for generators of positivity preserving semigroups, Indiana Univ. Mat. J., 26 (1977), 10671073. MR 57:1194
 12.
 B. Simon, Kato's inequality and the comparison of semigroups, J. Funct. Anal., 32 (1979), 97101. MR 80e:47036
 13.
 B. Simon, Maximal and minimal Schrödinger forms, J. Operator Theory, 1 (1979), 3747. MR 81m:35104
 14.
 M. Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equations, SpringerVerlag, New York e.a., 1996. MR 98b:35003
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Additional Information
Ognjen Milatovic
Affiliation:
Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Address at time of publication:
Department of Mathematics, Fitchburg State College, Fitchburg, Massachusetts 01420
Email:
ogmilato@lynx.neu.edu, omilatovic@fsc.edu
DOI:
http://dx.doi.org/10.1090/S0002993903070758
PII:
S 00029939(03)070758
Received by editor(s):
August 20, 2002
Published electronically:
April 24, 2003
Communicated by:
David S. Tartakoff
Article copyright:
© Copyright 2003
American Mathematical Society
