On a domain characterization of Schrödinger operators with gradient magnetic vector potentials and singular potentials
Authors:
Jerome A. Goldstein and Roman Svirsky
Journal:
Proc. Amer. Math. Soc. 105 (1989), 317323
MSC:
Primary 47F05; Secondary 35J10, 81C10
MathSciNet review:
931731
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Abstract: Of concern are the minimal and maximal operators on associated with the differential expression where for some real function on and satisfies . In particular, for , reduces to the singular Schrödinger operator . Among other results, it is shown that the maximal operator (associated with the ) is the closure of the minimal operator, and its domain is precisely provided that .
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 P. Baras and J. A. Goldstein, Remarks on the inverse square potential in quantum mechanics, Differential equations (I. W. Knowles and R. T. Lewis, ed.), Elsevier (NorthHolland), Dordrecht, 1984, pp. 3135. MR 799330 (87a:35090)
 [2]
 E. B. Davies, Some norm bounds and quadratic form inequalities for Schrödinger operators. II, J. Operator Theory 12 (1984), 177196. MR 757118 (86a:35106)
 [3]
 J. Glimm and A. Jaffe, Singular perturbation of selfadjoint operators, Comm. Pure Appl. Math. 22 (1969), 401414. MR 0282243 (43:7955)
 [4]
 J. A. Goldstein, Semigroups of linear operators and applications, Oxford Univ. Press, New York and Oxford, 1985. MR 790497 (87c:47056)
 [5]
 J. A. Goldstein and R. Svirsky, Singular potentials and scaling, Houston J. Math. 13 (1987), 557566. MR 929292 (89e:35038)
 [6]
 H. Kalf, A note on the domain characterization of certain Schrödinger operators with strongly singular potentials, Proc. Roy. Soc. Edinburgh 97A (1984), 125130. MR 751183 (85m:47054)
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 H. Kalf, U.W. Schmincke, J. Walter and R. Wüst, On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, Lecture Notes in Math., No. 448, SpringerVerlag, Berlin, 1975, 182226. MR 0397192 (53:1051)
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 M. Reed and B. Simon, Methods of modern mathematical physics. I, II, Academic Press, New York, 1972, 1976. MR 751959 (85e:46002)
 [9]
 D. W. Robinson, Scattering theory with singular potentials. I, The twobody problem, Ann. Inst. H. Poincaré 21A (1974), 185215. MR 0377304 (51:13477)
 [10]
 D.W. Schmincke, Essential selfadjointness of a Schrödinger operator with strongly singular potential, Math. Z. 123 (1972), 4750.
 [11]
 B. Simon, Essential selfadjointness of Schrödinger operators with singular potential, Arch. Rational Mech. Anal. 52 (1973), 4448. MR 0338548 (49:3312)
 [12]
 B. Simon, Hardy and Rellich inequalities in nonintegral dimensions, J. Operator Theory 9 (1983), 143146 and 12 (1984), 197. MR 695943 (84e:35123a)
 [13]
 E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N.J., 1971. MR 0304972 (46:4102)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198909317318
PII:
S 00029939(1989)09317318
Keywords:
Essential self adjointness,
Schrödinger operator,
singular potential,
magnetic vector potential
Article copyright:
© Copyright 1989 American Mathematical Society
