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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

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), 317-323
MSC: Primary 47F05; Secondary 35J10, 81C10
MathSciNet review: 931731
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Abstract: Of concern are the minimal and maximal operators on $ {L^2}({{\mathbf{R}}^n})$ associated with the differential expression

$\displaystyle {\tau _Q} = \sum\limits_{j = 1}^n {(i\partial /\partial {x_j}} + {q_j}(x){)^2} + W(x)$

where $ (q, \ldots ,{q_n}) = \operatorname{grad}Q$ for some real function $ W$ on $ {{\mathbf{R}}^n}$ and $ W$ satisfies $ c{\left\vert x \right\vert^{ - 2}} \leq W(x) \leq C{\left\vert x \right\vert^{ - 2}}$. In particular, for $ Q = 0$, $ {\tau _Q}$ reduces to the singular Schrödinger operator $ - \Delta + W(x)$. Among other results, it is shown that the maximal operator (associated with the $ {\tau _Q}$) is the closure of the minimal operator, and its domain is precisely

$\displaystyle \operatorname{Dom}\left( {\sum\limits_{j = 1}^n {{{(i\partial /\partial {x_j} + {q_j}(x))}^2}} } \right) \cap \operatorname{Dom}(W),$

provided that $ C \geq c > - n(n - 4)/4$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0931731-8
PII: S 0002-9939(1989)0931731-8
Keywords: Essential self adjointness, Schrödinger operator, singular potential, magnetic vector potential
Article copyright: © Copyright 1989 American Mathematical Society