P. Kurasov, DISTRIBUTION-THEORY FOR DISCONTINUOUS TEST FUNCTIONS AND DIFFERENTIAL-OPERATORS WITH GENERALIZED COEFFICIENTS, Journal of mathematical analysis and applications, 201(1), 1996, pp. 297-323
Investigation of the differential operators with the generalized coeff
icients having singular support on a disjoint set of points requires t
he consideration of the distribution theory with the set of discontinu
ous test functions. Such a distribution theory for test functions havi
ng discontinuity at one point is developed. A four-parameter family of
Schrodinger operators, formed by the operators with singular potentia
l, singular metrics and singular gauge field, is considered. It is pro
ved that this family of singular interactions describes all possible s
elfadjoint extensions of the second derivative operator defined on the
functions vanishing in a neighbourhood of the point. Approximation by
operators with smooth coefficients is discussed. (C) 1996 Academic Pr
ess, Inc.