Positivity 2: 77–99, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
Schrödinger Operator Perturbed by Operators
Related to Null Sets
, V. KOSHMANENKO
and S. ÔTA
Institute of Theoretical Physics, University of Wroclaw, Wroclaw, Poland;
Mathematics, Ukrainian Academy of Sciences, Kyiv, Ukraine;
Kyushu Institute of Design,
(Accepted: 6 January 1998)
Abstract. We discuss the Schrödinger operator with positive singular perturbations given by opera-
tors which act in the space constructed by a positive measure supported by a null set. We construct
examples when perturbations are given by the one-dimensional Laplacian on a segment.
Mathematics Subject Classiﬁcations (1991): 47A10, 47A55
Key words: singular perturbation, self-adjoint extension, positive Radon measure
The theory of perturbing the Schrödinger operator by an object living on a set of
vanishing Lebesgue measure originated in physics. A typical problem treated in
quantum mechanics is to describe the motion of a non-relativistic particle moving
in the ﬁeld of external potential forces. The corresponding Hamiltonian is then
H =−+V (1.1)
where is the Laplace operator in
with the Dirichlet boundary condition at
inﬁnity and V a measurable function. If one deals with the short-range forces
around, say, point zero, then V is supported by a small vicinity of 0. It is then
natural to think of idealization leading to zero-range potential supported by a point.
Symbolically one would then have
or more generally
is the coupling constant describing intensity of the forces at y, δ
Dirac δ-function at y and Y is a ﬁnite or countably inﬁnite set of points in
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