Mediterr. J. Math. (2017) 14:189
published online August 16, 2017
Springer International Publishing AG 2017
Weyl Function of Sturm–Liouville Problems
with Transmission Conditions at Finite
Kun Li, Jiong Sun and Xiaoling Hao
Abstract. In this paper, a class of singular Sturm–Liouville problems
with transmission conditions at ﬁnite interior points is studied. The
equation, center and radius of Weyl circle are given. The necessary and
suﬃcient conditions for the function m(λ), which is in the Weyl circle,
are established. The deﬁnition of Weyl function for singular Sturm–
Liouville problems with transmission conditions in the case of limit circle
is given, and the properties of Weyl function are discussed.
Mathematics Subject Classiﬁcation. 34B24, 47E05.
Keywords. Sturm–Liouville problems, Weyl function, transmission con-
ditions, limit circle.
In recent years, more and more researchers are interested in the discontinuous
Sturm–Liouville problems for its wide applications in physics and engineer-
ing. Such problems are connected with discontinuous material properties,
such as heat and mass transfer (see ), vibrating string problems when the
string loaded additionally with point masses, the heat transfer problems of
the laminated plate of membrane (that is, the plate which is formed by over-
lap of materials with diﬀerent characteristics) and diﬀraction problems. To
deal with interior discontinuities, some conditions are imposed on the dis-
continuous points, which are often called transmission conditions, interface
conditions or point interactions (see [2–5]). The various physics applications
of this kind of problems are found in many literatures (see [5–13]) and cor-
responding references cited therein.
Regular Sturm–Liouville problems with discontinuities have been widely
researched, which include one discontinuity (see [5–7]), two discontinuities
(see [8,9]) and ﬁnite discontinuities (see [10–13]). In these papers, the asymp-
totic formulas of eigenvalues and eigenfunctions, inverse problems, Green