Boundary value problems with regular singularities and singular boundary conditions

where skm are real numbers, pk0(t) ∈ C2[a,b], p00(t)p20(t) = 0, p00(t)/p20(t) > 0 for t ∈ [a,b]. Let s2m < s0m + 2, s2m ≤ s1m + 2, m = 0,1, that is, we consider the case of so-called regular singularities. Operators with irregular singularities possess different qualitative properties and require different investigations. Since the solutions of (1.1) may have singularities at the endpoints of the interval, and since in general the values of the solutions and their derivatives at the endpoints are not defined, an important question is how to introduce singular two-point boundary conditions in the general case under consideration. For some particular cases this problem has been studied in [4, 5, 6, 15, 21, 23] and other works. For example, in [4] singular boundary conditions were constructed in the case when the endpoints are of limit-circle type. In this paper, we provide a general method for defining two-point singular boundary conditions in the above-mentioned general case. In Section 2, we construct singular boundary conditions and formulate the corresponding boundary value problems. In Section 3, properties of the spectrum are studied for boundary value problems with