Inverse problems of spectral analysis for the Sturm-Liouville operator with regular boundary conditions

We consider the Sturm-Liouville operator Lu = u ′′ − q(x)u defined on (0, π) with regular but not strongly regular boundary conditions. Under some supplementary assumptions we prove that the set of potentials q(x) that ensure an asymptotically multiple spectrum is everywhere dense in L 1 (0, π). In the present paper we consider eigenvalue problems for the Sturm-Liouville equation u ′′ − q(x)u + λu = 0 (1) with two-point boundary conditions B i (u) = a i1 u ′ (0) + a i2 u ′ (π) + a i3 u(0) + a i4 u(π) = 0, (2) where B i (u) (i = 1, 2) are linearly independent forms with arbitrary complex-valued coefficients. Function q(x) is an arbitrary complex-valued function of class L 2 (0, π). It is convenient to write conditions (2) in the matrix form A = a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 and denote the matrix composed of the ith and jth columns of A (1 ≤ i < j ≤ 4) by A(ij); we set A ij = det A(ij). Let the boundary conditions (2) be regular but not strongly regular [1, pp. 71-73], which, by [1, p. 73] is equivalent to the conditions To investigate this class of problems it is appropriate [2] to divide conditions (2) satisfying (3) into 4 types: