On the finite spectrum of three-point boundary value problems

The article is devoted to the solution of one of John Locker's problems, namely, to clarify whether boundary-value problem can have a finite spectrum. It is shown that if a differential equation does not have multiple roots of the characteristic equation, then the spectrum of the corresponding three-point boundary value problem cannot be finite. The proof of the theorem is based on the fact that the corresponding characteristic determinant is an entire function of class K and the results of V.B. Lidsky and V.A. Sadovnichy, from which it follows that the number of its roots (if any) of an entire function of class K is infinite and, if the characteristic determinant is not identical to zero, has asymptotic representations written out in the works of V.B. Lidsky and V.A. Gardener. If the roots of the characteristic equation are multiple, then the spectrum can be finite. Moreover, there are boundary value problems with a predetermined finite spectrum. The problems of determining the finite spectrum from a predetermined finite spectrum can be viewed as the inverse problems of determining the coefficients of a differential equation from a given spectrum.