On Degenerate Boundary Conditions for Operator $$D^4$$ D 4

Abstract

The common form for degenerate boundary conditions for the operator \(D^4\) (\(D^n\)) is found. It is shown that the matrix for coefficients of degenerate boundary conditions has a two diagonal form and the elements for one of the diagonal are units. Operator \(D^4\) whose spectrum fills the entire complex plane are studied, too. Earlier, examples of eigenvalue problems for the differential operator of even order with common boundary conditions (not containing a spectral parameter) whose spectrum fills the entire complex plane were given. However, in connection with this, another question arises whether there are other examples of such operators. In this paper we show that such examples exist. Moreover, all eigenvalue boundary problems for the operator \(D^4\) whose spectrum fills the entire complex plane are described. It is proved that the characteristic determinant is identically equal to zero if and only if the matrix of coefficients of boundary conditions has a two diagonal form. The elements of this matrix for one of the diagonal are units, and the elements of the other diagonal are 1, \(-1\) and an arbitrary constant.