Deficiency indices and spectral theory of third order differential operators on the half line

Abstract

We investigate the spectral theory of a general third order formally symmetric differential expression of the form. A Kummer-Liouville transformation is introduced which produces a differential operator unitarily equivalent to L. By means of the Kummer-Liouville transformation and asymptotic integration, the asymptotic solutions of L[y] = zy are found. From the asymptotic integration, the deficiency indices are found for the minimal operator associated with L. For a class of operators with deficiency index (2, 2), it is further proved that almost all selfadjoint extensions of the minimal operator have a discrete spectrum which is necessarily unbounded below. There are however also operators with continuous spectrum.

Two singular point linear Hamiltonian systems with an interface condition

“…Further work was done by Behncke in [5,6] for fourth order equations and for higher order equations in [4,7]. Additional results may be found in [9,13]. Results on asymptotic integration which extend the classical Levinson theorem have been given by Behncke and Remling [10,11] and Behncke [8].…”

Spectral analysis of fourth order differential operators III

“…+ k r = l if the coefficients f = m, n, p, q, w are l times differentiable. The first of these conditions implies that the contribution to the diagonal from all matrix elements outside the (3,4) block is integrable from the second diagonalization onward. It is rather weak and superfluous in most cases.…”

“…P r o o f. As before apply the first diagonalizing transformation as in section 2. It follows from (2.20), (4.1), and (4.2) that the matrix elements outside the (1,2) and (3,4) block are integrable. For this reason we choose the second transformation to be of block form 1 0 0 S2 .…”

“…Thus all large eigenvalues square integrable solutions are uniform square integrable. Since all matrix elements outside the (1,2) and (3,4) block are O(x −2 ) the remaining diagonalizing tranformations may be chosen in block form. As expected the large eigenvalue submatrix can be diagonalized without problems.…”

Some singular third‐order boundary value problems

“…However, the same may not be so easy if at least one of the coefficients has a singularity at at least one point on the interval. The latter one has been a famous problem since the pioneering work was introduced by Weyl, 4 and at this point, we should note that there exist some ways to solve this, and more general problems in the literature (for example, see, previous studies [5][6][7][8][9][10][11][12][13][14]. A huge number of works are about even-order cases.…”