for certain classes of functions h on [0,1] . We show that for so-called strongly odd dominated functions r (including odd r ) these problems are equivalent. This allows us to apply known results from the theory of one problem to the others.Mathematics subject classification (2010): Primary 34B24; Secondary 26D10, 34L10, 47A75.

No abstract

In this note we construct an odd weight function w on (-1,l) with w ( x ) > 0 for x > 0 such that the eigenfunctions of the indefinite Sturm-Liouville problem -f" = Xwf with boundary conditions f(-1) = f(1) = 0 do not form a FLiesz basis of L f w , ( -l , 1).

No abstract

Let −DpD be a differential operator on the compact interval [−b, b] whose leading coefficient is positive on (0, b] and negative on [−b, 0), with fixed, separated, self-adjoint boundary conditions at b and −b and an additional interface condition at 0. The self-adjoint extensions of the corresponding minimal differential operator are non-semibounded and are related to non-semibounded sesquilinear forms by a generalization of Kato's representation theorems. The theory of non-semibounded sesquilinear forms is applied to this concrete situation. In particular, the generalized Friedrichs extension is obtained as the operator associated with the unique regular closure of the minimal sesquilinear form. Moreover, among all closed forms associated with the self-adjoint extensions, the regular closed forms are identified. As a consequence, eigenfunction expansion theorems are obtained for the differential operators as well as for certain indefinite Kreĭn–Feller operators with a single concentrated mass.