Consider the differential expression

It is shown here that given pn > 0 with pn ∈ C2n(0, ∞) there exist coefficients

such that all solutions of the equation Ry = 0 are in L2(0,∞). The pi for i<n can be explicitly obtained in terms of pn and a parameter function.

The purpose of this work is to consider some analytic aspects of the matrix functional differential equation, y′(x) = Ay(λx) + By(x), (0 ≦ x < ∞), where λ is a constant with 0 < λ < 1 and A and B are constant matrices.

For various classes of right noetherian rings it is shown that projective right modules are either finitely generated or free.

A differential operator (1) Ly = y(n) + an-1(t)y(n-1) +…+ao(t)y is said to be disconjugate on an interval I if every non-trivial solution of Ly = 0 has less than n zeros on I, multiple zeros being counted according to their multiplicity. An algorithm for the construction of disconjugate operators of type (1) is given here. All such operators are obtained at least in the case when the interval I is open or compact.

In the first part of the paper a criterion is given for two self-adjoint operators T, S in a Hilbert space to have the same essential spectrum, S being given in terms of T and a perturbation P. If P is a symmetric operator and the operator sum T+P is self-adjoint, then S = T+P. Otherwise, T is assumed to be semi-bounded and S is taken to be the form extension of T+P defined in terms of semi-bounded sesquilinear forms. In the case when S = T+P, the result obtained generalises the results of Schechter, and Gustafson and Weidmann for Tm- compact (m> 1) perturbations of T. In the second part of the paper a detailed study is made of the Dirichlet integral

associated with the general second-order (degenerate) elliptic differential expression in a domain Conditions under which t is closed and bounded below are established, the most significant feature of the results being that the restriction of q to suitable subsets of Ω can have large negative singularities on the boundary of Ω and at infinity. Lastly some examples are given to illustrate the abstract theory.

Sufficient conditions are derived for every solution of a nonlinear Schrödinger equation (or inequality) to be oscillatory in an exterior domain of En. Such results apply in particular to the n-dimensional Emden-Fowler equation. The method involves oscillatory behaviour of solutions of a nonlinear ordinary differential inequality satisfied by the spherical mean of a positive solution of the Schrödinger equation.