In an unbounded domain Ω we study the asymptotic decay (for | x |→∞) of functions u ∊ L2(Ω) which are solutions of the following problem –Δu + cu = 0. c denotes a strictly positive function. Upper bounds are easily found via the maximum principle. When c is rotationally invariant lower bounds are obtained via asymptotic expansion. In the general case we use a method of ‘commutation’ of operators. In particular we consider the case where . Applications to the asymptotic decay of the bound states of a Hamiltonian are given.

The paper presents a list of unsolved problems about operators on Hilbert space, accompanied by just enough definitions and general discussion to set the problems in a reasonable context. The subjects are: quasitriangular matrices, the resemblances between normal and Toeplitz operators, dilation theory, the algebra of shifts, some special invariant subspaces, the category (in the sense of Baire) of the set of non-cyclicoperators, non-commutative(i.e. operator) approximation theory, infinitary operators, and the possibility of attacking invariance problems by compactness or convexity arguments.

Second-order expressions –d2/dx2+q whose squares are of limit-3 type are constructed. The construction is simpler than others in the literature. The function q can be chosen to be infinitely differentiable and decreasing and to satisfy q(x)≧ –x2(log x)2+ε (x→∞) for a given ε>0.

This paper is concerned with some properties of an ordinary symmetric matrix differential expression M, denned on a certain class of vector-functions, each of which is defined on the real line. For such a vector-function F we have M[F] = −F“ + QF on R, where Q is an n × n matrix whose elements are reasonably behaved on R. M is classified in an equivalent of the limit-point condition at the singular points ± ∞, and conditions on the matrix coefficient Q are given which place M, when n> 1, in the equivalent of the strong limit-point for the case n = 1. It is also shown that the same condition on Q establishes the integral inequality for a certain class of vector-functions F.

Using a non-linear transformation of the state space, the dynamic behaviour of an adaptive thirdorder relay system is analysed. The control strategy yields state paths close to the time-optimal trajectories.

Let L denote the ordinary differential operator given by Lf = (pf″)″ + (qf′)′ + rf, with p″, q′ and r continuous functions on [0,∞), and with p>0, q ≦ 0, and r ≧ 0. It is proved that if the equation Lg = 0 possesses a non-oscillatory solution, then any non-trivial solution f to Lf = 0 such that f(0) = f′(0) = 0 is eventually bounded away from zero.

This theorem is used to prove that, for a general class of functions q and r containing the polynomials as a very special case, the equation Lg = 0 has at most two linearly independent square integrable solutions, when p is identically one, q ≦ 0 and r ≧ 0.

Finally, the main theorem is applied to show that certain sixth-order operators are limit-3.

Integral operators are used to solve the direct and inverse problems of the scattering of acoustic waves by a spherically stratified inhomogeneous medium of compact support. The results are valid for all values of the wave number and an arbitrarily large index of refraction. In the limiting case of small wave number or small inhomogeneities the results are in agreement with those of Rorres and Born.