TY - JOUR
AU - Eastham, M. S. P.
AB - ON THE DISCRETENESS OF THE SPECTRUM IN EIGENFUNCTION THEORY M. S. P. EASTHAM 1. Let S? denote th e spectrum of th e differential equation = O (1.1) holding in th e whole space in n dimensions, where q(x) is real and n^l. Since we shall make use of the theory of (1.1) as developed in [7], we assume throughout the paper (except in §7), without further mention, that, in any bounded region, q(x) is bounded and has continuous first derivatives except possibly on a finite number of piecewise differentiable surfaces (curves in two dimensions) (cf. [7; §§11.6 and 13.12]). Then the discreteness of Sf is related to the behaviour of q(x) for large values of x by th e following theorem. [7 ; Theorem 16.5]: THEOREM 1. If q(x)^a. for all x and q(x) ^ /? for all sufficiently large values of\x\, then S? is discrete for A < /?. In this paper we investigate the question of what can be said about the discreteness of S? when the condition that q(x)^f$ fails to hold in some unbounded region A. Suppose then tha t q(x) ^ y in A and q(x)^f$ in ^A, both when | x | 
TI - On the Discreteness of the Spectrum in Eigenfunction Theory
JF - Journal of the London Mathematical Society
DO - 10.1112/jlms/s1-42.1.309
DA - 1967-01-01
UR - https://www.deepdyve.com/lp/wiley/on-the-discreteness-of-the-spectrum-in-eigenfunction-theory-ID5ideRjQE
SP - 309
EP - 315
VL - s1-42
IS - 1
DP - DeepDyve
ER -