TY - JOUR AU - Eastham, M. S. P. AB - ON THE NATURE OF THE SPECTRUM IN THE THEORY OF EIGENFUNCTIONS M. S. P. EASTHAM 1. In the theory of the differential equation = O (1.1) in two dimensions, where we use the general notation of [4], the following theorem is known [4; §16.5]. THEOREM 1. // q(x) > a for all x and q(x)^f} for all sufficiently large values of |x| , the spectrum in the problem with (1.1) holding in the whole space is discrete over A < jS. The question arises as to whether the conclusion of this theorem continues to hold if q(x) ^j 8 only for some x with |x | large instead of for all such x. To make this more precise, let A(cf>) denote the region 0<0<, where 9 is the angular polar coordinate and 0<< 2n. Then we ask whether the following extension of Theorem 1 is true. (i) / / q{x) ^ a for all x, q(x)^y for all xin A (<£) with \ x \ large enough, and g(x) ^j 8 for all other x with \x\ large enough, where j8>y , then the spectrum in the problem with (1.1) holding in the whole space is discrete over A