THE ASYMPTOTIC-BEHAVIOR OF THE PRINCIPAL EIGENVALUE FOR SMALL PERTURBATIONS OF CRITICAL ONE-DIMENSIONAL SCHRODINGER-OPERATORS WITH V(X)=L(+-)/X(2) FOR +/-X-MUCH-GREATER-THAN-1/

Let H=-d(2)/dx(2)+V on R, where V(x)=l(1)/x(2), on x much less than 1, and V(x)=l(2)/x(2), on x much less than-1, for constants l(1),l(2). A ssume that H is a critical operator. It turns out that it is possible to realize a critical operator H of the above form if and only if min( l(1),l(2))greater than or equal to-1/4. Denote the ground state of H b y phi(0). Let W be a compactly supported function and define H-epsilon =H+epsilon W. It is known that H-epsilon will possess a negative eigen value for epsilon>0 if and only if I=integral(R) W phi(0)(2) dx less t han or equal to 0. This negative eigenvalue, lambda(epsilon), is uniqu e if epsilon>0 is sufficiently small. We obtain the leading order asym ptotics for lambda(epsilon), as E-->0. In particular, the order of dec ay depends on whether I=0 or I<0, and also varies continuously as min( l(1),l(2)) varies in the interval [-1/4,3/4]. The order of decay is in dependent of min(l(1),l(2)), for min(l(1),l(2))>3/4, but this order is not equal to the order when min(l(1),l(2))=3/4. (C) 1995 Academic Pre ss, Inc.