ABSTRACT
We first recall a standard formulation for the one-parameter case. Let q(x), a ≤ x <∞, be real and continuous. For the differential equation
y′′ + (λ− q(x))y = 0, a ≤ x <∞, (11.1.1)
we introduce the following basic classification:
(i) the equation is in the “limit-circle” condition at ∞ if all solutions are in L2(a, ∞),
(ii) the equation is in the limit-point condition at ∞ if there is a solution not in L2(a, ∞). It is known, from the fundamental work of Weyl, that the classification does not depend on the choice of the parameter λ; if all solutions are in L2(a, ∞) for some λ, real or complex, then this is the case for all λ.
