We study a linear cocycle over irrational rotation s?(x) = x+? of a circle T1. It is supposed that the cocycle is generated by a A? : T1 to SL(2, R) that depends on a small parameter ? « 1 and has the form of the Poincaré map corresponding to a singularly perturbed Schrödinger equation. Under assumption that the eigenvalues of A?(x) are of the form exp (±?(x)/?), where ?(x) is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect to the parameter ?. We show that in the limit ? ? 0 the cocycle exhibits ED for the most parameter values only if it is exponentially close to a constant cocycle. In the other case, when the cocycle is not close to a constant one and, thus, it does not possess ED, the Lyapunov exponent is typically large.