Suppose $f: {\mathbb R}\to {\mathbb R}$ is a continuous function satisfying $$f(s) = c_- >0,\forall s<-T; f(s) = c_+<0, \forall s>T.$$ (For simplicity we may even assume that $f$ is smooth, decreasing, etc).

Then consider $$D: L_1^2 \to L^2,\ \eta\mapsto {d\over ds} \eta + f \eta, $$ which is Fredholm with index -1. It is injective and its range is the $L^2$-complement of $$\eta_0:= \exp \Big(\\,\int_0^s f(z) dz\Big) .$$

Hence we have related estimates: there exists constant $C_i>0$ with

1) $\| \eta\|_{L_1^2} \leq C_1 \| D \eta \|_{L^2},$ because $D$ is injective.

2) On $\eta_0^\bot$, we have a bounded right inverse $$Q: \eta_0^\bot \to L_1^2,\ D Q = P$$ where $P$ is the projection onto $\eta_0^\bot$ and $ \|Q \| \leq C_2$.

My question is, for any real number $\lambda>0$, if we replace $f$ by $$f_\lambda(s) =f(\lambda s)$$ then $D$ still have all above qualitative properties. But the constants will vary. As $\lambda \to 0$ or $\lambda \to \infty$, how do they vary? Linearly in $\lambda$ or exponentially?