Consider a one-dimensional Schrodinger operator of the form

$$

H_h = – frac {d ^ 2} {dx ^ 2} + V_h (x), quadro> 0,

$$

sure $ L ^ 2 ([-1,1]$. Suppose further that mapping $ (0, infty) times[-1,1] ni (h, x) mapsto V_h (x) $ is smooth. Let $ lambda_0 (h): = inf sigma (H_h) $ denotes the smallest spectral element of $ H_h $. According to Sturm-Liouville operators theory, it is known that $ lambda_0 (h) $ is in fact a nondegenerate eigenvalue (under appropriate boundary conditions, for example, Dirichlet).

I am now interested in $ frac {d} {d}} lambda_0 (h) $. The approach of a hitchhiker would be the following:

$$

frac {d} {d}} lambda_0 (h) = frac {d} {d} {min} {min} { substack {f in mathcal {D} (H_h) \ | f | = 1}} int _ {- pi} ^ pi f ^ prime (x) ^ 2 + V_h (x) f (x) ^ 2 , dx = min _ { n n n mathcal {D} (H_h) \ | f | = 1}} int _ {- pi} ^ pi f ^ prime (x) ^ 2 + f (x) ^ 2 frac { partial} { partial h} V_h (x) , dx.

$$

Can this manipulation be made rigorous? A reference to relevant literature would suffice.