# Differentiation of the eigenvalues ​​of a Schrodinger operator

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.

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