Consider a one-dimensional Schrodinger operator of the form $$ H_h=-\frac{d^2}{dx^2}+V_h(x), \quad h>0, $$ on $L^2([-1,1])$. Let us further assume that the mapping $(0,\infty)\times[-1,1]\ni(h,x)\mapsto V_h(x)$ is smooth. Let $\lambda_0(h):=\inf\sigma(H_h)$ denote the smallest spectral element of $H_h$. By the theory of Sturm-Liouville operators, it is known that $\lambda_0(h)$ is in fact an eigenvalue which is non-degenerate (under suitable boundary conditions, say Dirichlet).
I'm now interested in $\frac{d}{dh}\lambda_0(h)$. A hitchhiker's approach would be the following: $$ \frac{d}{dh}\lambda_0(h)=\frac{d}{dh}\min_{\substack{f\in\mathcal{D}(H_h)\\\|f\|=1}}\int_{-\pi}^\pi f^\prime(x)^2+V_h(x)f(x)^2\,dx=\min_{\substack{f\in\mathcal{D}(H_h)\\\|f\|=1}}\int_{-\pi}^\pi f^\prime(x)^2+f(x)^2\frac{\partial}{\partial h}V_h(x)\,dx. $$ Can this manipulation be made rigorous? A reference to relevant literature would suffice.