Let $N\geq1$ be an integer and let $H:[0,1]^2\to\mathbb C^{N\times N}$ be a pointwise hermitean matrix valued function.
For $y\in[0,1]$ and $0\leq a\leq b\leq 1$, let $U\_y(b,a)$ be the time evolution operator (solution operator) to the Schrödinger equation $i\psi'(x)=H(x,y)\psi(x)$.
More precisely, the operator is defined so that for any $\phi\in\mathbb C^N$ and $a\in[0,1]$ the solution of the initial value problem
$$
\begin{cases}
i\psi'(x)=H(x,y)\psi(x)\\
\psi(a)=\phi
\end{cases}
$$
is $\psi(x)=U\_y(x,a)\phi$.
(Alternatively, we could define $U\_y(b,a)$ to be the solution of $U\_y(a,a)=I$, $\partial\_aU\_y(a,b)=-iH(a,y)U\_y(a,b)$ and $\partial\_bU\_y(a,b)=iU\_y(a,b)H(b,y)$.)


I'm interested in differentiating this time evolution operator with respect to the parameter $y$.
A sketchy calculation and physical intuition suggest that
$$
\partial\_yU\_y(1,0)
=
-i\int\_0^1U\_y(1,x)\partial\_yH(x,y)U\_y(x,0)dx.
$$
What regularity of $H$ do I need to assume for this to be true?
Is there a reference for this result?
I feel that giving a proof instead of a reference would be a distraction from what I want (readers of what I'm writing) to focus on, unless there is a particularly short and neat proof.



Derivative of a time evolution operator w.r.t. a parameter
Come to think of it, though, since you are in the ODE world, isn't what you wrote a direct consequence of variation of parameters?


I write it in a slightly easier (for me) notations. 


**Proposition**: Let $S(t\_1,t\_2)$ be the solution operator for $\phi'(t) = A(t) \phi(t)$ with initial data prescribed at $t\_2$, and let $T(t\_1,t\_2)$ be the solution operator for $\phi'(t) = B(t) \phi(t)$ with initial data prescribed at $t\_2$, we have
$$ S(t\_1,t\_2)x - T(t\_1,t\_2)x = \int\_{t\_2}^{t\_1} T(t\_2, s) [A(s) - B(s)] S(s,t\_1) ~\mathrm{d}s .$$


*Proof*:


$$ \begin{align} (S(t,0) x - T(t,0)x)' &= A(t)S(t,0) x - B(t) T(t,0)x \\ &= [A(t)- B(t)]S(t,0) x + B(t) (S(t,0)x - T(t,0)x) \end{align}$$
so by Duhamel's principle we have
$$ S(t,0) x - T(t,0)x = \int\_0^t T(t,s) [A(s) - B(s)] S(s,0) x ~\mathrm{d}s $$




---


Now if $A(t) = B(t) + D(t) y + O(y^2)$, you immediately have for each fixed $t$


$$ S(t,0) x - T(t,0) x = y \int\_0^t T(t,s) D(s) T(s,0) \mathrm{d}s + O(y^2) $$


which should imply the result you want. 


