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I'm trying to understand the derivation of a formula which gives the probability of a quantum forced harmonic oscillator to transit to the state $n$, at instant $t$, if it was at state $m$, at instant $t=0$. That formula can be find here, at page 5. It is:

$$P_{mn}(t)=|b_{mn}(t)|^2=m!n!e^{-epsilon_0}epsilon_0^{m+n}S_{mn}^2$$

where $epsilon_0(t)=frac{frac{1}{2}Mdot{u}^²(t)+frac{1}{2}ku^2(t)}{hbar w}$, with $w=sqrt{frac{k}{M}}$ and $S_{mn}(t)=sum_{j=0}^mufrac{left(-1right)^jepsilon_0^{-j}}{(n-j)!j!(m-j)!}$, with $mu=text{min}(m,n)$.

The coefficient $b_{mn}$ is given by:

begin{equation}
b_{mn}(t)=int_{-infty}^{infty}psi_m(x,t)v_n^*(x,t)dx
tag{1}
end{equation}

where:

$$psi_m(x,t)=N_mexpleft{frac{i}{hbar}left[Mdot{u}(t)x-int_{0}^tleft[delta(t)+E_mright]dtright]-frac{1}{2}alpha^2left[x-u(t)right]^2right}H_mleft{alphaleft[x-u(t)right]right}$$

and

$$v_n(x,t)=N_nexpleft(-frac{1}{2}alpha^2x^2right)H_nleft(alpha xright)expleft(-frac{i}{hbar}E_ntright)$$

with $N_m^2=frac{alpha}{pi^frac{1}{2}2^m m!}$ and $alpha^2=frac{1}{hbar}left(Mkright)^{frac{1}{2}}$. More details are given in the paper.

My doubt is about the integral $(1)$ which involves Hermit polynomials with different arguments. I don't know how to solve it, and it isn't explained in the paper. Can you help me?