Let $\psi(x,t)$ be a solution of the Schroedinger on the line
$$i\frac{\partial \psi}{\partial t}=-\frac{1}{2m}\frac{\partial^2 \psi}{\partial x^2}.$$
One assumes that $\psi(x,0)$ "behaves well" as $|x|\to \infty$, e.g. square integrable on the line. If necessary one may impose some extra assumptions.



> 
> For a fixed bounded interval $[a,b]$ I would be interested to estimate the behaviour of 
>  $$\int\_a^b|\psi(x,t)|^2dx$$
>  when $t\to +\infty$. The above integral is expected to decay as $t\to +\infty$, and the question is to give a more explicit estimate. Eventually I would like to show that
>  $$\int\_{0}^{+\infty}dt\int\_a^b|\psi(x,t)|^2dx<\infty.$$
> 
> 
> 



Estimate of a solution of Schroedinger equation for a free particle
For the free evolution specifically, in relation to [Mateusz's comment](https://mathoverflow.net/questions/346077/estimate-of-a-solution-of-schroedinger-equation-for-a-free-particle#comment866168_346077): on the Fourier side you can write the solution as (for $m = -1/2$; you can rescale/invert time to get other scalings) 


$$ \phi(t,x) = \frac{1}{\sqrt{2\pi}} \int e^{it\xi^2} e^{ix\xi} \hat{\phi}\_0(\xi) ~\mathrm{d\xi} $$


where $\hat{\phi}\_0$ is the Fourier transform of the initial data. Stationary phase tells you that locally uniformly in $y$ you have (notice also that the limit is independent of $y$)


$$ \lim\_{t\to\infty} t^{1/2} \phi(t,y) = \frac{e^{i\pi/4}}{\sqrt{2}} \hat{\phi}\_0(0) = \frac{e^{i\pi/4}}{2\sqrt{2}} \int \phi\_0(x) ~\mathrm{d}x $$


Similarly, you have that if $\hat{\phi}\_0(0)$ has vanishing first $k$ derivatives, then locally uniformly 


$$ \lim\_{t\to\infty} t^{1/2 + k+1} \phi(t,y) \propto \hat{\phi}\_0^{(k+1)}(0) \propto \int x^{k+1} \phi\_0(x) ~\mathrm{d}x $$


This estimate holds, in particular, for all Schwartz initial data. 


