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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 $$\int_{0}^{+\infty}dt\int_a^b|\psi(x,t)|^2dx<\infty.$$

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.

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 $$\int_{0}^{+\infty}dt\int_a^b|\psi(x,t)|^2dx<\infty.$$

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asv
asv
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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.

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 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.

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.

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asv
asv
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Estimate of a solution of Schroedinger equation for a free particle

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 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.