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  1. The necessary condition for $\int\limits_{-\infty}^{+\infty}|\psi(x)|^2dx$ to be integrable is that $\psi(x)\rightarrow 0$ as $x\rightarrow\pm\infty$. But this is not the sufficient condition. For example, $\delta(x)$ vanishes as $x\rightarrow \pm \infty$, but it is not square-integrable. What is the sufficient criterion for $\int\limits_{-\infty}^{+\infty}|\psi(x)|^2dx$ to converge?

  2. In the previous question $\psi(x)$ was an arbitrary function. If now $\psi(x,t)$ be an arbitrary solution (not necessarily a stationary state) of the time-dependent Schrodinger equation what is the sufficiency condition for $\int\limits_{-\infty}^{+\infty}|\psi(x,t)|^2dx$ to be finite?

  3. Does this conditions remain valid in three-dimensions?

EDIT: Can we have a continuous function $\psi(x)$ which does not go to zero as $x\rightarrow \pm\infty$ and yet the integral $\int\limits_{-\infty}^{+\infty}|\psi(x)|^2dx$ converges?

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    $\begingroup$ This is a math question. Consider Math SE $\endgroup$ – ClassicStyle Apr 4 '16 at 14:09
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    $\begingroup$ Also, see this. As it turns out, going to zero isn't even necessary physics.stackexchange.com/q/75527 $\endgroup$ – ClassicStyle Apr 4 '16 at 14:17
  • $\begingroup$ Yeah, going to zero is not necessary. A function that is $e^{-x^2}$ for $x\in \mathbb{R}\setminus \mathbb{Z}$ and $1$ for $x\in \mathbb{Z}$ is perfectly square-integrable wrt Lebesgue measure. $\endgroup$ – yuggib Apr 4 '16 at 14:21
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    $\begingroup$ Qmechanic gives a smooth counterexample to the condition being necessary here. I'm not sure what your question is - in general, there is no better characterization of square-integrable functions other than them being, well, square-integrable. Also, $\delta(x)$ doesn't "vanish" as $x\to \pm\infty$, that doesn't make sense to begin with - the Dirac delta is not a function. $\endgroup$ – ACuriousMind Apr 4 '16 at 14:50
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    $\begingroup$ Also, consider replacing "the necessary condition" by "a necessary condition". There never is a unique necessary (or sufficient) condition. $\endgroup$ – Antoine Apr 4 '16 at 14:51