For energy eigenstates $\psi(x)$, does $\psi(x)\to0$ as $|x|\to\infty$ imply $ELet $\psi(x)$ be a solution of the time-independent Schrodinger equation (TISE) in one-dimension $$\psi''(x)+\gamma\left[E-V(x)\right]\psi(x)=0$$ where $\psi(x)$ is called an energy eigenfunction, $\gamma=2m/\hbar^2$ and $E$ is a fixed parameter, called the energy.
Claim 1
If an energy eigenfunction $\psi(x)$ vanishes as $|x|\to\infty$, then examples point out that the energy $E$ of that state satisfies $E<V(\pm\infty)$.
Claim 2
If the energy $E$ of an energy eigenfunction $\psi(x)$  satisfies $E<V(\pm\infty)$, then examples point out that $\psi(x)$ vanishes as $|x|\to\infty$.
Can we prove the above claims mathematically, at least under reasonable assumptions, from the time-independent Schrodinger equation?
 A: The condition that the wave function vanishes at $\pm\infty$ can be formally stated as
$$
\int dx |\psi(x)|^2 =const,
$$
where the constant is by convention taken to be one, i.e., the wave function is normalized. The integration here and everywhere below is implied to be from $-\infty$ to $+\infty$.
We can now calculate
$$
\int dx \psi^*(x)\left[-\frac{\hbar^2}{2m}\partial_x^2 + V(x)\right]\psi(x) = E\int dx |\psi(x)|^2=E
$$
Furthermore
$$ 
\int dx \psi^*(x)\partial_x^2\psi(x) = \psi^*(x)\partial_x\psi(x)|_{x=-\infty}^{+\infty} - \int dx |\partial_x\psi(x)|^2 = - \int dx |\partial_x\psi(x)|^2,
$$
where I used the fact that the wave function vanishes in $+\infty$.
Thus, we can write
$$
E = \frac{\hbar^2}{2m}\int dx |\partial\psi(x)|^2 + \int dx V(x)|\psi(x)|^2
\leq \int dx V(x)|\psi(x)|^2 \leq \max_{x\in\mathbb{R}}V(x)=V_{max}.
$$
This answers claim 1.
Claim 2 is actually not totally true. In the region where $E<V(x)$ the solution of the Schrödinger equation
$$
\psi''(x) + \gamma(E-V_{max})\psi(x)=0
$$
has an approximately exponentially decaying and exponentially growing solutions. Thus, we may have a finite energy solution with a wave function diverging in the infinity. Scattering theory deals with the solutions that do not decay in the infinity, i.e., which are not normalizable (but correspond to a finite particle flux). The solutions growing are mostly discarded on th grounds of physical interpretation, as a mathematical artefact.
Now, let bring up the Ehrenfest theorem, which in this case means that the expectation value of energy is the expectation values of the kinetic energy and the potential energy, which generalizes my derivation to multiple dimensions, including magnetic field, etc. In other words, our "classical" knowledge about finite and infinte motion in a potential is directly transferable to the quantum case.
