How is a bound state defined in quantum mechanics? How is a bound state defined in quantum mechanics for states which are not eigenstates of the Hamiltonian i.e. which do not have definite energies? Can a superposition state like $$\psi(x,t)=\frac{1}{\sqrt{2}}\phi_1(x,t)+\frac{1}{\sqrt{2}}\phi_2(x,t),  $$ where $\phi_1$ and $\phi_2$ are energy eigenstates be a bound state? How to decide?
 A: The bound state is defined such that the probability density average will be finite at some particular space region when time passes. While for unbounded states, as time passes, the probability density will tends to zero. See Landau Quantum Mechanics section 10.
This can be understand as this, if the state is bounded, i.e. it is exist only within some particular region, so the probability density should be finite in that region as time passes. On the contrary, when the state is a free motion, the wave package will spread out as time passes, thus the probability density at any point will tends to zero as time tends to infinity.
Edit: now I want to say that discrete eigenstates OR superpositions of these are bound states; while otherwise is an unbounded one.
Note that in this definition of bound state, the average energy $E<V(\pm\infty)$ always holds. However, $E<V(\pm\infty)$ cannot guarantee a state to be bounded. For example, a state (like this one) composed of both discrete spectra and continuum spectra, can have its average energy $E$ either larger or smaller than $V(\pm\infty)$. You might say it's an unbounded one, it depends.
The criterion $E<V(\pm\infty)$ guarantees a bounded state if and only if by $E$ you refer to the energy of an eigenstate.
A: Bound states are usually understood to be square-integrable energy eigenstates; that is, wavefunctions $\psi(x)$ which satisfy
$$
\int_{-\infty}^\infty|\psi(x)|^2\text dx<\infty
\quad\text{and}\quad
\hat H \psi=E\psi.
$$
This is typically used in comparison to continuum states, which will (formally) obey the eigenvalue equation $\hat H\psi=E\psi$, but whose norm is infinite. Because their norm is infinite, these states do not lie inside the usual Hilbert space $\mathcal H$, typically taken to be $L_2(\mathbb R^3)$, which is why the eigenvalue equation is only formally true if taken naively - the states lie outside the domain of the operator. (Of course, it is possible to deal rigorously with continuum states, via a construct known as rigged Hilbert spaces, for which a good reference is this one.)
A: Since states which are not eigenstates of the Hamiltonian are also not eigenstates of the time evolution, it does not make sense to talk about "bound states" for these states, as they are continually changing into other states. For energy eigenstates, it makes sense to speak of "a bound state", since that state will stay the same forever unless acted upon.
A: Bound states of a system are the states for which the particle(s) remains localised in a bounded region of space for all times. Let us consider the case for a single particle. $\mathcal{H} := \mathrm{L}^2(\mathbb{R}^3)$ is the Hilbert space of a single particle where $\mathbf{r} \in \mathbb{R}^3$. A pure state $|\psi\rangle \in \mathcal{H}$ is called a bound state iff for every $\epsilon 
> 0$, there exists a bounded set $A \subset \mathbb{R^3}$ such that $$\int_A |\psi(\mathbf{r}, t)|^2 \, \mathrm{d}^3\mathbf{r} \geq 1 - \epsilon$$ for all $t \in \mathbb{R}$. The integral captures the probability of the particle to be found in the region $A$ at time $t$. If this probability remains arbitrarily close to $1$ for all times, then we can say that the particle remains in this region fo  all times. Note that we first fix $A$ and then evolve the states in time. Our choice of $A$ should be valid for all times.
In scattering states, the particle can be thought of as escaping to infinity as time evolves. Similarly, if for every bounded set $A \subset \mathbb{R}^3$ $$\int_A |\psi(\mathbf{r}, t)|^2 \, \mathrm{d}^3\mathbf{r} \rightarrow 0,$$ as $t \rightarrow +\infty$, we say that the states are scattering states. Now, for whatever choice of our bounded set, the integral, i.e. the probability of finding the particle inside our region $A$ should vanish as $t \rightarrow +\infty$. If we assume our Hamiltonian to be time-independent, we can replace $t$ by $|t|$.
The definitions can be suitably extended to the multiparticle case.
References:

*

*Blanchard, Philippe; Brüning, Edward (2015). "Some Applications of the Spectral Representation". Mathematical Methods in Physics: Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics (2nd ed.). Switzerland: Springer International Publishing. p. 431. ISBN 978-3-319-14044-5.

*Gustafson, Stephen J.; Sigal, Israel Michael (2011). "Spectrum and Dynamics". Mathematical Concepts of Quantum Mechanics (2nd ed.). Berlin, Heidelberg: Springer-Verlag. p. 50. ISBN 978-3-642-21865-1.

*Ruelle, David (9 January 2016). "A Remark on Bound States in Potential-Scattering Theory". Nuovo Cimento A (1965-1970). 61 (June 1969): 655–662. doi: 10.1007/BF02819607. pdf.

