How is the Schrödinger equation solved for time varying curved potential barriers? How would the Schrödinger equation be solved for curved barriers which change as a function of time, e.g., a paraboloid potential barrier with maximum height, $V$ changing with time into a Hyperboloid potential barrier (with the same constant height, $V$, at its saddle point), which further changes into an ellipsoidal barrier. What would be the mathematical tools required for analysis? Are such systems practically found?
Mathematical formulation:

Consider a n-dimensional Schrödinger equation of the form:
  $$\left[\sum_{k=1}^{n}\frac{\partial^{n}}{\partial{x_{k}^{2}}}-V(x,t)\right]\psi(x,\alpha)=\lambda(\alpha)\psi(x,\alpha)$$
  where the potential $V(x,t)$ depends on the column vector $x$ belonging to the n-dimensional complex space $C^{n}$
Now let the elliptic potential be: the 2-gap Lamé potential 
  $$V_{e}(x,t)=2\wp(x-x_{1}(t))+2\wp(x-x_{2}(t))+2\wp(x-x_{3}(t))$$
Now this potential varies with time and changes into a hyperbolic potential of the form:
  $$V_{h}(x,t)=aV_{0}coth(\alpha x)+bV_{1}coth^{2}(\alpha x)-cV_{2}cosech(\alpha x)+d-cos(\alpha t)$$ where $a,b,c,d$ and $V_{0},V_{1},V_{2}$ are constants.
  Here is a picture of the graph of the potential of only time independent variables:
  

How would I solve such a system of time varying potential?
 A: You seem to have a very clear idea about the shape of the potential, but on the other hand you ask: Are such systems practically found?
For general time dependent potential you will not solve the Schrödinger equation analytically.
You can probably find approximate solutions if:


*

*the potential changes very slowly (adiabatic approximation) or very quickly


or


*if the typical dimensions of the potential are much smaller or much bigger than the wave length.

A: You may have to simulate this situation and get have to settle for a numerical answer.  This would involve storing the wave function and a function for its first derivative.  
Each time step you calculate a the function and its time derivative.
$\frac{\partial \psi}{\partial t}={\frac{\partial \psi}{\partial t}}_0+\frac{\partial^2 \psi}{\partial t^2} \delta t$
$\psi=\psi_0+\frac{\partial \psi}{\partial t} \delta t$
Where $\delta t$ is your time step.
You then get the next value for $\frac{\partial^2 \psi}{\partial t^2}$ from the Schrödinger Equation.
And then you repeat.
A: The time-dependent Schrödinger equation for a single particle with time-dependent potential $U(r,t)$ is:
$$i\hbar\frac{\partial}{\partial t}\Psi(r,t)=\Big[-\frac{\hbar^2}{2m}\nabla^2+U(r,t)\Big]\Psi(r,t)\tag{1}$$
If we replace $U(r,t)$ with a time-independent potential $U(r)$, we get:
$$i\hbar\frac{\partial}{\partial t}\Psi(r,t)=\Big[-\frac{\hbar^2}{2m}\nabla^2+U(r)\Big]\Psi(r,t)\tag{2}$$
$(2)$ can be separated into two ODEs, if we assume:
$$\Psi(r,t)=\psi(r)T(t)$$
Substituting into $(2)$ we get two ODEs:
$$i\hbar\frac{T'(t)}{T(t)}=E\tag{3}$$
$$-\frac{\hbar^2}{2m}\frac{d^2\psi(r)}{dr^2}+U(r)\psi(r)=E\psi(r)\tag{4}$$
Where $E$ is the particle's total energy.
$(4)$ is of course the time-independent Schrödinger equation and $(3)$ is the time evolution ODE for the wave function.
This method of separating variables only works for a time-independent potential $U(r)$. It does not work for a $U(r,t)$ potential.

How would the Schrödinger equation be solved for curved barriers which change as a function of time

For that reason the usual methods of computing the wave function $\Psi(r,t)$ don't really work with a time-dependent potential and you'll find far fewer examples of such derivations in most textbooks.
