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?

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.

How would the 3-d graph of the Lamé potential look like?

How do I handle this system, as I want the change of the potential functions(as a function of time)to be continuous, I.e., the elliptic 2-gap lamé potential changing to the hyperbolic potential and further to a parabolic potential with the wavefunction being continuous in every $\epsilon$ part of the barrier for every $\delta$ change in time. How would I solve such a system of time varying potential?