On solving the Schrödinger equation for time varying potential barriers

Question

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?

Comments

Enough symmetries to solve a PDE do not always exist. Those for which they exist are called integrable, and form a very small subclass of all! There is also no systematic way that i know of of checking whether a given SE is integrable; so one needs to check if it is in a list of known cases (see, e.g., https://en.wikipedia.org/wiki/Integrable_system#Exactly_solvable_models ) of find (somehow; there are no general rules) a symmetry oneself.

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The symmetries organize the quantities $I(\psi(x,t),t)$  invariant under a time shift, which are needed to find a closed-form solution. Without symmetries, it is almost impossible to guess the closed form solution, except in essentially trivial, directly separable cases. The detailed story of integrable systems is quite complicated - some people spend their whole life working on these! Thus if you don't find your equation in the list, prepare for a long study!

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One looks at what others did and finds it there or tries similar things. As I said, there are no fixed rules, and it takes some time to know enough of the literature to get a feeling whether a particular problem falls into the integrable class. And most systems don't, but simple systems have a higher change to be integrable than complicated ones. The Wikipedia article contains references. For a warm-up you might look at the PDE book by Olver http://link.springer.com/book/10.1007/978-3-319-02099-0

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Your "Schrödinger equation" does not contain the time partial derivative $\partial/\partial t$, which is strange. If you consider an eigen-state problem, then $t$ is just a formal constant parameter in your eigen-state problem (differential equation).

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there are some technics suitable for simple cases. More details in Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral. the Introduction chapter is public , read the § 1.6. The 2-gap Lamé potential brings new intrinsic difficulties.

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Even if the equation happens to to itegrable, what is the reason for the dependence of these equations on symmetries? Why would it be different equations? Is this to do with the time-dependent evolution of the equation in space?

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Thanks a lot! Just one more question, how are these studies done and are they based on? What methods are in practice for figuring out a symmetry(or symmetries) of a system?

Answer 1

There are no general rules for solving PDEs. Exact solutions depend on symmetries that one must discover.The symmetries organize the quantities I(ψ(x,t),t)  invariant under a time shift, which are needed to find a closed-form solution. For a warm-up you might look at the PDE book by Olver http://link.springer.com/book/10.1007/978-3-319-02099-0.

Enough symmetries to solve a PDE do not always exist. Those for which they exist are called integrable, and form a very small subclass of all! There is no systematic way that I know of of checking whether a given SE is integrable; so one needs to check if it is in a list of known cases (see, e.g., https://en.wikipedia.org/wiki/Integrable_system#Exactly_solvable_models ) of find (somehow; there are no general rules) a symmetry oneself. Thus one looks at what others did and finds it there or tries similar things.

The detailed story of integrable systems is quite complicated - some people spend their whole life working on these! It therefore takes some time to know enough of the literature to get a feeling whether a particular problem falls into the integrable class. Most systems don't, but simple systems have a higher change to be integrable than complicated ones. The Wikipedia article mentioned contains references. 

Without symmetries, it is almost impossible to guess the closed form solution, except in essentially trivial, directly separable cases. When not enough symmetries can be found one must resort to numerical methods or perturbation theory. The principles are explained in many places, but applying it to a particular equation of interest is always a bit of work.

A class of integrable PDEs involving the Weierstrass elliptic functions figuring in your concrete questions are the elliptic Calogero-Moser (or Sutherland) systems; enter the words Calogero-Moser and elliptic into scholar.google.com to get a ton of papers about these.