Why separation of variables works in PDEs?

The method of separation of variables is used in many occasions in the upper level physics courses such as QM and EM. But when it is used there is no clear reason why using it is permitted it except that it works, or that let us try and see.

Is there a theorem which provides the condition(s) that if satisfied then the method of separation of variables will be guaranteed to work in PDEs?

Solution 1:

To the best of my knowledge:

1. No, there is no general theorem that tells you how to start from an arbitrary partial differential equation and conclude whether that partial differential equation can be solved by separation of variables. (I should note here that one of the problems is to start from an arbitrary partial differential equation, and deduce a change of variables relative to which one can perform the separation of variables. The conditions given by naryb below is certainly sufficient [use the trivial change of variables], but definitely not necessary.)

2. On the other hand, if you looked through the literature, there are a lot of criteria given for individual partial differential equations of specific forms. A particularly well-known example is that of Eisenhart's classification of potential functions for which the associated Schrodinger operator is separable. See this link.

3. It is undeniable that separation of variables have something to do with symmetries of the (system of) partial differential equations you are looking at; but the connection is still poorly understood. For example, it was observed long ago that the linear wave equation on certain curved space-times (more precisely the Kerr black hole backgrounds) are separable, but the space-time does not appear to have enough infinitesimal symmetries for this to have been possible. (By an infinitesimal symmetry I mean that which is associated to a Lie group action on the PDE.) It turns out that this separability has to do with the so-called Carter constant, which is defined by some notion of a higher order symmetry. As far as I know, the connection between these higher order symmetries and separability of PDEs is still a field of active investigation.

4. I don't know if there is even a universally accepted definition of what "separation of variables" mean!

Some suggested further readings:

• Miller's 1977 treatise studying the relation between symmetries and separation of variables for second order partial differential equations.

• Koornwinder wrote a very detailed book review about the previous treatise, which has many historical points and shows, at least 30 years ago, how muddled the notions are.

• If you have access to it, a brief survey of Miller's book is available as a short paper.

• And here is the article alluded to in Koornwinder's review, which addresses the issue of defining the notion of separation of variables.

• For an explanation of the notions of symmetry and generalised symmetry for differential equations, Olver's book is by now a somewhat standard resource.

Solution 2:

Note that the OP posted this to physics.se (now closed) as well, and I'm giving my rough-n-ready reasoning for using the method.

In many of the systems that physicists deal with there is a uniqueness theorem for solutions to the partial differential equation that satisfy certain classes of boundary conditions.

This allows you to know that having found a solution to the physical problem (which includes boundary conditions) you have found all the solutions.

Once you have that the pure ease of the method makes it worth trying if you have any symmetry that suggests how the solution might factor.

Guess-n-check is a legitimate problem solving technique.

Solution 3:

To extend on the answer posted by dmckee, there are uniqueness theorems for Laplace's equation, Poisson's equation, the wave equation, and Schroedinger's equation (for the Hamiltonians of most simple physical systems), and of course more, though these are probably the ones you are referring to.

Another thing to keep it mind is that this does not imply solutions are necessarily not separable, as linear differential equations allow us to take sums of solutions and remain in the solution space. Thus functions like $x+y+z$ might still be solutions even though they are not separable, so long as they can be built out of separable solutions. I would venture to guess that the separable solutions constitute a basis for the solution space in just about all common cases.

Also have a look at this and this too.