When does Separation of Variables yield basis of solution set?

Solution 1:

The basic equations for which separation of variables works are classical types of linear equations.

• Laplace equation $$ \nabla^{2}u = 0. $$
• Wave equation $$ \frac{\partial^{2}u}{\partial t^{2}} = c^{2}\nabla^{2}u + Vu $$
• Heat equation $$ \frac{\partial u}{\partial t} = k\nabla^{2}u + Vu $$
• Schrodinger Equation $$ i\hbar \frac{\partial u}{\partial t} = -\frac{\hbar^{2}}{2\mu}\nabla^{2}u + Vu $$

You have to be able to separate variables, which puts restrictions on $V$. You can separate variables in different coordinate systems, but normally only using orthogonal coordinate systems, of which there are a couple of dozen reasonable ones where the Laplacian separates. The domains can be infinite or semi-infinite in some cases, or bounded with faces that are part of a constant coordinate face. Boundary conditions must be consistent with separation of variables.

When all of these conditions are in place so that separation of variables can succeed, then basically you get what you want. The decompositions are complete in $L^{2}$ spaces. General solutions will either be built from discrete sums or integral sums of separated solutions with respect to separation parameters. (It's theoretically possible to end up with really pathological decompositions requiring general measures to build up full solutions, but not for Physically realistic functions $V$.) However, you may need both Fourier sums and integrals in the general case. $L^{2}$ theory comes from general Spectral Theory for unbounded selfadjoint linear operators on a Hilbert space, even though the heat equation is better suited for $L^{1}$. You'd be surprised how much it takes to justify the expansions for the Sturm-Liouville ODEs arising out of separation of variables problems.

Keep in mind that all bets are off if you don't choose the conditions correctly to end up with selfadjoint problems for the ODEs, but well-posed Physical problems can be expected to lead to well-posed Mathematical problems. Basically, if your conditions are correct, and the coordinate systems are separable, then everything works out, at least so far as $L^{2}$ theory is concerned. A thorough examination of pointwise results is probably not realistic, however. Pointwise convergence results for the ordinary trigonometric Fourier series would fill a small library.

Solution 2:

Here is a large class of counterexamples. Let $F(x,t)$ be a smooth nonvanishing function $\mathbb{R}^2 \to \mathbb{R}$ which $F$ does not factor as $g(x) h(t)$. As an example, I'll take $\exp(-x^2-xt-t^2)$. Consider the differential equation $$\frac{\partial u}{\partial t} = \frac{\partial F/\partial t}{F} u.$$ So, in my example, $\tfrac{\partial u}{\partial t} = -(x+2t) u$.

The solutions to this differential equation are of the form $F(x,t) v(x)$. So, if $F(x,t)$ does not factor, then no solution of this differential equation factors.

The second part of this answer is going to be very imprecise because I am terrible at functional analysis, but I am hoping someone else will write up an answer that fills in the gaps. Separation of variables should work when the system has time symmetry. This includes every example that DisintegratingByParts gives.

Let $X$ be a manifold and let's say we have a differential operator $D$ on $C^{\infty}(X \times \mathbb{R})$ which commutes with $\partial/(\partial t)$.

Then the kernel of $D$ (space of solutions to the equation) is closed under $\partial/\partial t$. Therefore, this follows from proving some statement along the lines of "a nice subspace $V$ of $C^{\infty}(X \times \mathbb{R})$ which is closed under $\partial/\partial t$ obeys $V=\bigoplus_{\omega \in \mathbb{R}} V \cap C^{\infty}(X) e^{i \omega t} $." This direct sum should presumably be in the sense of integrating against some sort of measure on $\mathbb{R}$.

The finite dimensional analogue (which is true) is that, if $H$ is a finite dimensional vector space, $A: H \to H$ is a diagonalizable operator such that $H = \bigoplus H_{\lambda}$ with $A$ acting by $\lambda$ on $H_{\lambda}$ and $V \subset H$ is an $A$-invariant subspace, then $V = \bigoplus_{\lambda} (V \cap H_{\lambda})$. The relevance is that the $i \omega$ eigenspace of $\partial/\partial t$ on $C^{\infty}(X \times \mathbb{R})$ is $C^{\infty}(X) e^{i \omega t}$.

As I said, I am terrible at functional analysis, so I don't know what the careful formulation of this statement would be. But I hope someone will come along to record it and give a reference for it.