Hi everone,

Ok so to begin with I know that for Dirichlet, Neuman and Robyn boundary conditions you can use the method of fourier series to solve the heat and wave equations given Cauchy data on $[0,L]$. This essentially uses the fact that the bounary conditions are *symmetric* meaning that if I'm considering the operator $-d^2/dx^2$ with boundary conditions on $0$ and $L$ then I need $XX^{\prime}|_0^L = 0$ for the operator to be self adjoint so that I can use spectral theory on the solution operator (which is compact).

So when I'm given *non-homogeneous* boundary conditions like $u(0,t)=h(t)$ and $u(L,t)=g(t)$ then one method is to subtract off these boundary values by some function so that I can reduce it to solving $u_{tt} - u_{xx} = f$ with $u(0,t)=0=u(L,t)$. The same holds true for Neuman and Robyn boundary conditions. So my question becomes:

**Question:** Exactly what type of boundary conditions can I pose on the wave/heat equation on $[0,L]$ so that the method of Fourier series works to generate a solution? Is a necessary and sufficient condition that the boundary conditions be *symmetric*?

There are obvious compatibility conditions with the Cauchy data/boundary data but this is not my concern here. I'm simply aiming to understand when we can write out the solution was $u(t,x) = \sum_{k} c_k(t)e^{ikx}$. My feeling is that you need the boundary conditions to be symmetric *for all times* so that you may indeed write $c_k(t)$ in terms of the eigenvalues of $-d^2/dx^2$ for those given boundary conditions.

I hope this is clear. Best, Dorian