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Given some linear differential operator ##L##, I'm trying to solve the eigenvalue problem ##L(u) = \lambda u##. Given basis functions, call them ##\phi_i##, I use a variational procedure and the Ritz method to approximate ##\lambda## via the associated weak formulation

$$\langle L(\phi_i),\phi_j\rangle = \lambda \langle \phi_i,\phi_j\rangle.$$

As you can see, this expression is now a matrix equation, solutions to which are straightforward. For my particular problem, the basis functions are $$\phi_j = \cos\left( \frac{\pi j}{2}(x+1) \right) \cosh\left( \frac{\pi j}{2}(y+h) \right).$$

However, this solution, when inputted into the weak formulation equation, does not output correct eigenvalues. However, ##\phi_j## can be split into even and odd components:

$$ \phi_j^o = \sin \left( \pi(j-1/2)x \right)\cosh\left( \pi(p-1/2)(y+h) \right)\\

\phi_j^e = \cos \left( \pi j x \right)\cosh\left( \pi j(y+h) \right)

$$

Now to obtain eigenvalues I solve two separate equations, one for even eigenvalues and one for odd:

$$\langle L(\phi_i^e),\phi_j^e\rangle = \lambda \langle \phi_i^e,\phi_j^e\rangle\\

\langle L(\phi_i^o),\phi_j^o\rangle = \lambda \langle \phi_i^o,\phi_j^o\rangle.$$

This latter approach gives correct solutions: why? Any insight or direction is greatly appreciated.