Hi there. It is well known from the theory of differential equation the Sturm-Liouville theory, concerning with the eigenvalue problem:(adsbygoogle = window.adsbygoogle || []).push({});

##\displaystyle \frac{d}{dx} \left ( p(x) \frac{dy_n}{dx} \right ) - q(x) y_n(x)=\lambda_n w(x) y_n(x)##

with prescribed boundary condtitions for ##a\leq x \leq b##, lets say ##y_n(a)## and ##y_n(b)##.

##w(x)## is the weight function. The eigenfunctions ##y_n## obeys the weighted orthonormality relation:

##\int_a^b w(x) y_n(x) y_m(x)dx=\delta_{n,m}##,

Also the solutions of the Sturm-Liouville problem provide a complete set of functions.

What I want to know is if all the theory regarding the Sturm-Liouville problem in one dimension holds in multidimensional problems, id est, for the eigenvalue problem:

##\displaystyle \nabla \cdot \left ( p(\mathbf{r}) \nabla \phi(\mathbf{r}) \right ) - q(\mathbf{r}) \phi(\mathbf{r})=\lambda w(\mathbf{r}) \phi(\mathbf{r})##,

Are the solutions to this problem orthogonal for prescribed boundary conditions? do they form a complete set?

In the affirmative case, I would also like to know if there are any references in the bibliography to the Sturm-Liouville problem in multiple dimensions (specially in 2D and 3D, which I think are the most important).

Thanks in advance.

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# I Sturm-Liouville theory in multiple dimensions

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