Chapter 9: Separation of Variables

### Sturm-Liouville Theory (optional)

#### The Main Idea

When you use the separation of variables procedure on a PDE, you end up with one or more ODEs that are eigenvalue problems, i.e. they contain an unknown constant that comes from the separation constants. These ODEs are called Sturm-Liouville equations. By solving the ODEs for particular boundary conditions, we find particular allowed values for the eigenvalues. Furthermore, the solutions of the ODEs for these special boundary conditions and eigenvalues form an orthogonal set. These solutions can be used to form a basis of solutions of the original PDE. The rest of this section states the technical mathematics theorems for the most important cases where this process works. If you are a math geek, please read on and enjoy. If not, then rest assured that the mathematics guarantees that the separation of variables process will work whenever you are asked to use it to in undergraduate physics.

#### Sturm-Liouville Theory for Boundary-Value Problems

A second order Sturm–Liouville problem is a homogeneous boundary value problem of the form \begin{eqnarray*} [ P(x)\, y'] '+Q(x)\, y+\lambda\, w(x)\,y&=&0\cr \alpha_1\, y(a) +\beta_1\, y'(a)&=&0\cr \alpha_2\, y(b) +\beta_2\, y'(b)&=&0\cr \end{eqnarray*} where $P, P', Q, w$ are continuous and real on $[a,b]$, and $P$ and $w$ are positive.

Theorem: For $y_1$ and $y_2$ two linearly independent solutions of the homogeneous differential equation, nontrivial solutions of the homogeneous boundary value problem exist iff \begin{vmatrix} \alpha_1\, y_1(a) +\beta_1\, y'_1(a)&\alpha_1\, y_2(a)+\beta_1\, y'_2(a)\\ \noalign{\smallskip} \alpha_2\, y_1(b) +\beta_2\, y'_1(b)&\alpha_2\, y_2(b) +\beta_2\, y'_2(b) \end{vmatrix}=0

Definition: Values of $\lambda$ for which nontrivial solutions exist are called eigenvalues. The corresponding solutions are called eigenfunctions.

Theorem: The eigenvalues of a homogeneous Sturm-Liouville problem are real and non-negative and can be arranged in a strictly increasing infinite sequence $$0\le \lambda_1<\lambda_2<\lambda_3<\dots$$ and $\lambda_n\rightarrow\infty$ as $n\rightarrow\infty$.

Theorem: For each eigenvalue, there exists exactly one linearly independent eigenfunction, $y_n$. These eigenfunctions for differing eigenvalues are orthogonal with respect to the inner product: $$(y_n,y_m)_w=\int_a^b y_n(x)\, y_m(x)\, w(x)\, dx =N_n\delta_{n,m}$$

Theorem: The eigenfunctions $y_n$ span the vector space of piecewise smooth functions satisfying the boundary conditions of the Sturm-Liouville problem. (Convergence in the mean, not pointwise.) $$f(x)=\sum_{n=1}^\infty c_n\, y_n(x)$$ where the $c_n$'s are given by: $$c_n={1\over N_n} (y_n,f)_w={1\over N_n}\int_a^b y_n^*(x)\, f(x)\, w(x)\, dx$$

#### Sturm-Liouville Theory for Periodic Systems

A second order periodic Sturm–Liouville problem is a homogeneous problem of the form $$[ P(x)\, y'] '+Q(x)\, y+\lambda\, w(x)\,y=0$$ where $P, P', Q, w$ are continuous and real on $[a,b]$, and $P$ and $w$ are positive, and $$\left.\left[ P(x)\left( f^*(x)\, g'(x)-{f^*}'(x)\, g(x)\right)\right] \right|_a^b=0$$ for $f(x)$ and $g(x)$ and two vectors in the vector space.

Definition: Values of $\lambda$ for which nontrivial solutions of the periodic Sturm-Liouville problem exist are called eigenvalues. The corresponding solutions are called eigenfunctions.

Theorem: The eigenvalues of a periodic Sturm-Liouville problem are real.

Theorem: For each eigenvalue, there exist linearly independent eigenfunctions, $y_n$. These eigenfunctions for differing eigenvalues are orthogonal with respect to the inner product: $$(y_n,y_m)_w=\int_a^b y_n(x)\, y_m(x)\, w(x)\, dx =N_n\delta_{n,m}$$ Eigenfunctions with the same eigenvalue can be orthogonalized using Gram-Schmidt orthogonalization.

Theorem: The eigenfunctions $y_n$ span the vector space of piecewise smooth functions satisfying the boundary conditions of the Sturm-Liouville problem. (Convergence in the mean, not pointwise.) $$f(x)=\sum_{n=1}^\infty c_n\, y_n(x)$$ where the $c_n$'s are given by: $$c_n={1\over N_n} (y_n,f)_w={1\over N_n}\int_a^b y_n^*(x)\, f(x)\, w(x)\, dx$$