Sturm-Liouville theory

For $\mathcal{L}y=-y^{\prime \prime }-2a{y}^{\prime }$, setting $y=e^{-ax}v$ gives $\mathcal{L}y=e^{-ax}(-v^{\prime \prime }+a^{2}v)$. Dirichlet conditions are unchanged. The weighted self-adjoint form is $-e^{-2ax}(e^{2ax}{y}^{\prime }{)}^{\prime }$, with weight $e^{2ax}$. On an interval of length $L$, eigenvalues are $(n\pi /L{)}^2+a^2$ and eigenfunctions are $e^{-ax}\sin (n\pi x/L)$.

For a regular self-adjoint Sturm-Liouville problem with positive weight $w$ and self-adjoint boundary conditions, eigenfunctions with distinct eigenvalues are orthogonal in the weighted inner product $\int w\overline{u}v$. Integration by parts proves this by equating $\int \overline{u}Lv$ and $\int \overline{Lu}v$.

For $\alpha >0$, the Gegenbauer polynomial $C_n^{\alpha }$ is the degree-$n$ polynomial solution ofThese polynomials are orthogonal on $(-1,1)$ with weight $(1-x^2{)}^{\alpha -1/2}$.

The Gegenbauer differential equation is the displayed Sturm–Liouville eigenvalue equation defining the Gegenbauer polynomial. Multiplication by $(1-x^2{)}^{\alpha -1/2}$ puts it in self-adjoint form.

For $\alpha >0$, all $n$ roots of $C_n^{\alpha }$ are simple and lie in $(-1,1)$. If the product $q$ of its interior linear factors had degree below $n$, orthogonality would make $\int C_n^{\alpha }qw=0$, while $C_n^{\alpha }q=q^2(C_n^{\alpha }/q)$ has a constant strict sign away from finitely many points.

A scalar second-order differential operator is in Sturm–Liouville form when it is written aspossibly together with a positive weight in an eigenvalue problem.

The weighted operator $L=W^{-1}[-(P\xi \prime )\prime +Q\xi ]$, with real coefficients and $P,W>0$ in the interior, defines a Sturm-Liouville problem. Its realization as a self-adjoint differential operator requires a domain whose endpoint conditions annihilate the boundary form $[P(\eta {\prime }^{\ast }\xi -{\eta }^{\ast }\xi \prime )]$. Its Rayleigh quotient uses the weight $W$ in the denominator.

A regular Sturm-Liouville eigenvalue problem has the formon a bounded interval with self-adjoint boundary conditions and positive weight $w$.

A Dirichlet eigenvalue is an eigenvalue for an eigenfunction constrained to vanish at the endpoints of its interval or on the boundary of its domain.

For a bounded domain $\Omega$, the first Dirichlet eigenvalue of $-\Delta$ has the variational characterization

For a regular self-adjoint Sturm-Liouville problem, eigenfunctions are orthogonal in the weighted inner productand the coefficient of $y_n$ is $⟨f,y_n{⟩}_w/⟨y_n,y_n{⟩}_w$.

A differential operator is self-adjoint when integration by parts makes its inner products symmetric on its domain.

For real coefficient functions, the fourth-order scalar operatoris formally self-adjoint exactly when $q=p^{\prime }$. Indeed, $(p{D}^2{)}^{\ast }=p{D}^2+2p^{\prime }D+p^{\prime \prime }$ and $(qD{)}^{\ast }=-qD-q^{\prime }$. Under clamped boundary conditions, its quadratic form is

A second-order scalar differential equation is in self-adjoint form when it can be writtenMultiplication by $y$ and integration by parts then gives an energy identity.

If $L(y_0;{\lambda }_0)=0$ and $y_0,y$ satisfy homogeneous separated boundary conditions, the Lagrange identity givesThus $L(y;{\lambda }_0)=f$ can be solved only if $f$ is orthogonal to the null mode $y_0$.

At a simple eigenvalue of a self-adjoint linear operator, an inhomogeneous equation is solvable only when its forcing is orthogonal to the null eigenfunction. In weakly nonlinear expansions, this condition determines the amplitude equation.

For $L(y;\lambda )=y^{m+1}$ near a normalized simple null mode $y_0$, put $y=\epsilon y_0+O({\epsilon }^2)$ and $\lambda -{\lambda }_0={\epsilon }^m\mu$. Orthogonality to $y_0$ yields