# How to show an operator is Hermitian?

I have Sturm-Liouville problem: $$x^2u''+xu'+\lambda u=0$$

With conditions

$$1

$$u'(1)=0$$

$$u'({e^\pi})=0$$

I wrote this SL problem is self adjoint form:

$$\frac{d}{dx}[x \frac{du}{dx}]=-\lambda w(x)u$$, where $$w(x)=\frac{1}{x}$$

Now I have to show on the given interval and subject to given boundary conditions, operator $$L$$ is Hermitian and find eigenvalues and eigenfunction to verify that the eigenfunctions are orthogonal on the interval $$1. At last, I have to find expansion function $$\delta(x-e)(1 in a series of eigenfunctions. I don't know anyting abou expansion function. Also i cannot show orthogonality and operator is Hermitian. How can I do that?

I won't do the entire problem, but I'll provide some help. First, you need to write your equation in the form of an eigenvalue problem, that is $$Lu=\lambda u$$. To do this, divide your equation by $$-w(x)$$, so $$Lu=\frac{-1}{w(x)}\left[\frac{d}{dx}\left(x\frac{du}{dx}\right)\right] =\lambda u.$$ So you now have your $$L$$, which should be viewed as an operator in a Hilbert space with inner product $$(\psi, \phi)= \int_{1}^{e^{\pi}}\psi^{*}(x)\phi(x) \ w(x)dx,$$ where $$\phi(x)$$ and $$\psi(x)$$ are smooth functions satisfying the boundary conditions of your problem. You wanna show $$L$$ is hermitian, meaning $$(L\psi, \phi)=(\psi, L\phi)$$ for arbitrary functions of the aforementioned Hilbert space. Indeed you can use integration by parts twice to show \begin{align}(L\psi, \phi) &= -\int_{1}^{e^\pi}\frac{d}{dx}\left(x\frac{d\psi^*}{dx}\right)\phi dx=\int_{1}^{e^\pi}\left(x\frac{d\psi^*}{dx}\right)\frac{d\phi}{dx} dx \\&=\int_{1}^{e^\pi}\psi^*\left[-\frac{1}{w(x)}\frac{d}{dx}\left(x\frac{d\phi}{dx} \right)\right] w(x)dx =(\psi,L\phi), \end{align} where the boundary terms vanish by virtue of the boundary conditions, and in the last step I multiplied and divided by $$w(x)$$. Thus, $$L$$ is hermitian. To verify the eigenfunctions are orthogonal you are gonna have to solve this differential equation. You should then find a set of permissible $$\lambda$$ (this equation may not be solvable for every possible $$\lambda$$), and a related solution (the eigenfunction) for each $$\lambda$$. When you obtain this set of functions (let's call it $$\{\phi_n \}$$), you must verify that $$(\phi_n,\phi_m)=0$$ for $$n\neq m$$, and that $$(\phi_n,\phi_n)= c^2$$ for some number $$c$$.