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

With conditions




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 $1<x<e^{\pi}$. At last, I have to find expansion function $\delta(x-e)(1<x<e^{\pi})$ 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 $c$.

  • $\begingroup$ Tomorrow I'll try these. Thanks a lot! $\endgroup$ May 23 at 0:15

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