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My self-adjount differential operator $L$ is defined by $$L f(x) \equiv u(x) \frac{\partial^2}{\partial x^2} \left( u(x) f(x) \right)$$ where $u(x)$ is a known but arbitrary smooth function that satisfies $u(x) > 0$ $\forall x$.

I seek a basis of eigenfunctions of $L$, and their eigenvalues.

Are there any results known that help with this problem?

Are there any specific forms for $u(x)$ (other than $u\equiv 1$) that make the problem easier?

Edit: I am looking for eigenfunctions that do not explode, and form a useful basis in which I could in principle expand other "well behaved" functions. For example, if $u(x)\equiv 1$ I would select $e^{i\lambda x}$ as eigenfunctions, with eigenvalues $-\lambda^2$.

Edit 2: In practice, $u(x)$ is a function that is close to $1$. Its shape might be roughly $1$ at $-\infty$ smoothly reducing to $0.5$ at $0$ and back to $1$ at $\infty$.

Edit 3: As I said it is self-adjoint, I have a specific inner product in mind, namely $<f,g> = \int dx f(x) g(x)$. In that case, I think we can write $L = - D^T D$. Does this help?

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    $\begingroup$ What space of functions...? Any boundary conditions...? Even things that "look" self-adjoint (some people call them "formally self-adjoint") can fail to have any self-adjoint extensions at all, or have infinitely-many, ... Can you clarify? $\endgroup$
    – paul garrett
    Commented Sep 21 at 21:28
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    $\begingroup$ Just for clarity, are there parentheses missing after the derivative? $\endgroup$
    – Bertoldo Baccalà
    Commented Sep 22 at 8:12
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    $\begingroup$ if $u(x)=x$, eigenfunctions are $f(x)=x^q$ with eigenvalue $q(q+1)$ $\endgroup$
    – Carlo Beenakker
    Commented Sep 22 at 20:05
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    $\begingroup$ I assume the domain is $x\in \mathbb{R}$? Is the function $u$ bounded? Is it bounded away from zero uniformly? If it is allow to approach zero, what is the asymptotic rate? $\endgroup$
    – Willie Wong
    Commented Sep 22 at 21:51
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    $\begingroup$ With the inner product you mentioned, you are formally looking for critical points of the Rayleigh quotient $ \frac{\int [\partial_x (uf)]^2 \,dx}{\int f^2\,dx}$, but be cautious with the integrability and BCs. This gives some intuition on the obvious fact that $1/u$ is an eigenfunction to $\lambda=0$ if this is an admissible function in the game. $\endgroup$
    – Bertoldo Baccalà
    Commented Sep 23 at 6:25

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