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There are a couple of questions that I have not find in the book. if I have a linear operator acting $L^2$ to $L^2$ as identity minus Laplacian with a variable coefficient: $L:= I-a(x)Dxx$, assume a is a smooth function, what would be now the eigenvalue problem for $?$

I first try to make a guess regarding eigenfunctions: $e^{ikx}$ and get the eigenvalue. If $a$ is constant so is an eigenvalue, but if $a(x)$ is a function I get that the eigenvalue is a function of $x$. Don't eigenvalues have to be only constants? Or we consider them as such for every $x$?

And as for Laplacian itself, $-a(x)Dxx$, if for constant case we know it is unbounded, would be nice to know if there are some $a(x)$ can make it bounded! It is not self-adjoint for variable $a(x)$, therefore the norm of $A:=-a(x)Dxx$ would be the maximum eigenvalue of $A^*A$, but how to identify what $A^*$ in this case? Or there is another way to approach to bound it? thanks for any hint!

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An eigenvalue must be a constant. To find eigenvalues and eigenfunctions, you would want to solve the differential equation $a(x) D_{xx} y = (1+\lambda) y$, and then see for what $\lambda$ the solution is in $L^2$. Depending on $a(x)$, this may be easy or difficult.

$-a(x) D_{xx}$ is never bounded except in the trivial case $a(x)=0$. The adjoint (formally: you really have to worry about domains and boundary conditions) is $y \mapsto D_{xx}(-a(x) y)$.

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