Eigenfunctions of the operator $Au:=-u''+ix^2u$ in $L^2(\mathbb{R})$

Question

I need to find the eigenfunctions of the operator $$Au:=-u''+ix^2u$$ if the domain of $A$ is the set of functions in $L^2(\mathbb{R})$ with absolutely continuos derivatives $u,u'$ in $L^2(\mathbb{R})$. Can you give me any hint, please?.

Answer 1

Well we want $Au = \lambda u$.

$-u'' + ix^2 u = \lambda u$ is a second order homogeneous ODE with linear coefficients which we can solve.

http://www.math.psu.edu/tseng/class/Math251/Notes-2nd%20order%20ODE%20pt1.pdf

Some notes on second order linear ODEs