To keep things simple, let us consider the following: $L$ is a positive, unbounded S.A. operator on $L\_2(\mathbb{R},f(x))$, where $f(x)$ is a Gaussian. Assume that we know the spectrum and eigenfunctions of $L$ well.


Consider $L\_{per}=L+ikx$, where $k$ is an integer. What can we say about its spectrum ? Is there a systematic way to study this problem ?
If we cannot say much in general, is there any specific system of the above type that has been studied in the literature ?



Sum of positive self-adjoint operator and an imaginary "potential": literature request
Let me consider the case of the Schrödinger equation, $L=-\nabla^2+V(x)$. Then the operator $L+ikx$ has [PT-symmetry,](https://en.wikipedia.org/wiki/Non-Hermitian_quantum_mechanics) meaning that it is invariant under the combined action of inversion $x\mapsto-x$ (parity P) and complex conjugation (time-reversal T). Such operators may have a fully real spectrum. The particular case that $V(x)=\lambda|x|$ has been studied in [The linear PT-symmetric complex potential](http://www.nipne.ro/rjp/2017_62_5-6/RomJPhys.62.112.pdf) (2017).


There is an extensive literature on PT-symmetric operators in the quantum physics context, for a review, see [PT-symmetric quantum theory](https://iopscience.iop.org/article/10.1088/1742-6596/631/1/012002/pdf).


