Give me an example of a differential operator that is normal, not self-adjoint and has application in Physics

Question

I am looking for an example of a differential operator whose domain is the Hilbert space of square integrable functions with real $N$-dimensional domain $L^2(\mathbb R^N)$ having the following characteristics:

1. the operator is a linear combination of derivatives up to order 2;
2. the operator is normal, but not self-adjoint;
3. the operator's adjoint is known and is provided in the answer with an adjointness proof (or at least a sketch of the proof).
4. the operator has application in Physics (please, provide a sketch of the application).
5. the operator has non-trivial eigenfunctions.

Thanks in advance.