Self-adjointness and spectrum of Stark operators on finite intervals

In this paper, we study self-adjointness and spectrum of operators of the form $$H=\displaystyle -\frac{d^2}{dx^2}+Fx, F>0 \quad\text{on} \quad \mathcal{H}=L^{2}(-L,L).$$ $H$ is called Stark operator and describes a quantum particle in a quantum asymmetric well. Most of known results on mathematical physics does not take in consideration the self-adjointness and the operating domains of such operators. We focus on this point and give the parametrization of all self-adjoint extensions. This relates on self-adjoint domains of singular symmetric differential operators. For some of these extensions, we numerically, give the spectral properties of $H$. One of these examples performs the interesting phenomenon of splitting of degenerate eigenvalues. This is done using the a combination of the Bisection and Newton methods with a numerical accuracy less than $10^{-8}$.