  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 $10^{-8}$.
