It is then possible to show that formula_88 is not a symmetric operator, which certainly implies that formula_3 is not essentially self adjoint. Indeed, formula_88 has eigenvectors with pure imaginary eigenvalues, which is impossible for a symmetric operator. This strange occurrence is possible because of a cancellation between the two terms in formula_88: There are functions formula_68 in the domain of formula_88 for which neither formula_94 nor formula_95 is separately in formula_96, but the combination of them occurring in formula_88 is in formula_96. This allows for formula_88 to be nonsymmetric, even though both formula_100 and formula_101 are symmetric operators. This sort of cancellation does not occur if we replace the repelling potential formula_84 with the confining potential formula_103.