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A delta well with a mass jump

In this paper, it is shown that a one-dimensional Hamiltonian with an attractive delta potential at the origin plus a mass jump at the same point cannot have a bound state, as is the case with the ordinary attractive delta potential with constant mass, unless another term is added into the potential in the form of a derivative of the Dirac delta at the origin. The consistency of this singular potential with two terms is guaranteed by choosing suitable matching conditions at the singular point for the wavefunctions. Furthermore, it is proved that the self-adjointness of the Hamiltonian with both singular interactions determines the coefficient of the derivative of the delta in a unique manner. Under these conditions, the bound state and its energy are obtained and it is checked that the correct results in the limit of equal masses are obtained.