The discrete spectrum of the spinless one-dimensional Salpeter Hamiltonian perturbed by δ-interactions

We rigorously define the self-adjoint one-dimensional Salpeter Hamiltonian perturbed by an attractive delta-interaction, of strength beta, centred at the origin, by explicitly providing its resolvent. Our approach is based on a 'coupling constant renormalization', a technique used first heuristically in quantum field theory and implemented in the rigorous mathematical construction of the self-adjoint operator representing the negative Laplacian perturbed by the delta-interaction in two and three dimensions. We show that the spectrum of the self-adjoint operator consists of the absolutely continuous spectrum of the free Salpeter Hamiltonian and an eigenvalue given by a smooth function of the parameter pi/beta. The method is extended to the model with two twin attractive deltas symmetrically situated with respect to the origin in order to show that the discrete spectrum of the related self-adjoint Hamiltonian consists of two eigenvalues, namely the ground state energy and that of the excited anti-symmetric state. We investigate in detail the dependence of these two eigenvalues on the two parameters of the model, that is to say both the aforementioned strength beta and the separation distance. With regard to the latter, a remarkable phenomenon is observed: differently from the well-behaved Schrodinger case, the 1D-Salpeter Hamiltonian with two identical Dirac distributions symmetrically situated with respect to the origin does not converge, as the separation distance shrinks to zero, to the one with a single delta-interaction centred at the origin having twice the strength. However, the expected behaviour in the limit (in the norm resolvent sense) can be achieved by making the coupling of the twin deltas suitably dependent on the separation distance itself.