Perturbation of Schrödinger Hamiltonians by measures—Self‐adjointness and lower semiboundedness

We study the Hamiltonians for nonrelativistic quantum mechanics in one dimension, in terms of energy forms ∫‖d f/d x‖2 d x+∫‖ f ‖2 d( μ −ν), where μ and ν are positive, not necessarily finite measures on the real line. We cover, besides regular potentials, cases of very singular interactions (e.g., a particle interacting with an infinite number of fixed particles by ‘‘delta function potentials’’ of arbitrary strengths). We give conditions for lower semiboundedness and closability of the above energy forms, which are sufficient and, for certain classes of potentials (e.g., μ−ν a signed measure), also necessary. In contrast to the results in other approaches, no regularity conditions and no restrictions on the growth of the measures μ and ν at infinity are needed.