Hamiltonians for Quantum Systems with Contact Interactions

Alessandro Teta () and Daniele Ferretti ()
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Alessandro Teta: Sapienza Università di Roma
Daniele Ferretti: Gran Sasso Science Institute

Chapter 97 in Operator Theory, 2026, pp 2965-2987 from Springer

Abstract: Abstract We discuss the problem of constructing self-adjoint and lower bounded Hamiltonians for a system of n > 2 $$n>2$$ nonrelativistic quantum particles in dimension three with contact (known also as zero range or δ $$\delta $$ ) interactions. Such interactions are described by (singular) boundary conditions satisfied at the coincidence hyperplanes, i.e., when the coordinates of two particles coincide. Following the line of recent works appeared in the literature, we introduce a boundary condition slightly modified with respect to the usual boundary condition one has in the one-body problem. With such new boundary condition we can show that the instability property due to the fall to the center phenomenon described by Minlos and Faddeev in 1962 is avoided and then one obtains a physically reasonable Hamiltonian for the system. We apply the method to the case of a gas of N interacting bosons and to the case of N distinguishable particles of equal mass M interacting with a different particle. In the latter case we also discuss the limit of the model for M → + ∞ $$M \longrightarrow {\scriptstyle +} \infty $$ . We show that in the limit one obtains the one-body Hamiltonian for the light particle subject to N (nonlocal) point interactions placed at fixed positions. We will verify that such nonlocal point interactions do not exhibit the ultraviolet pathologies that are present in the case of standard local point interactions.

Keywords: Zero-range interactions; Many-body Hamiltonians; Schrödinger operators (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16356-1_92

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