We introduce \emph{contact (zero range) interactions } , a special class of self-adjoint extensions of the N-body Schrödinger free hamiltonian $ H_0$ restricted to functions with support away from the \emph{contact manifold} $ Γ\equiv \cup Γ_{i,j} \;\;\; Γ_{i,j}\equiv \{x_i = x_j \; i \not= j \} \;,\; x_i \in R^3 $. These extensions are defined by boundary conditions at $ Γ$. We discuss the spectral properties as function of the masses and the statistics The (Efimov) spectrum is entirely due "conspiracy" of the contact interactions of two pairs. These states are called in Theoretical Physics \emph{trimers} if the two pairs have an element in common, \emph{quadrimers} otherwise. The analysis can be extended to the case in which there is a regular two-body potential, but then the spectral properties cannot be given explicitly. We prove that these interactions are limit (in strong resolvent sense) when $ ε\to 0$ of N-body hamiltonians $ H_ε= H_0+ \sum_{i,j} \frac {1}{ε^3}V_{i,j}( \frac {|x_i-x_j|}ε) ,\; V_{i,j} \in L^1 $. The result is \emph{ independent of the shape of the potential} Strong resolvent convergence implies convergence of spectra.This makes contact interaction a valuable tool in the study of the spectrum of a system of
N particles interacting through potentials of very short range.
Abstract We discuss what kind of quantum dynamics can yield thermalization. We show that any temperature measurement protocol must be sensitive to the system’s local energy scale, not just its state. Specifically, we illustrate this by showing that any temperature measurement protocol that does not resolve the system’s local Hamiltonian (such as, e.g. full state tomography) is susceptible to being fooled into measuring any value for the temperature. Since placing a system in thermal contact with a reference system at a known temperature is a way of measuring temperature, any robust notion of thermal contact must be resistant to this ‘attack’. This provides us with a necessary conditions for thermal contact between quantum systems. We then apply this condition to a scenario where one expects to find thermalization: a system repeatedly interacting with the microconstitutents of a thermal reservoir. We show such models of thermalization cannot be correct without fine-tuning.
Abstract In Dell’Antonio (Eur Phys J Plus 13:1–20, 2021), we explored the possibility to analyse contact interaction in Quantum Mechanics using a variational tool, Gamma Convergence. Here, we extend the analysis in Dell’Antonio (Eur Phys J Plus 13:1–20, 2021) of joint weak contact of three particles to the non-relativistic case in which the free one particle Hamiltonian is
H0=−2MΔ H 0 = - Δ 2 M . We derive the Gross–Pitaevskii equation for a system of three particles in joint weak contact. We then define and study strong contact and show that the Gross–Pitaevskii equation is also the variational equation for the energy of the Bose–Einstein condensate (strong contact in a four-particle system). We add some comments on Bogoliubov’s theory. In the second part, we use the non-relativistic Pauli equation and weak contact to derive the spectrum of the conduction electrons in an infinite crystal. We prove that the spectrum is pure point with multiplicity two and eigenvalues that scale as
logn1 1 logn .