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The European Physical Journal Plus

Contact interactions and strong resolvent convergence, a partly variational approach

Gianfausto Dell’Antonio

Published May 8, 2023• 0 citations
Publisher

Abstract

Abstract In Dell’Antonio (Eur. Phys. J. Plus 136:392, 2021) we considered several types of contact (zero range) interactions. Their Hamiltonians are limit, in strong resolvent topology, of a sequence of potentials with decreasing support. Here we review and improve these results and provide a new analysis Bose–Einstein condensation and of the Fermi sea.

DOI
10.1140/epjp/s13360-023-03989-0

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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 $$ H_0 = - \frac{\Delta }{2M} $$ 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 $$ \frac{1}{log {n}}$$ 1 logn .

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Zero range (contact) interactions conspire to produce Efimov trimers and quadrimers

Gianfausto Dell’Antonio

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.

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Nonlinear Dynamics of Bose-Einstein Condensates with Long-Range Interactions

G. Wunner, Holger Cartarius, Tomaž Fabčič +3 more

Abstract. The motto of this paper is: Let's face Bose-Einstein condensation through nonUnear dynamics. We do this by choosing variational forms of the condensate wave functions (of given symmetry classes), which convert the Bose-Einstein condensates via the time-dependent Gross-Pitaevskii equation into Hamiltonian systems that can be studied using the methods of nonUnear dynamics. We consider in particular cold quantum gases where long-range interactions between the neutral atoms are present, in addition to the conventional short-range contact interaction, viz. gravity-Uke interactions, and dipole-dipole interactions. The residts obtained serve as a useful giude in the search for nonlinear dynamics effects in numerically exact quantum calcinations for Bose-Einstein condensates. A main residt is the prediction of the existence of stable islands as well as chaotic regions for excited states of dipolar condensates, which coidd be checked experimentally. Keywords: Bose-Einstein condensation, Gross-Pitaevskii equation, nonlinear dynamics effects PACS: 03.75.Kk, 34.20.Cf, 02.30.-f, 47.20.Ky 1.

•Sep 16, 2009•1 citation
Abstract In Dell’Antonio (Eur. Phys. J. Plus 136:392, 2021) we considered several types of contact (zero range) interactions. Their Hamiltonians are limit, in strong resolvent topology, of a sequence of potentials with decreasing support. Here we review and improve these results and provide a new analysis Bose–Einstein condensation and of the Fermi sea.
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=−Δ2M H_0 = - \frac{\Delta }{2M} 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 1logn \frac{1}{log {n}}logn1​ 1 logn .
In Classical Mechanics constraints describe forces restricting the motion of two systems when they are in contact. In Quantum Mechanics in the Schrodinger representation the state of the system is described by a (probability) wave and the Schrodinger equation is dispersive (it does not preserve locality of the wave function): at any time the wave function is spread out over all space. It is difficult to define "contact". To avoid this difficulty it is convenient to use the Heisenberg representation and describe the system by means of self-adjoint operators on some function space. Each self-adjoint operator operator has a domain of definition. We consider first in some detail the dynamics in R^3 and later consider the case of dimension two and dimension one. In three dimension contact interactions gives the Efimov spectrum of trimers and quadrimers in low energy physics and the Bose-Einstein condensate, both in low density and in high density. The case of dimension one is particularly interesting because the system may represent three particles on a Y-shaped graph with contact interaction at the vertex. We consider both the case of Bose particles which satisfy the Schrodinger equation and the case of spin 1\2 fermions which satisfy the Pauli equation. They form respectively Bose crystals and Fermi crystals (not necessary periodic). In both cases there is only one bound state at each vertex. In an extended crystal, due to Fermi-Dirac statistics, all bound states may be occupied to form the Fermi sea. The particles on the surface have a Dirac spectrum. We prove that in the semiclassical limit the dynamics of particles on the surface of the Fermi sea is the (classical) motion studied in detail by Novikov and Maltsev. In this semiclassical description the Fermi surface can be deformed by a magnetic field and one may have topological resonances.
We consider a Hamiltonian describing three quantum particles in dimension one interacting through two-body short-range potentials. We prove that, as a suitable scale parameter in the potential terms goes to zero, such a Hamiltonian converges to one with zero-range (also called delta or point) interactions. The convergence is understood in the norm resolvent sense. The two-body rescaled potentials are of the form vσε(xσ)=ε−1vσ(ε−1xσ), where σ = 23, 12, 31 is an index that runs over all the possible pairings of the three particles, xσ is the relative coordinate between two particles, and ε is the scale parameter. The limiting Hamiltonian is the one formally obtained by replacing the potentials vσ with ασδσ, where δσ is the Dirac delta-distribution centered on the coincidence hyperplane xσ = 0 and ασ=∫Rvσdxσ. To prove the convergence of the resolvents, we make use of Faddeev’s equations.
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 NNN particles interacting through potentials of very short range.
Abstract. The motto of this paper is: Let's face Bose-Einstein condensation through nonUnear dynamics. We do this by choosing variational forms of the condensate wave functions (of given symmetry classes), which convert the Bose-Einstein condensates via the time-dependent Gross-Pitaevskii equation into Hamiltonian systems that can be studied using the methods of nonUnear dynamics. We consider in particular cold quantum gases where long-range interactions between the neutral atoms are present, in addition to the conventional short-range contact interaction, viz. gravity-Uke interactions, and dipole-dipole interactions. The residts obtained serve as a useful giude in the search for nonlinear dynamics effects in numerically exact quantum calcinations for Bose-Einstein condensates. A main residt is the prediction of the existence of stable islands as well as chaotic regions for excited states of dipolar condensates, which coidd be checked experimentally. Keywords: Bose-Einstein condensation, Gross-Pitaevskii equation, nonlinear dynamics effects PACS: 03.75.Kk, 34.20.Cf, 02.30.-f, 47.20.Ky 1.