Ground State Energy of the One-Component Charged Bose Gas

The model considered here is the “jellium” model in which there is a uniform, fixed background with charge density −eρ in a large volume V and in which N=ρV particles of electric charge +e and mass m move – the whole system being neutral. In 1961 Foldy used Bogolubov's 1947 method to investigate the ground state energy of this system for bosonic particles in the large ρ limit. He found that the energy per particle is −0.402 in this limit, where . Here we prove that this formula is correct, thereby validating, for the first time, at least one aspect of Bogolubov's pairing theory of the Bose gas. Received: 23 August 2000 / Accepted: 5 October 2000     

Upper Bounds to the Ground State Energies of the One- and Two-Component Charged Bose Gases

We prove upper bounds on the ground state energies of the one- and two-component charged Bose gases. The upper bound for the one-component gas agrees with the high density asymptotic formula proposed by L. Foldy in 1961. The upper bound for the two-component gas agrees in the large particle number limit with the asymptotic formula conjectured by F. Dyson in 1967. Matching asymptotic lower bounds for these systems were proved in references [10] and [11]. The formulas of Foldy and Dyson which are based on Bogolubov’s pairing theory have thus been validated.     

The Ground State Energy of a Dilute Two-Dimensional Bose Gas

The ground state energy per particle of a dilute, homogeneous, two-dimensional Bose gas, in the thermodynamic limit is shown rigorously to be E₀/N=(2²/m)|ln(a²)|^–1, to leading order, with a relative error at most O(|ln(a²)|^–1/5). Here N is the number of particles, =N/V is the particle density and a is the scattering length of the two-body potential. We assume that the two-body potential is short range and nonnegative. The amusing feature of this result is that, in contrast to the three-dimensional case, the energy, E₀ is not simply N(N–1)/2 times the energy of two particles in a large box of volume (area, really) V. It is much larger.     

Ground State Energy of Dilute Bose Gas in Small Negative Potential Case

It is well known that the ground state energy of a three dimensional dilute Bose gas in the thermodynamic limit is E=4π a ρ N when the particles interact via a non-negative, finite range, spherically symmetric, two-body potential. Here, N is the number of particles, ρ is the density of the gas, and a is the scattering length of the potential. In this paper, we prove the same result without the non-negativity condition on the potential, provided the negative part is small.     

The Ground State Energy of the Weakly Interacting Bose Gas at High Density

We prove the Lee-Huang-Yang formula for the ground state energy of the 3D Bose gas with repulsive interactions described by the exponential function, in a simultaneous limit of weak coupling and high density. In particular, we show that the Bogoliubov approximation is exact in an appropriate parameter regime, as far as the ground state energy is concerned.     

The Ground State Energy of Dilute Bose Gas in Potentials with Positive Scattering Length

The leading term of the ground state energy/particle of a dilute gas of bosons with mass m in the thermodynamic limit is 2p(^h/_2p)² a r/m{2\pi \hbar^2 a \varrho/m} when the density of the gas is r{\varrho}, the interaction potential is non-negative and the scattering length a is positive. In this paper, we generalize the upper bound part of this result to any interaction potential with positive scattering length, i.e, a > 0 and the lower bound part to some interaction potentials with shallow and/or narrow negative parts.     

The Second Order Upper Bound for the Ground Energy of a Bose Gas

Consider N bosons in a finite box Λ=[0,L]³?R ³ interacting via a two-body smooth repulsive short range potential. We construct a variational state which gives the following upper bound on the ground state energy per particle

$\overline{\lim}_{\rho\to0}\overline{\lim}_{L\to\infty,\,N/L^3\to \rho}\biggl(\frac{e_0(\rho)-4\pi a\rho}{(4\pi a)^{5/2}(\rho)^{3/2}}\biggr )\leq\frac{16}{15\pi^2},$

where a is the scattering length of the potential. Previously, an upper bound of the form C16/15π ² for some constant C>1 was obtained in (Erdös et al. in Phys. Rev. A 78:053627, 2008). Our result proves the upper bound of the prediction by Lee and Yang (Phys. Rev. 105(3):1119–1120, 1957) and Lee et al. (Phys. Rev. 106(6):1135–1145, 1957).

     

Ground State Energy of Unitary Fermion Gas with the Thomson Problem Approach

The dimensionless universal coefficient ε defines the ratio of the unitary fermions energy density to that for the ideal non-interacting ones in the non-relativistic limit with T = 0. The classical Thomson problem is taken as a nonperturbative quantum many-body arm to address the ground state energy including the low energy nonlinear quantum fluctuation/correlation effects. With the relativistic Dirac continuum field theory formalism, the concise expression for the energy density functional of the strongly interacting limit fermions at both finite temperature and density is obtained. Analytically, the universal factor is calculated to be ε= 4/9. The energy gap is△= 5/8k^2f/（2m）.     

Exact Ground State Energy of the Strong-Coupling Polaron

The polaron has been of interest in condensed matter theory and field theory for about half a century, especially the limit of large coupling constant, α. It was not until 1983, however, that a proof of the asymptotic formula for the ground state energy was finally given by using difficult arguments involving the large deviation theory of path integrals. Here we derive the same asymptotic result, , and with explicit error bounds, by simple, rigorous methods applied directly to the Hamiltonian. Our method is easily generalizable to other settings, e.g., the excitonic and magnetic polarons. Received: 6 May 1996/Accepted: 20 May 1996     

Ground State of a Quantum Particle Coupled to a Scalar Bose Field

We use methods from functional integration to prove the existence and uniqueness of the ground state of a confined quantum particle coupled to a scalar massless Bose field. For an external potential growing at infinity, the ground state exists under fairly general conditions while, for a decaying potential, an unphysical condition on the coupling strength is still needed.     

Free Energy of a Dilute Bose Gas: Lower Bound

A lower bound is derived on the free energy (per unit volume) of a homogeneous Bose gas at density \(\varrho\) and temperature T. In the dilute regime, i.e., when \(a^3\varrho \ll 1\) , where a denotes the scattering length of the pair-interaction potential, our bound differs to leading order from the expression for non-interacting particles by the term \(4{\pi}a ( 2{\varrho^2}-[\varrho-\varrho_c]_+^2 )\) . Here, \(\varrho_c(T)\) denotes the critical density for Bose-Einstein condensation (for the non-interacting gas), and \([\, \cdot \, ]_+ = \max\{ \, \cdot\, , 0\}\) denotes the positive part. Our bound is uniform in the temperature up to temperatures of the order of the critical temperature, i.e., T ~ \(\varrho\) ^2/3 or smaller. One of the key ingredients in the proof is the use of coherent states to extend the method introduced in [17] for estimating correlations to temperatures below the critical one.     

Ground State Energy of the Low Density Hubbard Model

We derive a lower bound on the ground state energy of the Hubbard model for given value of the total spin. In combination with the upper bound derived previously by Giuliani (J. Math. Phys. 48:023302, [2007]), our result proves that in the low density limit the leading order correction compared to the ground state energy of a non-interacting lattice Fermi gas is given by 8π a ? _u ? _d , where ? _u(d) denotes the density of the spin-up (down) particles, and a is the scattering length of the contact interaction potential. This result extends previous work on the corresponding continuum model to the lattice case.     

Potential Energy Surfaces of Nitrogen Dioxide for the Ground State

The potential energy function of nitrogen dioxide with the C2v symmetry in the ground state is represented using the simplified Sorbie-Murrell many-body expansion function in terms of the symmetry of NO2. Using the potential energy function, some potential energy surfaces of NO2（C2v, X^-^2A1）, such as the bond stretching contour plot for a fixed equilibrium geometry angle θ and contour for O moving around N-O （R1）, in which R1 is fixed at the equilibrium bond length, are depicted. The potential energy surfaces are analysed. Moreover, the equilibrium parameters for NO2 with the C2v, Cs and Dsn symmetries, such as equilibrium geometry structures and energies, are calculated by the ab initio （CBS-Q） method.     

Analyticity of the Ground State Energy for Massless Nelson Models

We show that the ground state energy of the translationally invariant Nelson model, describing a particle coupled to a relativistic field of massless bosons, is an analytic function of the coupling constant and the total momentum. We derive an explicit expression for the ground state energy which is used to determine the effective mass.     

A Remark on the Ground State Energy of Bosonic Atoms

Monotonicity properties of the ground state energy of bosonic atoms as established in a recent paper by M.K.H. Kiessling [J. Stat. Phys. 139:1063 (2009)] are studied. Symmetry and scaling arguments lead to a more direct proof of a slightly stronger result of this monotonicity and the behavior of the ground state energy as a function of the number of bosonic electrons. Furthermore, invoking appropriate lower and upper bounds on two-electron systems, the stability of the bosonics He⁻ ion is rigorously demonstrated.     

Gross-Pitaevskii Theory of the Rotating Bose Gas

 We study the Gross-Pitaevskii functional for a rotating two-dimensional Bose gas in a trap. We prove that there is a breaking of the rotational symmetry in the ground state; more precisely, for any value of the angular velocity and for large enough values of the interaction strength, the ground state of the functional is not an eigenfunction of the angular momentum. This has interesting consequences on the Bose gas with spin; in particular, the ground state energy depends non-trivially on the number of spin components, and the different components do not have the same wave function. For the special case of a harmonic trap potential, we give explicit upper and lower bounds on the critical coupling constant for symmetry breaking. Received: 1 December 2001 / Accepted: 19 April 2002 Published online: 6 August 2002     

The Ground State Energy of the Multi-Polaron in the Strong Coupling Limit

We consider the Fröhlich N-polaron Hamiltonian in the strong coupling limit and bound its ground state energy from below. In particular, our lower bound confirms that the ground state energy of the Fröhlich polaron and the ground state energy of the associated Pekar–Tomasevich variational problem are asymptotically equal in the strong coupling limit. We generalize the operator approach that was used to prove a similar result in the N =  1 case in Lieb and Thomas (Commun. Math. Phys. 183:511–519, 1997) and apply a Feynman–Kac formula to obtain the same result for an arbitrary particle number N ≥  1.