Narrow Results

In the “Many Interacting Worlds” (MIW) discrete Hamiltonian system approximation of Schrödinger’s wave equation, introduced in Ref. 11, convergence of ground states to the Normal ground state of the quantum harmonic oscillator, via Stein’s method, in Wasserstein-$1$ distance with rate ${𝒪}(\sqrt{logN}/N)$ has been shown Refs. 5, 13, and 15. In this context, we construct approximate higher energy states of the MIW system, and show their convergence with the same rate in Wasserstein-1 distance to higher energy states of the quantum harmonic oscillator. In terms of techniques, we apply the “differential equation” approach to Stein’s method, which allows to handle behavior near zeros of the higher energy states.

We consider a model of a quantum mechanical system coupled to a (massless) Bose field, called the generalized spin-boson model (A. Arai and M. Hirokawa, J. Funct. Anal. 151 (1997), 455–503), without infrared regularity condition. We define a regularized Hamiltonian H (ν) with a parameter ν ≥ 0 such that H = H (0) is the Hamiltonian of the original model. We clarify a relation between ground states of H (ν) and those of H by formulating sufficient conditions under which weak limits, as ν → 0, of the ground states of H (ν)'s are those of H. We also establish existence theorems on ground states of H (ν) and H under weaker conditions than in the previous paper mentioned above.

We consider two kinds of stability (under a perturbation) of the ground state of a self-adjoint operator: the one is concerned with the sector to which the ground state belongs and the other is about the uniqueness of the ground state. As an application to the Wigner–Weisskopf model which describes one mode fermion coupled to a quantum scalar field, we prove in the massive case the following: (a) For a value of the coupling constant, the Wigner–Weisskopf model has degenerate ground states; (b) for a value of the coupling constant, the Wigner–Weisskopf model has a first excited state with energy level below the bottom of the essential spectrum. These phenomena are nonperturbative.

We consider a model of quantum particles coupled to a massless quantum scalar field, called the massless Nelson model, in a non-Fock representation of the time-zero fields which satisfy the canonical commutation relations. We show that the model has a ground state for all values of the coupling constant even in the case where no infrared cutoff is made. The non-Fock representation used is inequivalent to the Fock one if no infrared cutoff is made.

We consider, in an abstract form, a system of "quantum particles" coupled to a Bose field. It is shown that, under suitable hypotheses, the composed system can have a ground state even if the uncoupled particle system has no ground state.

We prove in dimension n = 3 an asymptotic stability result for ground states of the Nonlinear Schrödinger Equation which contain one internal mode.

For a flow α on a C*-algebra one defines a symmetry as the group of automorphisms γ such that γαγ^-1 is a cocycle perturbation of α. We propose to define a core of this symmetry, which acts trivially on the set of equivalence classes of KMS state representations, but may act non-trivially on the set of equivalence classes of covariant irreducible representations. In particular this core acts transitively on the set of those which induce faithful representations of the crossed product by α.

The operator-theoretic renormalization group (RG) methods are powerful analytic tools to explore spectral properties of field-theoretical models such as quantum electrodynamics (QED) with non-relativistic matter. In this paper, these methods are extended and simplified. In a companion paper, our variant of operator-theoretic RG methods is applied to establishing the limiting absorption principle in non-relativistic QED near the ground state energy.

In this paper, we consider a model for a nucleon interacting with the ω and σ mesons in the atomic nucleus. The model is relativistic, but we study it in the nuclear physics non-relativistic limit, which is of a very different nature from the one of the atomic physics. Ground states with a given angular momentum are shown to exist for a large class of values for the coupling constants and the mesons' masses. Moreover, we show that, for a good choice of parameters, the very striking shapes of mesonic densities inside and outside the nucleus are well described by the solutions of our model.

An otherwise free classical particle moving through an extended spatially homogeneous medium with which it may exchange energy and momentum will undergo a frictional drag force in the direction opposite to its velocity with a magnitude which is typically proportional to a power of its speed. We study here the quantum equivalent of a classical Hamiltonian model for this friction phenomenon that was proposed in [11]. More precisely, we study the spectral properties of the quantum Hamiltonian and compare the quantum and classical situations. Under suitable conditions on the infrared behavior of the model, we prove that the Hamiltonian at fixed total momentum has no ground state except when the total momentum vanishes, and that its spectrum is otherwise absolutely continuous.

New numerical methods of the ground state and thermodynamic properties calculations of one-dimensional Generalized Wigner crystal on disordered host-lattice are proposed. Unlike computer simulation methods (for instance, Monte Carlo) these methods bring the exact results. Another attractive feature of the proposed methods is their speed: it is possible to study the systems with length about 10⁴–10⁵ nodes even on a personal computer. This is especially important in the case of weakly disordered systems and the long-range correlations. The gapless structure of low-energy excitation and breaking long-range correlations at arbitrary small disordering are established.

The half-lives of deformed nuclei are reported. We consider an angle-dependent potential which yields multipole approximations. We see that the results are in better agreement with the experimental data when the multipole approximation is solely considered for the daughter nuclei.

We will study the Dirac–Kepler problem plus a Coulomb-type scalar potential by generalizing the Lippmann–Johnson operator to D spatial dimensions. From this operator, we construct the supersymmetric generators to obtain the energy spectrum for discrete excited eigenstates and the radial spinor for the SUSY ground state.

The second (unphysical) critical charge in the three-body quantum Coulomb system of a nucleus of positive charge $Z$ and mass $m_p$, and two electrons, predicted by Stillinger has been calculated to be equal to $Z_B^{\infty }=0.904854$ and $Z_B^{m_p}=0.905138$ for infinite and finite (proton) mass $m_p$, respectively. It is shown that in both cases, the ground state energy $E(Z)$ (analytically continued beyond the first critical charge $Z_c$, for which the ionization energy vanishes, to $\mathrm{\text{Re}}Z$$<$$Z_c$) has a square-root branch point with exponent 3/2 at $Z=Z_B$ in the complex $Z$-plane. Based on analytic continuation, the second, excited, spin-singlet bound state of negative hydrogen ion H${}^{-}$ is predicted to be at −0.51554 a.u. (−0.51531 a.u. for the finite proton mass $m_p$). The first critical charge $Z_c$ is found accurately for a finite proton mass $m_p$ in the Lagrange mesh method, $Z_c^{m_p}=0.911069724655$.

We suggest that the dark matter halo in some of the spiral galaxies can be described as the ground state of the Bose–Einstein condensate of ultra-light self-gravitating axions. We have also developed an effective “dissipative” algorithm for the solution of nonlinear integro-differential Schrödinger equation describing self-gravitating Bose–Einstein condensate. The mass of an ultra-light axion is estimated.

We consider a two-component dark matter halo (DMH) of a galaxy containing ultra-light axions (ULA) of different mass. The DMH is described as a Bose–Einstein condensate (BEC) in its ground state. In the mean-field (MF) limit, we have derived the integro-differential equations for the spherically symmetrical wave functions of the two DMH components. We studied, numerically, the radial distribution of the mass density of ULA and constructed the parameters which could be used to distinguish between the two- and one-component DMH. We also discuss an interesting connection between the BEC ground state of a one-component DMH and Black Hole temperature and entropy, and Unruh temperature.

We consider a dark matter halo (DMH) of a spherical galaxy as a Bose–Einstein condensate (BEC) of the ultra-light axions (ULA) interacting with the baryonic matter. In the mean-field (MF) limit, we have derived the integro-differential equation of the Hartree–Fock type for the spherically symmetrical wave function of the DMH component. This equation includes two independent dimensionless parameters: (i) $\beta$ is the ratio of baryon and axion total mases and (ii) $\xi$ is the ratio of characteristic baryon and axion spatial parameters. We extended our “dissipation algorithm” for studying numerically the ground state of the axion halo in the gravitational field produced by the baryonic component. We calculated the characteristic size, $x_c$ of DMH as a function of $\beta$ and $\xi$ and obtained an analytical approximation for $x_c$.

The behavior of an electrically charged massive particle (an electron) is studied in a constant uniform magnetic field and a single attractive λδ(r) potential. A simple transcendental equation that determines the electron energy spectrum is derived. The approximate wave function of a loosely bound state is constructed in a very simple form. The model under consideration makes it possible to study the effect of magnetic fields on a loosely bound electron. It is shown that the sizes of the electron localization region change and the probability current density arises when the electron is in the loosely bound state in the presence of a constant uniform magnetic field. The above current must involve (and exercise influence on) the electron scattering. The probability current resembles a stack of "pancake vortices" whose circulating (around the z-axes) "currents" are mostly confined within the plane z = 0 in the weak magnetic field. The equation for determining the energy levels of the electron states is obtained for the model under study in two spatial dimensions and the energy of the loosely bound state is found for the two-dimensional model.

We study the ground state of the boundary Izergin–Korepin model. The boundary Izergin–Korepin model is defined by the so-called R-matrix and K-matrix for which satisfy Yang–Baxter equation and boundary Yang–Baxter equation. The ground state associated with identity K-matrix was constructed by W.-L. Yang and Y.-Z. Zhang in earlier study. We construct the free field realization of the ground state associated with nontrivial diagonal K-matrix.

In this work we have applied the optimal coupled-cluster approximation to study the ground state of the E⊗(b₁+b₂) Jahn–Teller effect. The effectiveness of the optimal coupled-cluster approximation has been investigated for the whole range of the asymmetry parameter and various coupling strengths. It is shown that our results up to the third level of approximation are in very good agreement with the exact numerical diagonalization results and are better than those from earlier variational treatments. Furthermore, unlike previous variational treatments, the optimal coupled-cluster approximation has the advantage that the accuracy of both the ground state energy and wavefunction estimates is being taken care of and can be systematically improved. Since the mathematical treatment in this work is simple, the optimal coupled-cluster approximation could be easily extended to the studies of other fermion-boson interacting systems, e.g. the extended Jahn–Teller system.