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A Lorentz-covariant system of wave equations is formulated for a quantum-mechanical three-body system in one space dimension, comprised of one photon and two identical massive spin one-half Dirac particles, which can be thought of as two electrons (or alternatively, two positrons). Manifest covariance is achieved using Dirac’s formalism of multi-time wave functions, i.e., wave functions Ψ (x ph , x e 1 , x e 2) where x ph , x e 1 , x e 2 are generic spacetime events of the photon and two electrons, respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifolds { x ph = x e 1 } and { x ph = x e 2 } compatible with conservation of probability current. The corresponding initial-boundary value problem is shown to be well-posed, and it is shown that the unique solution can be represented by a convergent infinite sum of Feynman-like diagrams, each one corresponding to the photon bouncing between the two electrons a fixed number of times. [ABSTRACT FROM AUTHOR]

The familiar second derivative test for convexity, combined with resolvent calculus, is shown to yield a useful tool for the study of convex matrix-valued functions. We demonstrate the applicability of this approach on a number of theorems in this field. These include convexity principles which play an essential role in the Lieb–Ruskai proof of the strong subadditivity of quantum entropy. [ABSTRACT FROM AUTHOR]

Recently, it has been proved that a set of spherically symmetric densities uniquely determines the Coulomb external potential. Utilizing this theorem, a previous method is extended to construct an orbital-free density functional theory. Auxilary spherical non-interacting systems and Kohn–Sham-like equations are constructed. A set of spherical generating functions are defined. These functions have two extra variables and are reduced to the set of spherically symmetric densities if the extra variables are equal to zero. Euler equations are rewritten as Schrödinger-like equations for the square root of the spherically symmetric generating functions. Generalized Pauli potentials are defined, and it is shown that these potentials can be calculated in the knowledge of the generating functions. The Euler equations can be solved with these Pauli potentials. [ABSTRACT FROM AUTHOR]

This paper concerns an approximation of the expectation values of the position and momentum of the solution to the semiclassical Schrödinger equation with a Gaussian as the initial condition. Of particular interest is the approximation obtained by our symplectic/Hamiltonian formulation of the Gaussian wave packet dynamics that introduces a correction term to the conventional formulation using the classical Hamiltonian system by Hagedorn and others. The main result is a proof that our formulation gives a higher-order approximation than the classical formulation does to the expectation value dynamics under certain conditions on the potential function. Specifically, as the semiclassical parameter ε approaches 0, our dynamics gives an O (ε 3 / 2) approximation of the expectation value dynamics, whereas the classical one gives an O (ε) approximation. [ABSTRACT FROM AUTHOR]

We consider a system of N bosons in the limit N → ∞ , interacting through singular potentials. For initial data exhibiting Bose–Einstein condensation, the many-body time evolution is well approximated through a quadratic fluctuation dynamics around a cubic nonlinear Schrödinger equation of the condensate wave function. We show that these fluctuations satisfy a (multi-variate) central limit theorem. [ABSTRACT FROM AUTHOR]

We consider the semiclassical Dirichlet Pauli operator in bounded connected domains in the plane. Rather optimal results have been obtained in previous papers by Ekholm–Kovařík–Portmann and Helffer–Sundqvist for the asymptotics of the ground state energy in the semiclassical limit when the magnetic field has constant sign. In this paper, we focus on the case when the magnetic field changes sign. We show, in particular, that the ground state energy of this Pauli operator will be exponentially small as the semiclassical parameter tends to zero and give lower bounds and upper bounds for this decay rate. Concrete examples of magnetic fields changing sign on the unit disk are discussed. Various natural conjectures are disproved, and this leaves the research of an optimal result in the general case still open. [ABSTRACT FROM AUTHOR]

We derive new expressions for the Rayleigh-Schrödinger series describing the perturbation of eigenvalues of quantum Hamiltonians. The method, somehow close to the so-called dimensional renormalization in quantum field theory, involves the Birkhoff decomposition of some Laurent series built up out of explicit fully non-resonant terms present in the usual expression of the Rayleigh-Schrödinger series. Our results provide new combinatorial formulae and a new way of deriving perturbation series in quantum mechanics. More generally we prove that such a decomposition provides solutions of general normal form problems in Lie algebras. [ABSTRACT FROM AUTHOR]

We derive an explicit expression for the kernel of the evolution group $${\exp(-\mathrm{i} t H_0)}$$ of the discrete Laguerre operator H (i.e., the Jacobi operator associated with the Laguerre polynomials) in terms of Jacobi polynomials. Based on this expression, we show that the norm of the evolution group acting from $${\ell^1}$$ to $${\ell^\infty}$$ is given by $${(1+t^2)^{-1/2}}$$ . [ABSTRACT FROM AUTHOR]

We show that the Siegel upper half space $${\Sigma_{d}}$$ is identified with the Marsden-Weinstein quotient obtained by symplectic reduction of the cotangent bundle $${T^{*} \mathbb{R}^{2d^{2}}}$$ with O(2 d)-symmetry. The reduced symplectic form on $${\Sigma_{d}}$$ corresponding to the standard symplectic form on $${T^{*} \mathbb{R}^{2d^{2}}}$$ turns out to be a constant multiple of the symplectic form on $${\Sigma_{d}}$$ obtained by Siegel. Our motivation is to understand the geometry behind two different formulations of the Gaussian wave packet dynamics commonly used in semiclassical mechanics. Specifically, we show that the two formulations are related via the symplectic reduction. [ABSTRACT FROM AUTHOR]

We consider the relativistic generalization of the quantum A Calogero-Sutherland models due to Ruijsenaars, comprising the rational, hyperbolic, trigonometric and elliptic cases. For each of these cases, we find an exact common eigenfunction for a generalization of Ruijsenaars analytic difference operators that gives, as special cases, many different kernel functions; in particular, we find kernel functions for Chalykh-Feigin-Veselov-Sergeev-type deformations of such difference operators which generalize known kernel functions for the Ruijsenaars models. We also discuss possible applications of our results. [ABSTRACT FROM AUTHOR]