We study the renormalization invariant trajectory of the $\phi^4$-perturbation of the free field fixed point in the hierarchical approximation. We parametrize it by a running $\phi^4$-coupling $g$ with linear step $\beta$-function. We rigorously control the non-perturbative corrections to finite order approximants from double perturbation theory in $g$ and $g^2\ln(g)$. The construction uses a contraction mapping for the extended renormalization group composed of a hierarchical block spin transformation with a flow of $g$.
In this article we prove several reciprocity theorems for some infinite-dimensional dual pairs of representations on Bargmann-Segal-Fock spaces.
We consider the nonlinear Schrodinger equation with a pure power repulsive nonlinearity on Schwarzschild manifolds. Equations of this type arise when a nonlinear wave equation on a Schwarzschild manifold is written in Hamiltonian form, cf. [2], [10]. For radial solutions with sufficiently localized initial data, we obtain global existence, L-p estimates, and the existence and asymptotic completeness of the wave operators. Our approach is based on a dilation identity and global space-time estimates.
We study the high energy limit of the Faddeev Scattering Amplitude for the Dirac operator associated with a potential Q. We prove that the Fourier transform of the potential and the limit of the scattering amplitude are related by an integral equation. Finally we apply these results to reconstruct the potential module a gauge transformation.
Considering the nonrelativistic Schrodinger operator for homonuclear diatomic molecular ions within the clamped nuclei approximation, we study the stability problem for increasing ratio Z/N of nuclear charge Z to number N of electrons. In particular, we derive improved bounds on the critical parameters that imply instability (upper bounds on the nuclear charge, or, equivalently, lower bounds on the number of electrons), viz. parameters that lead to dissociation of the molecular system into atomic fragments. The principal qualitative advantange of our estimates is the inclusion of electronic correlation, i.e., taking into account the effect of electron-electron repulsion on the molecular bond. Comparing; our rigorous results with empirical or computed data, we formulate a conjecture that should quantify the actual stability behaviour of realistic molecular species.
We present two possible ways for solving the so caned Gleason trouble in the framework of Bohm-Bub hidden variable theory : the so-called Wiener-Siegel method and a new method proposed by us. We study the non-local properties exhibited by these methods, and discuss the experimental possibility of measuring these non-local effects.
We present a Feynman path integral in the setting of geometric quantization of symplectic manifolds with Kahler polarization, where the Hamiltonian operator is given by Toeplitz quantization. We compute the quantum propagator as a limit of path integrals, involving Brownian motion in the phase space and geometricaly meaningful stochastic processes.
In the momentum space we find the bound eigenenergies and eigenfunctions of the ID Schrodinger equation for an asymmetric Coulomb potential. We find that eigenfunctions in the configuration space are expressed in terms of fractional derivatives. Our approach could provide qualitative features of the electronic states of an impurity located between two different quantum wires.
We start from the set of KMS-states for the global C-*-dynamics of a class of weakly inhomogeneous bipolaronic superconductors, which me have determined in a previous work. We discuss the spectral properties of the generator for the unitary implementation of the global Heisenberg dynamics in the GNS-representation over those Kh IS-states, which have minimal free energy density and unbroken internal symmetries. It is shown that the stable and hence macroscopically detectable part of such a spectrum is given by the spectrum of the homogenized model. The stable energy values depend on the temperature and doping of the system and lead to so-called spectral phase diagrams. The latter are meant to supplement the thermodynamic phase diagrams elaborated in earlier investigations. The different behaviour of the stable spectra for phase transitions of the first and second kind is especially significant. A classification of the factor types for the pure phase states - occurring in the central decomposition of the stable invariant KMS-states - is carried through. As a remarkable fact we found in certain phase regions a dense subset of pure phase states, which belong to factors of type III1, having type IIIlambda states with lambda not less than or equal to 1 in each neighborhood. Thus, in these weakly inhomogeneous quantum lattice systems one has representations with dynamical relaxation features.
When q is a root of unity, rwo mode extension of a single q-oscillator algebra was constructed. It was shown that this algebra (sl(q)(2)-covariant oscillator algebra) is covariant under sl(q)(2) The coherent states and coherence factor were computed.
We superpose one oscillator of bosonic type and another one of fermionic type, both of different angular frequencies, and show that this superposition is invariant under a (nonlinear) deformation of the Lie superalgebra osp(1/2; R). We also construct the unitary irreducible representations (positive discrete series) of this deformed structure.
In this paper we study the integrated density of surface states of a Schrodinger operator with an ergodic surface potential. We also introduce another generalised function similar to the spectral shift function known in the scattering theory. We show that these two quantities exist and coincide. We also study certain properties: their relation with the spectrum, smoothness, asymptotic behavior.
It is shown that propositional calculuses of both quantum and classical logics are noncategorical. We find that quantum logic is in addition to an orthomodular lattice also modeled by a weakly orthomodular lattice and that classical logic is in addition to a Boolean algebra also modeled by a weakly distributive lattice. Both new models turn out to be non-orthomodular. We prove the soundness and completeness of the calculuses for the models. We also prove that all the operations in an orthomodular lattice are five-fold defined. In the end we discuss possible repercussions of our results to quantum computations and quantum computers.
In this work we introduce the Killing-Yano symmetry on the phase space and we investigate the symplectic structure on the space of Killing-Yano tensors. We perform the detailed analyze of the $n$-dimensional flat space and the Riemaniann manifolds with constant scalar curvature. We investigate the form of some multipole tensors, which arise in the expansion of a system of charges and currents, in terms of second-order Killing-Yano tensors in the phase space of classical mechanics. We find some relations between these tensors and the generators of dynamical symmetries like the angular momentum, the mass-inertia tensor, the conformal operator and the momentum conjugate Runge-Lenz vector.
Q-algebras provide a non-boolean logical model and have been used to study quantum measurement, interference phenomena in quantum physics and the nature of the quantum probabilities. Each Q-algebra can be represented on a pre-Hilbert space, thus resulting in the standard model of quantum theory, but the representations considered by the author in recent papers involve unnecessarily large pre-Hilbert spaces (with an infinite dimension in all non-commutative cases even if the Q-algebra itself has a finite dimension).In the present paper, an "optimal" representation is constructed. It uses a pre-Hilbert space of minimum dimension, and is unique in a certain sense. A Q-algebra of finite dimension becomes isomorphic to a finite direct sum of matrix algebras.
A new predictor-corrector exponentially fitted Numerov-type method is developed for the numerical integration of the radial Schrodinger equation and of coupled differential equations arising from the Schrodinger equation. The Numerov-type method considered contains free parameters which allow if to be fitted to exponential functions. The new fourth algebraic order method is very simple and integrate more exponential functions than both the well known fourth order Numerov type exponentially fitted methods and the sixth algebraic order Runge-Kutta type methods. Numerical results also indicate that the new method is much more accurate than the other exponentially fitted methods. Based on the method developed in the present paper and on the method of Simos [24] a new variable-step procedure is developed for the numerical solution of the coupled differential equations arising from the Schrodinger equation. Numerical illustrations indicate that the new variable-step method is more efficient than other well known variable-step methods.
Unitary operations in Hilbert space of spin one half system for one qubit and two qubit systems are realized in terms of vertices of graphs of macroscopical automata realizing quantum logic. Examples of simple logical operations are analysed.
This review is devoted to the history formulation of standard Hilbert space quantum mechanics. We will give an overview over the basic ideas and concepts of the history approach. The consistent histories approach is usually formulated using the standard notions of observable and state. We will argue in the second part of this review that the natural notion of an observable in quantum mechanics is that of a positive-operator-valued measure (POV measure) and will show that the consistent history formalism can be generalized to incorporate POV measures in a natural and simple way.
We analyze a Wannier-Mott exciton in which the electron is constrained to move freely in a one-dimensional quantum wire (1DQW) and the hole moves freely in another perpendicular 1DQW. The resulting two-dimensional (2D) exciton Schrodinger equation in the laboratory frame of reference is solved in terms of the common 2D exciton equation in the center of mass frame when both electron and hole are in the same 2D quantum layer.
The peculiar role of the torsion in non-Riemannian gravity is elucidated in particular the fact it permits to obtain solutions for the field equations which have symmetries not obviously manifest in the action, the model then may have "Hidden symmetries" due to the fact the torsion acts as a compensating non symmetric term.
