The cauchy problem for non-linear Klein-Gordon equations

We consider in ⁿ⁺¹,n2, the non-linear Klein-Gordon equation. We prove for such an equation that there is a neighbourhood of zero in a Hilbert space of initial conditions for which the Cauchy problem has global solutions and on which there is asymptotic completeness. The inverse of the wave operator linearizes the non-linear equation. If, moreover, the equation is manifestly Poincaré covariant then the non-linear representation of the Poincaré Lie algebra, associated with the non-linear Klein-Gordon equation is integrated to a non-linear representation of the Poincaré group on an invariant neighbourhood of zero in the Hilbert space. This representation is linearized by the inverse of the wave operator. The Hilbert space is, in both cases, the closure of the space of the differentiable vectors for the linear representation of the Poincaré group, associated with the Klein-Gordon equation, with respect to a norm defined by the representation of the enveloping algebra.     

Representation of solutions of the Burgers equation using functional integrals

In the paper, a representation of a solution of the Burgers equation in ℝ ⁿ is obtained by using integrals with respect to the Wiener measure on the space of trajectories in ℝ ⁿ . The Burgers equation is considered in a rigged Hilbert space. It is proved that, in the infinite-dimensional case, there is an analog of the Cole-Hopf transformation relating the Burgers equation and an analog of the heat equation with respect to measures. The Feynman-Kac formula for the heat equation (with potential) with respect to measures in a rigged Hilbert space is obtained.     

Bound State Solutions of the Klein-Gordon Equation for the Mathews-Lakshmanan Oscillator

We study a boundary-value problem for the Klein-Gordon equation that is inspired by the well-known Mathews-Lakshmanan oscillator model. By establishing a link to the spheroidal equation, we show that our problem admits an infinite number of discrete energies, together with associated solutions that form an orthogonal set in a weighted L ²-Hilbert space.     

Relativity in the three-nucleon system

A Poincaré-invariant formulation of the three-body system is used. The two-body force embedded in the three-particle Hilbert space is generated out of the high-precision NN forces by solving a nonlinear equation. The solution of the relativistic 3N Faddeev equation for ³H reveals less binding energy than for the nonrelativistic one. The effect of the Wigner spin rotation on the binding energy is very small.     

Hilbert space representation of an algebra of observables forq-deformed relativistic quantum mechanics

Using a representation of theq-deformed Lorentz algebra as differential operators on quantum Minkowski space, we define an algebra of observables for a q-deformed relativistic quantum mechanics with spin zero. We construct a Hilbert space representation of this algebra in which the square of the massp ² is diagonal.     

Hamiltonian quantum dynamics with separability constraints

Schroedinger equation on a Hilbert space H, represents a linear Hamiltonian dynamical system on the space of quantum pure states, the projective Hilbert space PH. Separable states of a bipartite quantum system form a special submanifold of PH. We analyze the Hamiltonian dynamics that corresponds to the quantum system constrained on the manifold of separable states, using as an important example the system of two interacting qubits. The constraints introduce nonlinearities which render the dynamics nontrivial. We show that the qualitative properties of the constrained dynamics clearly manifest the symmetry of the qubits system. In particular, if the quantum Hamilton’s operator has not enough symmetry, the constrained dynamics is nonintegrable, and displays the typical features of a Hamiltonian dynamical system with mixed phase space. Possible physical realizations of the separability constraints are discussed.     

On Eigenfunction Expansion of Solutions to the Hamilton Equations

We establish a spectral representation for solutions to linear Hamilton equations with positive definite energy in a Hilbert space. Our approach is a special version of M. Krein’s spectral theory of J-selfadjoint operators in the Hilbert spaces with indefinite metric. Our main result is an application to the eigenfunction expansion for the linearized relativistic Ginzburg–Landau equation.     

Group representations in reproducing kernel Hilbert spaces

We present a simple construction which realises any cyclic representation of a unimodular group G in a reproducing kernel Hilbert space (i.e. a coherent state realisation). This construction in considered from a number of viewpoints, in particular, the main result characterises those reproducing kernel Hilbert subspaces of L²(G) invariant under the left regular representation using the Plancherel theorem. Some trace formulae are also discussed.     

Construction of physical states in Yang-Mills theory: The time-like gauge

We construct physical states in pure Yang-Mills theory in the time-like gauge A_α⁰ = 0. We also construct a complete basis in the physical subspace of Hilbert space. Comparison is made with a recent paper by Eylon.     

Entangled Quantum States

Entangled quantum states are an important component of quantum computingtechniques such as quantum error correction, dense coding, and quantumteleportation. We determine the requirements for a state in the Hilbert space C ⁹to be entangled and a solution to the corresponding factorization problem if thisis not the case.     

D = 1 Supergravity and Quantum Cosmology

Using a D = 1 supergravity framework I construct a super-Friedmann equation for an isotropic and homogenous universe including dynamical scalar fields. In the context of quantum theory this becomes an equation for a wave function of the universe of spinorial type, the Wheeler–DeWitt–Dirac equation. It is argued that a cosmological constant breaks a certain chiral symmetry of this equation, a symmetry in the Hilbert space of universe states, which could protect a small cosmological constant from being affected by large quantum corrections.

     

Algebraic Bethe ansatz for integrable Kondo impurities in the one-dimensional supersymmetric t-J model

An integrable Kondo problem in the one-dimensional supersymmetric t-J model is studied by means of the boundary supersymmetric quantum inverse scattering method. the boundary K matrices depending on the local moments of the impurities are presented as a nontrivial realization of the graded reflection equation algebras in a two-dimensional impurity Hilbert space. Further, the model is solved by using the algebraic Bethe ansatz method and the Bethe ansatz equations are obtained.     

Reggeon quantum mechanics: A critical discussion

The quantum-mechanical problem of reggeon field theory in zero transverse dimensions is re-examined in order to set up a precise mathematical framework for the case μ = α(0) ? 1 > 0. We establish a Hamiltonian formulation in a Hilbert space for (ifμ > 0) and we prove the equivalence of the related eigenvalue problem with a “radial” Schrödinger-type equation in an L²(0, ∞) space. We prove that the S-matrix and the pomeron Green functions, at fixed rapidity Y and triple-pomeron coupling λ ≠ 0, have a spectral decomposition and are analytic in μ for ?∞ < μ < + ∞. For μ > 0, we confirm most of the qualitative results found by previous authors, and in particular the tunnelling shift [～ exp(?μ²/2λ²)] setting the scale for the asymptotic behaviour in Y.In the classical limit of λ/μ small we find that the action, for μ > 0, develops a singularity in Y at some value Y_c. We give arguments to show that for Y ? Y_c the perturbative result is reached, while for $\text{Y ? Y}{}_{\mathrm{<mtext>c</mtext>}}$ perturbation theory breaks down. Most of these results are shown to be stable against the addition of a small quartic coupling of the simplest type [$\text{λ′(}\text{Ψ}\text{Ψ)}{}^2$] up to the “magic” vvalue λ′ = λ²/μ. The existence of a level crossing at this value is confirmed by an analytic continuation in λ′.     

Geometry and analysis on isospectral sets. I. Riemannian geometry,asymptotic case

A short introduction to the analytical and algebraic aspects of integrable systems is given. We consider the Riemannian geometry of the isospectral set belonging to the Dirichlet problem −y′' + q(x)y = λy, y(0) = y(1) = 0, where q is a square integrable function of the real Hilbert space L²_ℝ([0,1]). We derive the metric and the connection for the isospectral set, which is an infinite dimensional real analytic submanifold of LL²_ℝ([0,1 ]), in the case of large eigenvalues. The curvature in the asymptotic case is then derived and it is proved that the connection and the curvature are well defined if we take their coefficients in the discrete Sobolev spaces. We further give the explicit formulae for the parallel transport and a sufficiency condition is derived such that a curve on the isospectral set is a geodesic.     

Remark on the solution of the Schrödinger equation for anharmonic oscillators via the Feynman path integral

We give a simple proof of Feynman's formula for the Green's function of the n-dimensional harmonic oscillator valid for every time t with Im t ≤ 0. As a consequence the Schrödinger equation for the anharmonic oscillator is integrated and expressed by the Feynman path integral on Hilbert space.     

Local-potential approximation for the Brueckner G matrix and problem of optimally choosing model subspace

The validity of the local-potential approximation, which was proposed previously for the singlet-pairing problem in semi-infinite nuclear matter, is investigated in the Bethe-Goldstone equation for the Brueckner G matrix. For this purpose, use is made of the method developed earlier for solving this equation for a planar slab of nuclear matter in the case of a separable form of NN interaction. The ¹ S ₀ singlet and the ³ S ₁+³ D ₁ triplet channel are considered. The complete two-particle Hilbert space is split into a model and the complementary subspace that are separated by an energy E ₀. The two-particle propagator is calculated precisely in the first subspace, and the local-potential approximation is used in the second subspace. With an eye to subsequently employing the G matrix to calculate the Landau-Migdal amplitude, the total two-particle energy is fixed at E=2μ, where μ is the chemical potential of the system under consideration. Specific numerical calculations are performed at μ=?8 MeV. The applicability of the local-potential approximation is investigated versus the cutoff energy E ₀. It is shown that, in either channel being considered, the accuracy of the local-potential approximation is rather high for E ₀≥10 MeV.     

Reduced Dirac-Kähler lattice fermion action

It is shown that the lattice Dirac-Kähler action is reducible under a chiral-like transformation. This provides a new lattice fermion action for spinors that have 2^d−1 components (instead of 2^d), with the property that, in the free case, each component satisfies the lattice euclidean Klein-Gordon equation. Reflection positivity is satisfied on the lattice, thus assuring a (positive) physical Hilbert space. In d = 4 dimensions the spinors have 8 components, and the correct physical chiral anomaly in the continuum limit. The action is suitable for QCD quarks which, in the continuum limit, are described by Dirac spinors that occur in flavor doublets.     

Exact classical and quantum solutions for a covariant oscillator near the black hole horizon in Stueckelberg-Horwitz-Piron theory

We found exact solutions for canonical classical and quantum dynamics for general relativity in Horwitz general covariance theory. These solutions can be obtained by solving the generalized geodesic equation and Schrödinger-Stueckelberg-Horwitz-Piron (SHP) wave equation for a simple harmonic oscillator in the background of a two dimensional dilaton black hole spacetime metric. We proved the existence of an orthonormal basis of eigenfunctions for generalized wave equation. This basis functions form an orthogonal and normalized (orthonormal) basis for an appropriate Hilbert space. The energy spectrum has a mixed spectrum with one conserved momentum p according to a quantum number n. To find the ground state energy we used a variational method with appropriate boundary conditions. A set of mode decomposed wave functions and calculated for the Stueckelberg-Schrodinger equation on a general five dimensional blackhole spacetime in Hamilton gauge.     

Decay of isomers stimulated by laser radiation

A theory is presented for the induced decay of isomer levels for nuclei exposed to a laser field, which couples the initial isomer level to intermediate higher lying levels with short lifetimes. We took into account the electric dipole transitions between these levels in a one-photon approximation. The time-dependent Schrödinger equation with a periodic Hamiltonian is solved by the method of the composite Hilbert space. A simple expression for the broadening of the isomer level ΔΓ, caused by the laser radiation, is derived. As an example, we considered the decay of the isomer level 970.17 keV with the spin I ^π = 23/2^? of ¹⁷⁷Lu, coupled by the laser wave to the virtual level 1352.33 keV with I ^π = 21/2⁺. The ratio of the broadening ΔΓ to the natural width of the isomer level Γ is found to be about 40% for a 10¹⁷ W/cm² laser. Any influence of the electronic environment is not taken into account.