Complex energies in relativistic quantum theory

A new four-component spin-1/2 wave equation for ordinary mass is discussed. It is shown that this equation has a conserved current not easily identified with a transition probability, only pure imaginary energy states, and is covariant. A tachyon-like Klein-Gordon equation is satisfied by this equation, but rest states are explicitly constructed.     

Perturbation theory for the intensity of Stark lines of a hydrogen atom

Perturbation theory for the wave function of a hydrogen-like atom in a homogeneous electric field of strength F makes it possible to obtain the Rayleigh-Schrödinger series with the coefficients of F ^N (N=0, 1, 2,...) being linear combinations of the Sturm function, which represents the unperturbed state, with 8N ² functions of the corresponding complete set with indices adjacent to the parabolic quantum number of the initial level. A method for recursive analytic calculation of the coefficients of the linear combination for any order N is developed. General expressions for corrections to the matrix elements and intensities of the radiation transitions between Stark sublevels are obtained. Analytic formulas and numerical values of the corrections up to the fourth order for the Lyman and Balmer series are presented. A comparison with the available data for transitions between the Stark components of Rydberg states is given.     

An efficient high-order boundary element method for nonlinear wave–wave and wave-body interactions

A highly efficient high-order boundary element method is developed for the numerical simulation of nonlinear wave–wave and wave-body interactions in the context of potential flow. The method is based on the framework of the quadratic boundary element method (QBEM) for the boundary integral equation and uses the pre-corrected fast Fourier transform (PFFT) algorithm to accelerate the evaluation of far-field influences of source and/or normal dipole distributions on boundary elements. The resulting PFFT–QBEM reduces the computational effort of solving the associated boundary-value problem from O(N^2～3) (with the traditional QBEM) to O(N ln N) where N represents the total number of boundary unknowns. Significantly, it allows for reliable computations of nonlinear hydrodynamics useful in ship design and marine applications, which are forbidden with the traditional methods on the presently available computing platforms. The formulation and numerical issues in the development and implementation of the PFFT–QBEM are described in detail. The characteristics of accuracy and efficiency of the PFFT–QBEM for various boundary-value problems are studied and compared to those of the existing accelerated (lower- and higher-order) boundary element methods. To illustrate the usefulness of the PFFT–QBEM, it is applied to solve the initial boundary-value problem in the generation of three-dimensional nonlinear waves by a moving ship hull. The predicted wave profile and resistance on the ship are compared to available experimental measurements with satisfactory agreements.     

Schrödinger-Newton equation with a complex Newton constant and induced gravity

In the reversible Schrödinger-Newton equation a complex Newton coupling Gexp(−iα) is proposed in place of G. The equation becomes irreversible and all initial one-body states are expected to converge to solitonic stationary states. This feature is verified numerically. For two-body solutions we point out that an effective Newtonian interaction is induced by the imaginary mean-fields as if they were real. The effective strength of such induced gravity depends on the local wave functions of the participating distant bodies.     

Strings in the CP2N−1 model

The CP₂^N?1 model is discussed using strings as collective variables in the hamiltonian formulation. The large N limit is obtained as a semiclassical approximation. The mass gap and β-function are computed. In the limit N → ∞, it is shown that the singlet spectrum contains both bound states and scattering states whose energies and wave functions are calculated.     

On the left-hand cut in potential scattering

Partial wave $\text{N}\text{D}$ equations in potential scattering are solved for the exponential, Hulthén and Morse potentials. The driving terms are taken to be either the contributions of a finite number of Born terms or the total contributions of only the nearest singularities (first n poles). For repulsive potentials one observes ghosts, anomalous bound states or resonances if the order of approximation is small with respect to the potential strength. The origin and meaning of these unphysical phenomena are explained. For attractive potentials such anomalies occur only at very large potential strengths if at all. Input-equivalent Bargmann potentials are employed to determine the quality and nature of the approximate $\text{N}\text{D}$ solutions. Rough criteria for the validity of approximations within the $\text{N}\text{D}$ approach are formulated.     

Production of hypernuclei with hadronic and electromagnetic probes

We present an overview of a fully covariant formulation for describing hypernuclear production with hadronic and electromagnetic probes. This theory is based on an effective Lagrangian picture and it focuses on production amplitudes that are described via creation, propagation and decay into relevant channels of N^∗(1650), N^∗(1710) and N^∗(1720) intermediate baryonic resonance states in the initial collision of the projectile with one of the target nucleons. The bound state nucleon and hyperon wave functions are obtained by solving the Dirac equation with appropriate scalar and vector potentials. Specific examples are discussed for reactions which are of interest in current and future experiments on hypernuclear production.     

On the theory and methods of calculation of surface state energies in nearly free electron metals

Intrinsic Heine-type surface states are discussed analytically for the nearly free-electron metals. It is shown, that if one constructs the wavefunctions corresponding to the real loops exactly, while the wavefunctions corresponding to the real lines are constructed in an approximate, free electron-like manner, the general N-band, M-beam surface state matching determinant can be solved exactly in the closed form, and the problem is no more difficult than the N-band, 2-beam calculation. The resulting surface state equation has a very simple form and is easily adaptable to computer calculations. It includes all of the special cases discussed previously by other authors.     

Hartree theory for matter in a photon or phonon field

For a Dicke-type Hamiltonian describing M two-level systems coupled to N modes of a Boson field, the expectation value is minimized with respect to unrestricted products of 2^M-level states and field states. In the resulting Hartree ground state(s) ? ? Π, ? is ground state of a cubic Schrödinger equation in $\text{C}$^2M and Π is the product of N coherent one-mode states depending on ?. It is proved that ? factorizes as well, into M two-level states determined by a nonlinear equation in $\text{R}$^M; that for weak coupling the Hartree ground state is unique and independent of the parameters in H; and that for strong coupling there are consecutively 2,…,2^L (and possibly even more) Hartree ground states where 1≤L≤min{M, 2N} counts certain reflection symmetries. Details of this symmetry-breaking bifurcation (such as structural stability of relaxed two-level systems) and connections to the true ground state(s) of H are worked out.     

Spectrum (super-) symmetries of particles in a Coulomb potential

The Schrödinger equation for a spin-0 particle in the field of a dyon is obtained by dimensional reduction of the four-dimensional harmonic oscillator; the reduction is effected by imposing an equivariance condition on the wave functions of the latter system. This geometrical construction allows for a simple derivation of the SO(4, 2) spectrum symmetry of the dyon system. A supermultiplet of one $\text{spin}\text{-}\text{1}\text{2}$ and two spin-0 particles in a Coulomb potential is demonstrated to possess an N = 2 conformal supersymmetry through a generalization of the same method. The states and wave functions for these systems can be obtained from the representation theory of the corresponding symmetry algebras. A particular case for which this approach provides a complete group theoretical analysis is that of the Pauli equation for a $\text{spin}\text{-}\text{1}\text{2}$ particle in the field of a dyon.     

Microscopic Theory of the Two-Dimensional Quantum Antiferromagnet in a Paramagnetic Phase

We have developed a consistent theory of the Heisenberg quantum antiferromagnet in the disordered phase with a short range antiferromagnetic order on the basis of the path integral for spin coherent states. In the framework of our approach we have obtained the response function for the spin fluctuations for all values of the frequency ω and the wave vector k and have calculated the free energy of the system. We have also reproduced the known results for the spin correlation length in the lowest order in 1/N. We have presented the Lagrangian of the theory in a form which is explicitly invariant under rotations and found natural variables in terms of which one can construct a natural perturbation theory. The short wave spin fluctuations are similar to those in the spin wave theory and they are on the order of the smallness parameter 1/2s where s is the spin magnitude. The long-wave spin fluctuations are governed by the nonlinear sigma model and are on the order of the smallness parameter 1/N, where N is the number of field components. We also have shown that the short wave spin fluctuations must be evaluated accurately and the continuum limit in time of the path integral must be performed after the summation over the frequencies ω.     

Coupled channel T-operator equations with connected kernels: (II). N coupled two-body channels

A time-independent theory of rearrangement collisions involving transitions between two-body states is presented. It is assumed that the system of interest consists of particles that may be partitioned into two-body systems in N ways, including interchanges of particle labels without changing the kind of channel. An infinite family of sets of N coupled T-operator equations is derived by use of the channel coupling array, as in previous work on the three-body problem. Specialization to the channel-permuting arrays guaranteeing connected (N?1)th iterates of the kernel of the coupled equations is made in the N-channel case (N > 3) and the nature of the solutions to the coupled equations is discussed. Various approximation schemes to be used with numerical calculations are suggested. Since the transition operators for all rearrangement channels are coupled together, no problems concerning non-orthogonality of the eigenstates of different channel Hamiltonians are encountered; also the presence of the outgoing wave boundary condition in all channels is made explicit. The close resemblance of the equations in matrix form to those of one-channel scattering is exploited by introducing Møller wave operators and associated channel scattering states, an optical potential formalism that leads to rearrangement channel optical potential operators, and a variational formulation of the coupled equations using a Schwinger-like variational principle. A brief comparison with other many-body formalisms is also given.     

Supersymmetry in stochastic processes with higher-order time derivatives

A supersymmetric path-integral representation is developed for stochastic processes whose Langevin equation contains any number N of time derivatives, thus generalizing the presently available treatment of first-order Langevin equations by Parisi and Sourlas [Phys. Rev. Lett. 43 (1979) 744; Nucl. Phys. B 206 (1982) 321] to systems with inertia (Kramers' process) and beyond. The supersymmetric action contains N fermion fields with first-order time derivatives whose path integral is evaluated for fermionless asymptotic states.     

Stochastic theory of nonlinear rate processes with multiple stationary states

A comparison has been made between the deterministic and stochastic (master equation) formulation of nonlinear chemical rate processes with multiple stationary states. We have shown, via two specific examples of chemical reaction schemes, that the master equations have quasi-stationary state solutions which agree with the various initial condition dependent equilibrium solutions of the deterministic equations. The presence of fluctuations in the stochastic formulation leads to true equilibrium solutions, i.e. solutions which are independent of initial conditions as t → ∞. We show that the stochastic formulation leads to two distinct time scales for relaxation. The mean time for the reaction system to reach the quasi-stationary states from any initial state is of $\text{O}$(N⁰) where N is a measure of the size of the reaction system. The mean time for relaxation from a quasi-stationary state to the true equilibrium state is $\text{O}$(e^N). The results obtained from the stochastic formulation as regards the number and location of the quasi-stationary states are in complete agreement with the deterministic results.     

Determination of all Petrov type-N space-times on which the conformally invariant scalar wave equation satisfies Huygen's principle

It is shown that the conformally invariant wave equation on a Petrov type-N space-time satisfies Huygens' principle if and only if the space-time is conformally related to a plane wave space-time.     

On the existence of hydrogen atoms in higher dimensional Euclidean spaces

The question of whether hydrogen atoms can exist or not in spaces with a number of dimensions (D) greater than 3 is revisited. The lowest quantum mechanical stable states and the corresponding wave functions are determined by applying Numerov?s method to solve Schrödinger?s equation. States for different angular momentum quantum number and dimensionality are considered. One is lead to the result that hydrogen atoms in higher dimensions could actually exist. The most probable distance between the electron and the nucleus are then computed as a function of D showing the possibility of tiny confined states.     

Pairing vibrational and isospin rotational states in a particle number and isospin projected generator coordinate method

Pairing vibrational and isospin rotational states are described in different approximations based on particle number and isospin projected, proton-proton, neutron-neutron and proton-neutron pairing wave functions and on the generator coordinate method (GCM). The investigations are performed in models for which an exact group theoretical solution exists. It turns out that a particle number and isospin projection is essential to yield a good approximation to the ground state or isospin yrast state energies. For strong pairing correlations (pairing force constant equal to the single-particle level distance) isospin cranking (-ωT_x) yields with particle number projected pairing wave function also good agreement with the exact energies. GCM wave functions generated by particle number and isospin projected BCS functions with different amounts of pairing correlations yield for the lowest T = 0 and T = 2 states energies which are practically indistinguishable from the exact solutions. But even the second and third lowest energies of charge-symmetric states are still very reliable. Thus we conclude that also in realistic cases isospin rotational and pairing vibrational states may be described in the framework of the GCM method with isospin and particle number projected generating wave functions.     

Unitary massless representations of conformal superalgebras

Massless unitary irreducible representations of the conformal superalgebras SU(2, 2/N) are shown to be atypical representations. The existence of such representations appears restricted to N⩽4 if the spin condition s⩽2 is imposed on the states.