Nondegenerate Systems Research Articles

Degenerate Landau-Zener model: Analytical solution

The exact analytical solution of the degenerate Landau-Zener model, wherein two bands of degenerate energies cross in time, is presented. The solution is derived by using the Morris-Shore transformation, which reduces the fully coupled system to a set of independent nondegenerate two-state systems and a set of decoupled states. Due to the divergence of the phase of the off-diagonal element of the propagator in the original Landau-Zener model, not all transition probabilities exist for infinite time duration. In general, apart from some special cases, only the transition probabilities between states within the same degenerate set exist, but not between states of different sets. An illustration is presented for the transition between the magnetic sublevels of two atomic levels with total angular momenta $J=2$ and 1.

Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean space preliminaries

We show that second-order superintegrable systems in two-dimensional and three-dimensional Euclidean space generate both exactly solvable (ES) and quasiexactly solvable (QES) problems in quantum mechanics via separation of variables, and demonstrate the increased insight into the structure of such problems provided by superintegrability. A principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable. Most past separation of variables treatments of QES problems via partial differential equations have only incorporated separability, not multiseparability. Also, we propose another definition of ES and QES. The quantum mechanical problem is called ES if the solution of Schrödinger equation can be expressed in terms of hypergeometric functions Fnm and is QES if the Schrödinger equation admits polynomial solutions with coefficients necessarily satisfying a three-term or higher order of recurrence relations. In three dimensions we give an example of a system that is QES in one set of separable coordinates, but is not ES in any other separable coordinates. This example encompasses Ushveridze’s tenth-order polynomial QES problem in one set of separable coordinates and also leads to a fourth-order polynomial QES problem in another separable coordinate set.

Quantum methods for clock synchronization: Beating the standard quantum limit without entanglement

We introduce methods for clock synchronization that make use of the adiabatic exchange of nondegenerate two-level quantum systems: ticking qubits. Schemes involving the exchange of $N$ independent qubits with frequency $\ensuremath{\omega}$ give a synchronization accuracy that scales as ${(\ensuremath{\omega}\sqrt{N})}^{\ensuremath{-}1}$---i.e., as the standard quantum limit. We introduce a protocol that makes use of ${N}_{c}$ coherent exchanges of a single qubit at frequency $\ensuremath{\omega}$, leading to an accuracy that scales as ${(\ensuremath{\omega}{N}_{c})}^{\ensuremath{-}1}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{N}_{c}$. This protocol beats the standard quantum limit without the use of entanglement, and we argue that this scaling is the fundamental limit for clock synchronization allowed by quantum mechanics. We analyze the performance of these protocols when used with a lossy channel.

Nondegenerate 𝑞-biresolving textile systems and expansive automorphisms of onesided full shifts

We study nondegenerate, q q -biresolving textile systems and using properties of them, we prove a conjecture of Boyle and Maass on arithmetic constraints for expansive automorphisms of onesided full shifts and positively expansive endomorphisms of mixing topological Markov shifts. A similar result is also obtained for expansive leftmost-permutive endomorphisms of onesided full shifts.

The Physical Nature and Orbital Behavior of V523 Cassiopeiae

V523 Cassiopeiae is a dwarf contact binary with one of the shortest orbital periods among the nondegenerate systems. Its orbital history is marked by large period changes. The photometrically determined mass ratios have historically been inconsistent with those calculated from radial velocity curves. In 1998 we acquired high-precision and standardized Johnson-Cousins UBV light curves. Our simultaneous BV light curve/radial velocity curve solutions provided good fits to both the light curves and the radial velocity curves using standard gravity darkening coefficients with a mass ratio of ~0.5. Seven precision mean epochs of minimum light were determined from these observations. We combine these with all available published times of minimum light, along with 50 times of low light found from a search of the archival Harvard plate stacks in the interval 1901–1942 to give us 567 eclipse timings. Our period study covers nearly 160,000 orbits, or ~102 yr. We find a high-amplitude sinusoidal variation with a period of 101 ± 7 yr, overlaid on a strong continuous period increase. The quadratic term is common in contact binaries, but a sinusoidal one is not. This suggests a hierarchical three-star system. Assuming that this is the case and that the inclination from our orbital solution for the close pair is the same as the larger orbit, we obtain a mass for the third star of 0.41 M⊙. This is similar to the masses of the stars that comprise the contact binary. If this scenario is correct, V523 Cas consists of a trio of late K and early M-type dwarfs having a total mass of ~1.6 M⊙. We show that high-resolution imaging can confirm this suggestion.

EPR Studies of Poly(o‐Phenetidine)‐Poly(Styrenesulfonic Acid) and Poly(2‐Ethylaniline)‐Poly(Styrenesulfonic Acid) Complexes

AbstractThis study relates to electron paramagnetic resonance (EPR) of the poly(o‐phenetidine)‐poly(styrenesulfonic acid) [P(O‐P)‐PSSA] and poly(2‐ethylaniline)‐poly(styrenesulfonic acid) [P(2‐E)‐PSSA] complexes; the polyaniline [PANI], poly(o‐phenetidine) [POP], poly(2‐ethylaniline) [P2E], and polyaniline‐poly(styrenesulfonic acid) [PANI‐PSSA] were prepared to be the reference samples; the EPR spectra results were recorded at both 298 K and 77 K, and were analyzed on the basis of the polaron‐bipolaron model, which is typical for non‐degenerate systems. The EPR spectra results of the substituted polyaniline complexes P(O‐P)‐PSSA and P(2‐E)‐PSSA, show temperature‐dependent paramagnetism; the narrow line‐width arises due to polarons, and the charge transport takes place through both polarons and bipolarons. The narrower line‐width of the substituted polyaniline complexes can be attributed to the lower degree of structural disorder, the oxygen absorption of the polymer molecular complexes, and the steric effect by the alkyl group of the polyaniline; the reduced line‐widths were consistent with the lower molecular weight of the complexes.

Unified description of ballistic and diffusive carrier transport in semiconductor structures

A unified theoretical description of ballistic and diffusive carrier transport in parallel-plane semiconductor structures is developed within the semiclassical model. The approach is based on the introduction of a thermoballistic current consisting of carriers which move ballistically in the electric field provided by the band edge potential, and are thermalized at certain randomly distributed equilibration points by coupling to the background of impurity atoms and carriers in equilibrium. The sum of the thermoballistic and background currents is conserved and identified with the physical current. The current-voltage characteristic for nondegenerate systems and the zero-bias conductance for degenerate systems are expressed in terms of a reduced resistance. For arbitrary mean free path and arbitrary shape of the band edge potential profile, this quantity is determined from the solution of an integral equation, which also provides the quasi-Fermi level and the thermoballistic current. To illustrate the formalism, a number of simple examples are considered explicitly. The present work is compared with previous attempts towards a unified description of ballistic and diffusive transport.

Projective systems and perfect codes with a poset metric

The notion of a projective system, defined as a set X of n-points in a projective space over a finite field, was introduced by Tsfasman and Vlǎdut. By this notion, the weight distribution of a nondegenerate linear code can be computed by the configuration of X and hyperplanes in the dual projective space of C. In this paper, we construct a projective system X defined as a set of n-linear subspaces on the dual projective space of a poset code. This gives a natural one-to-one correspondence between the set of equivalence classes of nondegenerate poset projective systems and the set of nondegenerate poset codes. By this correspondence, the weight distribution of a nondegenerate poset code can be also computed by the configuration of X and hyperplanes in the dual projective space of C. Finally, we provide an algorithm for finding perfect poset codes.

Finite energy foliations of tight three-spheres and Hamiltonian dynamics

Surfaces of sections are a classical tool in the study of 3-dimensional dynamical systems. Their use goes back to the work of Poincare and Birkhoff. In the present paper we give a natural generalization of this concept by constructing a system of transversal sections in the complement of finitely many distinguished periodic solutions. Such a system is established for nondegenerate Reeb flows on the tight 3-sphere by means of pseudoholomorphic curves. The applications cover the nondegenerate geodesic flows on T1S2 - RP3 via its double covering S3, and also nondegenerate Hamiltonian systems in IR4 restricted to sphere-like energy surfaces of contact type.

Local tracking and stability for degenerate viscous shock waves

We study the pointwise behavior of perturbed degenerate (sonic) shock waves for scalar conservation laws with nonconstant diffusion. Building on the pointwise Green's function approach of Zumbrun and Howard (Indiana U. Math. J. 47(3) (1998) 741) we extend the linear analysis to an equation with nonintegrable coefficients. In lieu of working with the integrated equation, we employ a tracking mechanism that we expect will allow degenerate waves to be incorporated into the general framework for nondegenerate systems (Indiana U. Math. J. 47(3) (1998) 741).

Closed characteristics on non-degenerate star-shaped hypersurfaces in R2n

In this paper, firstly we establish the relation theorem between the Maslov-type index and the index defined by C. Viterbo for star-shaped Hamiltonian systems. Then we extend the iteration formula ofC. Viterbo for non-degenerate star-shaped Hamiltonian systems to the general case. Finally we prove that there exist at least two geometrically distinct closed characteristics on any non-degenerate star-shaped compact smooth hypersurface on R2n with n > 1. Here we call a hypersurface non-degenerate, if all the closed characteristics on the given hypersurface together with all of their iterations are non-degenerate as periodic solutions of the corresponding Hamiltonian system. We also study the ellipticity of closed characteristics whenn = 2

Pseudo-supersymmetric quantum mechanics and isospectral pseudo-Hermitian Hamiltonians

We examine the properties and consequences of pseudo-supersymmetry for quantum systems admitting a pseudo-Hermitian Hamiltonian. We explore the Witten index of pseudo-supersymmetry and show that every pair of diagonalizable (not necessarily Hermitian) Hamiltonians with discrete spectra and real or complex-conjugate pairs of eigenvalues are isospectral and have identical degeneracy structure except perhaps for the zero eigenvalue if and only if they are pseudo-supersymmetric partners. This implies that pseudo-supersymmetry is the basic framework for generating non-Hermitian PT-symmetric and non- PT-symmetric Hamiltonians with a real spectrum via a Darboux transformation, and shows that every diagonalizable Hamiltonian H with a discrete spectrum and real or complex-conjugate pairs of eigenvalues may be factored as H= L ♯ L where L is a linear operator with pseudo-adjoint L ♯. In particular, this factorization applies to PT-symmetric and Hermitian Hamiltonians. The non-degenerate two-level systems provide a class of Hamiltonians that are pseudo-Hermitian. We demonstrate the implications of our general results for this class in some detail.

Adiabatic approximation in the density matrix approach: non-degenerate systems

We study the adiabatic limit in the density matrix approach for a quantum system coupled to a weakly dissipative medium. The energy spectrum of the quantum model is supposed to be non-degenerate. In the absence of dissipation, the geometric phases for periodic Hamiltonians obtained previously by Berry are recovered in the present approach. We determine the necessary condition satisfied by the coefficients of the linear expansion of the non-unitary part of the Liouvillian in order that the imaginary phases acquired by the elements of the density matrix, due to dissipative effects, be geometric. The results derived are model independent. We apply them to spin 1 2 model coupled to reservoir at thermodynamic equilibrium.

On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case

We prove a result of existence and uniqueness of solutions to forward–backward stochastic differential equations, with non-degeneracy of the diffusion matrix and boundedness of the coefficients as functions of x as main assumptions. This result is proved in two steps. The first part studies the problem of existence and uniqueness over a small enough time duration, whereas the second one explains, by using the connection with quasi-linear parabolic system of PDEs, how we can deduce, from this local result, the existence and uniqueness of a solution over an arbitrarily prescribed time duration. Improving this method, we obtain a result of existence and uniqueness of classical solutions to non-degenerate quasi-linear parabolic systems of PDEs. This approach relaxes the regularity assumptions required on the coefficients by the Four-Step scheme.

Independent N-Electron Method and High Accuracy Calculations of the Ground State Energy of the Nonrelativistic Light Atoms (3 ≤ Ne ≤ 10)**The project partially supported by the Grant LWTZ-1298 of the Chinese Academy of Sciences

A new method based on the independent electron-pair approximation is developed to calculate the correlation energy and the basis size truncation error. The formalisms for the nondegenerate systems are described in this paper. Because all important terms are very efficiently selected and the calculations can be separated into several parts, this new method gives very accurate results and does not require expensive computing facilities. As a test, the ground state energies (GSE) of the nonrelativistic atoms for (3 ≤ Ne ≤ 10) are calculated using this new method. The calculated results are found to be in close agreement with the most recent available version of the exact GSE which is derived from the experimental data after subtracting the contributions from the relativistic effect, QED radiative corrections, finite nucleon mass corrections, and the finite radius of the nuclear size. Those accurate results have not been reproduced for Ne ≥ 5 by any direct theoretical calculations in literatures. Since this method is not restrained to nonrelativistic or Coulomb systems, it should become powerful to study other quantum interacting systems.

GROUPS OF TYPE E 7 OVER ARBITRARY FIELDS

Freudenthal triple systems come in two flavors, degenerate and nondegenerate. The best criterion for distinguishing between the two which is available in the literature is by descent. We provide an identity which is satisfied only by nondegenerate triple systems. We then use this to define algebraic structures whose automorphism groups produce all adjoint algebraic groups of type E 7 over an arbitrary field of characteristic ≠ 2, 3. The main advantage of these new structures is that they incorporate a previously unconsidered invariant (a symplectic involution) of these groups in a fundamental way. As an application, we give a construction of adjoint groups with Tits algebras of index 2 which provides a complete description of this involution and apply this to groups of type E 7 over a real-closed field.

Temporally stable coherent states in energy-degenerate systems: The hydrogen atom

Klauder's recent generalization of the harmonic oscillator coherent states [J. Phys. A 29, L293 (1996)] is applicable only in non-degenerate systems, requiring some additional structure if applied to systems with degeneracies. The author suggests how this structure could be added, and applies the complete method to the hydrogen atom problem. To illustrate how a certain degree of freedom in the construction may be exercised, states are constructed which are initially localized and evolve semi-classically, and whose long time evolution exhibits "fractional revivals."

An Extension of the Bolsinov-Fomenko Theorem on Orbital Classification of Integrable Hamiltonian Systems

The main result of the paper is an extension of the Bolsinov-Fomenko theorem on topological orbital classification of nondegenerate integrable Hamiltonian systems with two degrees of freedom on three-dimensional constant energy manifolds (1994). Namely, it is shown that their restriction that the integral has no critical circles with nonorientable separatrix diagrams can be omitted. Our proof is based on an analogue of obstruction theory for certain types of Seifert fibrations.

On integrable coupled KdV-type systems

In this paper we describe a new method for constructing integrable systems of differential equations. We are looking for systems in two variables in such forms that the reduction v = u leads us to a single equation in u. We give a complete classification of such systems that reduce to Korteweg-de Vries-type equations. Furthermore, we present an extensive (and complete for the systems of the Sawada-Kotera and Kaup-Kupershmidt types) classification of fifth-order equations in the same weighting. We show that the scalar integrable equations give rise to large classes of integrable systems. Moreover, we present a previously unknown example of a system that can be written in biHamiltonian form in infinitely many different ways, thereby solving the problem of the number of biHamiltonian forms that can have a differential equation. Finally, we present examples of nondegenerate systems possessing degenerate symmetries, which is impossible in the scalar case.

The open path phase for degenerate and non-degenerate systems and its relation to the wave-function modulus

We calculate the open path phase in a two state model with a slowly (nearly adiabatically) varying time-periodic Hamiltonian and trace its continuous development during a period. We show that the topological (Berry) phase attains π or 2π depending on whether there is or is not a degeneracy in the part of the parameter space enclosed by the trajectory. Oscillations are found in the phase. As adiabaticity is approached, these become both more frequent and less pronounced and the phase jump becomes increasingly more steep. Integral relations between the phase and the amplitude modulus (having the form of Kramers-Kronig relations, but in the time domain) are used as an alternative way to calculate open path phases. These relations attest to the observable nature of the open path phase.