Floquet Theory For Differential Equations Research Articles

Effects of spacetime curvature on spin-1/2 particle zitterbewegung

This paper investigates the properties of spin-1/2 particle zitterbewegung in the presence of a general curved spacetime background described in terms of Fermi normal coordinates, where the spatial part is expressed using general curvilinear coordinates. Adopting the approach first introduced by Barut and Bracken for zitterbewegung in the local rest frame of the particle, it is shown that non-trivial gravitational contributions to the relative position and momentum operators appear due to the coupling of zitterbewegung frequency terms with the Ricci curvature tensor in the Fermi frame, indicating a formal violation of the weak equivalence principle. Explicit expressions for these contributions are shown for the case of quasi-circular orbital motion of a spin-1/2 particle in a Vaidya background. Formal expressions also appear for the time derivative of the Pauli–Lubanski vector due to spacetime curvature effects coupled to the zitterbewegung frequency. Also, the choice of curvilinear coordinates results in non-inertial contributions in the time evolution of the canonical momentum for the spin-1/2 particle, where zitterbewegung effects lead to stability considerations for its propagation, based on the Floquet theory of differential equations.

Positive particle frequencies in some time-periodic universes

We consider the Klein-Gordon equation in FRW-like universes which are periodic in a suitable scale of time. In a remarkable sector of solutions, the Floquet theory of differential equations allows for generalizing the notion of particle frequency. In this sense, the sesquilinear form associated with Gordon's current is positive definite when restricted to the sub-sector of positive frequencies.

Design and analysis of recursive periodically time-varying digital filters with highly quantized coefficients

A design for recursive filters with coefficients epsilon (1, 0, -1) that can change at each filter update is introduced. This flexibility of allowing the multipliers to be functions of time can be used to offset the reduction in degrees of freedom when such quantization is used. This result will be especially useful in filtering applications in which a controllable or adaptive filter is required and high-speed multipliers are not available. A general transition-matrix state-variable formulation similar to Floquet theory for differential equations is developed that can be used to design IIR (infinite-impulse response) filters characterized by a periodically time-varying feedback matrix. By defining a time-varying similarity transformation and introducing circulant feedback matrices, a significant simplification of the highly quantized time-varying state recursion equation is realized. This leads to the enumeration of the filter transfer function and to a simple eigenvalue structure that facilitates the design of filters with desired pole locations. These analytical results are verified in simulations of lowpass and bandpass filter designs. >

Doubly-periodic Floquet theory

The purpose of this paper is to develop a theory for differential equations with doubly-periodic coefficients analogous to the classic Floquet theory for differential equations with singly periodic coefficients. Unlike the classical theory the role of the exponent v of the differential equation is fundamental. If v takes integral values the analogous theory is well known and goes back to the work of Hermite in 1877. When v is rational the theory depends essentially on whether a certain number theoretic conjecture proposed by F. M. Arscott and G. P. Wright in 1969 is true. The paper resolves the conjecture and brings the doubly-periodic Floquet theory to some degree of completion.