Related papers: Continued Fractions and the Rabi Model

The fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the…

In the present work the Rabi Problem, involving the response of a spin 1/2 particle subjected to a magnetic field, is considered in a second quantized approach. In this concrete physical scenario, we show that the second quantization…

We are used to thinking of an operator acting once, twice, and so on. However, an operator acting integer times can be consistently analytic continued to an operator acting complex times. Applications: (s,r) diagrams and an extension of…

We present several continued fraction algorithms, each of which gives an eventually periodic expansion for every quadratic element of ${\mathbb Q}_p$ over ${\mathbb Q}$ and gives a finite expansion for every rational number. We also give,…

Classical results on Diophantine approximation, such as Roth's theorem, provide the most effective techniques for proving the transcendence of special kinds of continued fractions. Multidimensional continued fractions are a generalization…

We study the quantum Rabi model within the framework of the analytical solution developed in Phys. Rev. Lett. 107,100401 (2011). In particular, through time-dependent correlation functions, we give a quantitative criterion for classifying…

We investigate a collection of orthonormal functions that encodes information about the continued fraction expansion of real numbers. When suitably ordered these functions form a complete system of martingale differences and are a special…

Jordan Normal Forms serve as excellent representatives of conjugacy classes of matrices over closed fields. Once we knows normal forms, we can compute functions of matrices, their main invariant, etc. The situation is much more complicated…

We introduce the notion of matrices graph, defining continued fraction algorithms where the past and the future are almost independent. We provide an algorithm to convert more general algorithms into matrices graphs. We present an algorithm…

The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These…

In this paper we define a new type of continued fraction expansion for a real number $x \in I_m:=[0,m-1], m\in N_+, m\geq 2$: \[x = \frac{m^{-b_1(x)}}{\displaystyle 1+\frac{m^{-b_2(x)}}{1+\ddots}}:=[b_1(x), b_2(x), ...]_m. \] Then, we…

Multidimensional continued fractions generalize classical continued fractions with the aim of providing periodic representations of algebraic irrationalities by means of integer sequences. However, there does not exist any algorithm that…

A recently introduced numerical approach to quantum systems is analyzed. The basis of a Fock space is restricted and represented in an algebraic program. Convergence with increasing size of basis is proved and the difference between…

This paper concerns extension of the classical Lagrange theorem, on the eventual periodicity of continued fraction expansions of quadratic surds, and the versions of it found in the literature in the case of complex numbers. In this…

It is desirable that a given continued fraction algorithm is simple in the sense that the possible representations can be characterized in an easy way. In this context the so-called finite range condition plays a prominent role. We show…

We investigate the emergence of semiclassical dynamics in the quantum Rabi model using a recently developed limiting procedure that formally establishes correspondence with the semiclassical Rabi Hamiltonian [E. K. Twyeffort Irish and A. D.…

We consider series of the form $$ \frac{p}{q} +\sum_{j=2}^\infty \frac{1}{x_j}, $$ where $x_1=q$ and the integer sequence $(x_n)$ satisfies a certain non-autonomous recurrence of second order, which entails that $x_n|x_{n+1}$ for $n\geq 1$.…

We develop the geometry of Hurwitz continued fractions, a major tool in understanding the approximation properties of complex numbers by ratios of Gaussian integers. Based on a thorough study of the geometric properties of Hurwitz continued…

We give a construction of a real number that is normal to all integer bases and continued fraction normal. The computation of the first n digits of its continued fraction expansion performs in the order of n^4 mathematical operations. The…

It is widely believed that the continued fraction expansion of every irrational algebraic number $\alpha$ either is eventually periodic (and we know that this is the case if and only if $\alpha$ is a quadratic irrational), or it contains…