Table of contents

We generalize the Milne quantization condition to non-Hermitian systems. In the general case the underlying nonlinear Ermakov–Milne–Pinney equation needs to be replaced by a nonlinear integral differential equation. However, when the system is PT-symmetric or/and quasi/pseudo-Hermitian the equations simplify and one may employ the original energy integral to determine its quantization. We illustrate the working of the general framework with the Swanson model and two explicit examples for pairs of supersymmetric Hamiltonians. In one case both partner Hamiltonians are Hermitian and in the other a Hermitian Hamiltonian is paired by a Darboux transformation to a non-Hermitian one.

We focus on the algebraic area probability distribution of planar random walks on a square lattice with ${m}_{1},{m}_{2},{l}_{1}$ and l₂ steps right, left, up and down. We aim, in particular, at the algebraic area generating function ${Z}_{{m}_{1},{m}_{2},{l}_{1},{l}_{2}}({\rm{Q}})$ evaluated at ${\rm{Q}}={{\rm{e}}}^{\displaystyle \frac{2{\rm{i}}\pi }{q}},$ a root of unity, when both ${m}_{1}-{m}_{2}$ and ${l}_{1}-{l}_{2}\;\mathrm{are}$ multiples of q. In the simple case of staircase walks, a geometrical interpretation of ${Z}_{m,0,l,0}({{\rm{e}}}^{\frac{2{\rm{i}}\pi }{q}})$ in terms of the cyclic sieving phenomenon is illustrated. Then, an expression for ${Z}_{{m}_{1},{m}_{2},{l}_{1},{l}_{2}}(-1),$ which is relevant to the Stembridge case, is proposed. Finally, the related problem of evaluating the nth moments of the Hofstadter Hamiltonian in the commensurate case is addressed.

We present a simple and efficient method to calculate the damping for the excitation spectrum of a uniform D-dimensional Bose gas. Starting from the original Popov's hydrodynamic description and integrating out phase variables, we obtained the effective action of amplitude fluctuations. Within this approach, the lifetime of quasi-particles with a finite momentum is calculated at a wide temperature range. It is shown that the correct use of the hydrodynamic approach leads to the damping rate, which coincides with results obtained by means of the perturbation theory.

The well known sandpile model of self-organized criticality generates avalanches of all length and time scales, without tuning any parameters. In the original models the external drive is randomly selected. Here, we investigate a drive which depends on the present state of the system, namely the effect of favoring sites with a certain height in the deposition process. If sites with height three are favored, the system stays in a critical state. Our numerical results indicate the same universality class as the original model with random deposition, although the stationary state is approached very differently. In contrast, when favoring sites with height two, only avalanches which cover the entire system occur. Furthermore, we investigate the distributions of sites with a certain height, as well as the transient processes of the different variants of the external drive.

We study percolation on self-dual hypergraphs that contain hyperedges with four bounding vertices, or 'four-edges', using three different generators, each containing bonds or sites with three distinct probabilities p, r, and t connecting the four vertices. We find explicit values of these probabilities that satisfy the self-duality conditions discussed by Bollobás and Riordan. This demonstrates that explicit solutions of the self-duality conditions can be found using generators containing bonds and sites with independent probabilities. These solutions also provide new examples of lattices where exact percolation critical points are known. One of the generators exhibits three distinct criticality solutions (p, r, t). We carry out Monte-Carlo simulations of two of the generators on two different hypergraphs to confirm the critical values. For the case of the hypergraph and uniform generator studied by Wierman et al, we also determine the threshold p = 0.441 374 ± 0.000 001, which falls within the tight bounds that they derived. Furthermore, we consider a generator in which all or none of the vertices can connect, and find a soluble inhomogeneous percolation system that interpolates between site percolation on the union-jack lattice and bond percolation on the square lattice.

The escape of particles from the phase space produced by a two-dimensional, nonlinear and area-preserving, discontinuous map is investigated by using both numerical simulations and the explicit solution of the corresponding diffusion equation. The mapping, given in action-angle variables, is parameterized by K, which controls a transition from integrability to non-integrability. We focus on the two dynamical regimes of the map: slow diffusion ($K\lt {K}_{c}$) and quasilinear diffusion ($K\gt {K}_{c}$) regimes, separated by the critical parameter value K_c = 1. When a hole is introduced in the action axis, we find the histogram of escape times ${P}_{{\rm{E}}}(n)$ and the survival probability of particles ${P}_{{\rm{S}}}(n)$ to be scaling invariant in both the slow and the quasilinear diffusion regimes, with scaling laws proportional to the corresponding diffusion coefficients, namely, proportional to ${K}^{5/2}$ and K², respectively. Our numerical simulations agree remarkably well with the analytical results obtained from the explicit solution of the diffusion equation, hence giving robustness to the escape formalism.

The general form of an integral of motion that is a polynomial of order N in the momenta is presented for a Hamiltonian system in two-dimensional Euclidean space. The classical and the quantum cases are treated separately, emphasizing both the similarities and the differences between the two. The main application will be to study Nth order superintegrable systems that allow separation of variables in the Hamilton–Jacobi and Schrödinger equations, respectively.

We analyze qubit decoherence in the framework of geometric quantum mechanics. In this framework the qubit density operators are represented by probability distributions which are also the Kähler functions on the Bloch sphere. Interestingly, the complete positivity of the quantum evolution is recovered as ellipticity of the second order differential operator (deformed Laplacian) which governs the evolution of the probability distribution.

From an algebraic construction of the mKdV hierarchy we observe that the space component of the Lax operator plays the role of a universal algebraic object. This fact induces the universality of a gauge transformation that relates two field configurations of a given member of the hierarchy. Such gauge transformation generates the Bäcklund transformation (BT). In this paper we propose a systematic construction of BT for the entire mKdV hierarchy from the known type-II BT of the sinh-Gordon theory. We explicitly construct the BT of the first few integrable models associated to positive and negative grade-time evolutions. Solutions of these transformations for several cases describing the transition from vacuum–vacuum and the vacuum to one-soliton solutions which determines the value for the auxiliary field and the Bäcklund parameter respectively, independently of the model. The same follows for the scattering of two one-soliton solutions. The resultant delay is determined by a condition independent of the model considered.

This paper reports on new aspects of the so-called V-line Radon transforms (RTs) complementing those reported in an earlier work. These new properties are nicely uncovered and described with Cartesian coordinates. In particular, we show that the V-line RT belongs to the class of RTs on curves in the plane which can be mapped onto the standard RT on straight lines and thereby are fully characterizable and invertible. Next, we show that the effect of geometric inversion on the V-line RT is to produce a new RT on a pair of supplementary circular arcs, which provides a new access to image reconstruction in the so-called Norton's modality of Compton scatter tomography, a front runner in the race for alternatives to current emission imaging.

We give new results concerning the Frobenius integrability and solution of evolution equations admitting travelling wave solutions. In particular, we give a powerful result which explains the extraordinary integrability of some of these equations. We also discuss 'local' conservations laws for evolution equations in general and demonstrate all the results for the Korteweg–de Vries equation.

In this paper a discretization of Toda equations is analyzed. The correspondence between these Δ-Toda equations for the coefficients of the Jacobi operator and its resolvent function is established. It is shown that the spectral measure of these operators evolve in t like ${(1+x)}^{1-t}\;{\rm{d}}\mu (x)$ where ${\rm{d}}\mu$ is a given positive Borel measure. The Lax pair for the Δ-Toda equations is derived and characterized in terms of linear functionals, where orthogonal polynomials which satisfy an Appell condition with respect to the forward difference operator Δ appear in a natural way. In order to illustrate the results of the paper we work out two examples of Δ-Toda equations related with Jacobi and Laguerre orthogonal polynomials.

We show that the position operator in a class of f-deformed oscillators has a fractal spectrum, homeomorphic to the Cantor set, via a unitary transformation to Harper's model. The class corresponds to a choice of ergodic operators for the deformation function. Hofstadter's butterfly is plotted by direct diagonalization of a position operator with an originally vanishing diagonal. This is equivalent to a one-dimensional hamiltonian without potential.

We investigate the tripartite relationship between the collapse and revival, the Q function splitting and the energy level structure in the Jaynes–Cummings (JC) model with an intensity-dependent level shift whose magnitude is tuned to give rise to periodic collapse and revivals. We show that this constitutes a clearer demonstration of the mechanism of Q function splitting and its relation with the collapse and revival than the standard JC model itself. The eigenstates form two groups, both of which form equidistant ladders with differing energy intervals. This structure gives rise to the periodic splitting and reunion of the Q function. Only when the reunion happens, a non-vanishing mutual interference between the two groups is possible and gives rise to observable Rabi oscillations. The possibility of observing the phenomena using a Rubidium atom in a cavity is also discussed. We believe the present work could contribute to the understanding of the collapse and revival and the Q function-splitting phenomena.

We formulate Grover's unstructured search algorithm as a chiral quantum walk, where transitioning in one direction has a phase conjugate to transitioning in the opposite direction. For small phases, this breaking of time-reversal symmetry is too small to significantly affect the evolution: the system still approximately evolves in its ground and first excited states, rotating to the marked vertex in time $\pi \sqrt{N}/2$. Increasing the phase does not change the runtime, but rather changes the support for the 2D subspace, so the system evolves in its first and second excited states, or its second and third excited states, and so forth. Apart from the critical phases corresponding to these transitions in the support, which become more frequent as the phase grows, this reveals that our model of quantum search is robust against time-reversal symmetry breaking.