Table of contents

We propose new algebraic invariants that distinguish and classify entangled states. Considering qubits as well as higher spin systems, we obtained complete entanglement classifications for cases that were either unsolved or only conjectured in the literature.

The development in the study of supersymmetric many-particle quantum systems with inverse-square interactions is reviewed. The main emphasis is on quantum systems with dynamical OSp(2|2) supersymmetry. Several results related to the exactly solved supersymmetric rational Calogero model, including shape invariance, equivalence to a system of free superoscillators and non-uniqueness in the construction of the Hamiltonian, are presented in some detail. This review also includes a formulation of pseudo-Hermitian supersymmetric quantum systems with a special emphasis on the rational Calogero model. There are quite a few number of many-particle quantum systems with inverse-square interactions which are not exactly solved for a complete set of states in spite of the construction of infinitely many exact eigenfunctions and eigenvalues. The Calogero–Marchioro model with dynamical SU(1, 1|2) supersymmetry and a quantum system related to the short-range Dyson model belong to this class and certain aspects of these models are reviewed. Several other related and important developments are briefly summarized.

We derive the first-passage-time statistics of a Brownian motion driven by an exponential time-dependent drift up to a threshold. This process corresponds to the signal integration in a simple neuronal model supplemented with an adaptation-like current and reaching the threshold for the first time represents the condition for declaring a spike. Based on the backward Fokker–Planck formulation, we consider the survival probability of this process in a domain restricted by an absorbent boundary. The solution is given as an expansion in terms of the intensity of the time-dependent drift, which results in an infinite set of recurrence equations. We explicitly obtain the complete solution by solving each term in the expansion in a recursive scheme. From the survival probability, we evaluate the first-passage-time statistics, which itself preserves the series structure. We then compare theoretical results with data extracted from numerical simulations of the associated dynamical system, and show that the analytical description is appropriate whenever the series is truncated in an adequate order.

We present exact stationary distribution of the asymmetric simple exclusion process with Langmuir kinetics and its related model on a ring. The distribution is expressed in simple factorized forms, that is supported by detailed balance conditions satisfied in the attachment and detachment transitions. Using the equations, we derive the density of particles, the current and the variance of particle numbers in each lane. The expressions for the density are well associated with the ones found in equilibrium theories. Moreover, a simple relation between the variance of particle number and the current is proved

We consider a partially directed walk model in three dimensions to study the problem of a homopolymer adsorbing on an inhomogeneous surface while subject to a force applied at the terminal monomer. The force has components in the surface parallel and perpendicular to the direction of the inhomogeneity as well as a component perpendicular to the surface. Depending on the relative values of these components, the force can either enhance the adsorption transition or lead to desorption of an adsorbed polymer. We solve the problem exactly and also discuss a simple approximate treatment of this problem, applicable at low temperatures. The approximation is exact at zero temperature and models the exact results extremely well for low temperatures. In addition, the treatment leads to an improved physical understanding of the shapes of the force–temperature curves at low temperatures.

The Ising model on a class of infinite random trees is defined as a thermodynamic limit of finite systems. A detailed description of the corresponding distribution of infinite spin configurations is given. As an application, we study the magnetization properties of such systems and prove that they exhibit no spontaneous magnetization. Furthermore, the values of the Hausdorff and spectral dimensions of the underlying trees are calculated and found to be, respectively, $\bar{d}_h=2$ and $\bar{d}_s=4/3$.

We apply the potential group method to a family of n-dimensional quantum Smorodinsky–Winternitz systems. The Hamiltonians of the systems are associated with first-order Casimir operators of the unitary group U(3n) restricted to certain subspaces of carrier space of the symmetric representation. Hence, the group U(3n) describes fixed energy states of a family of Smorodinsky–Winternitz systems with different potential strength. Moreover, it is shown that 2n − 1 integrals of motions (including the Hamiltonian) are related to Casimir operators of U(3n) and its subgroups.

Superposition rules form a class of functions that describe general solutions of systems of first-order ordinary differential equations in terms of generic families of particular solutions and certain constants. In this work, we extend this notion and other related ones to systems of higher order differential equations and analyse their properties. Several results concerning the existence of various types of superposition rules for higher order systems are proved and illustrated with examples extracted from the physics and mathematics literature. In particular, two new superposition rules for the second- and third-order Kummer–Schwarz equations are derived.

We prove an uncertainty relation for energy and arrival time, where the arrival of a particle at a detector is modeled by an absorbing term added to the Hamiltonian. In this well-known scheme the probability for the particle's arrival at the counter is identified with the loss of normalization for an initial wave packet. Under the sole assumption that the absorbing term vanishes on the initial wavefunction, we show that $\Delta T\Delta E\ge \sqrt{p}\hbar /2$ and $\langle T\rangle \Delta E\ge 1.37\sqrt{p}\hbar$, where 〈T〉 denotes the mean arrival time and p is the probability for the particle to be eventually absorbed. Nearly minimal uncertainty can be achieved in a two-level system, and we propose a trapped ion experiment to realize this situation.

The expansion of wavefunction into Sturmian functions is used to compute nonrelativistic energies for the Yukawa and Hulthén potentials. Using nonrelativistic solutions, the first-order relativistic corrections for bound states with n ⩽ 4 Dirac particles bounded in these potentials are calculated in the framework of direct perturbation theory. The dependence of the relativistic energy shifts and the fine structure splittings on the parameters of the potentials for several states are examined.

Using a complex time method with the formalism of Stokes lines, we establish a generalization of the Davis–Dykhne–Pechukas formula which gives in the adiabatic limit the transition probability of a lossy two-state system driven by an external frequency-chirped pulse-shaped field. The conditions that allow this generalization are derived. We illustrate the result with the dissipative Allen–Eberly and Rosen–Zener models.

We provide a classification of entangled states that uses new discrete entanglement invariants. The invariants are defined by algebraic properties of linear maps associated with the states. We prove a theorem on a correspondence between the invariants and sets of equivalent classes of entangled states. The new method works for an arbitrary finite number of finite-dimensional state subspaces. As an application of the method, we considered a large selection of cases of three subspaces of various dimensions. We also obtain an entanglement classification of four qubits, where we find 27 fundamental sets of classes.

The study of a particle with position-dependent effective mass (pdem), within a double heterojunction is extended into the complex domain—when the region within the heterojunctions is described by a non-Hermitian ${{{\mathcal P}{\mathcal T}}}$-symmetric potential. After obtaining the exact analytical solutions, the reflection and transmission coefficients are calculated and plotted as a function of the energy. It is observed that at least two of the characteristic features of non-Hermitian ${{{\mathcal P}{\mathcal T}}}$-symmetric systems—namely left/right asymmetry and anomalous behaviour at spectral singularity, are preserved even in the presence of pdem. The possibility of charge conservation is also discussed.

We specify for a general correlation scenario a particular type of local quasi hidden variable (LqHV) model (Loubenets 2012 J. Math. Phys.53 022201)—a deterministic LqHV model, where all joint probability distributions of a correlation scenario are simulated via a single measure space with a normalized bounded real-valued measure not being necessarily positive and random variables, each depending only on a setting of the corresponding measurement at the corresponding site. We prove that an arbitrary multipartite correlation scenario admits a deterministic LqHV model if and only if all its joint probability distributions satisfy the consistency condition, constituting the general nonsignaling condition formulated in Loubenets (2008 J. Phys. A: Math. Theor.41 445303). This mathematical result specifies a new probability model that has a measure-theoretic structure resembling the structure of the classical probability model but incorporates the latter only as a particular case. The local version of this quasi-classical probability model covers the probabilistic description of each nonsignaling correlation scenario, in particular, each correlation scenario on a multipartite quantum state.

We investigate the semi-classical dynamics of massless Dirac fermions in 2+1 dimensions in the presence of external electromagnetic fields. By generalizing the α matrices by two generators of the SU(2) group in the (2S + 1)-dimensional representation and doing a certain scaling, we formulate an S → ∞ limit where the orbital and the spinor degrees become classical. We solve for the classical trajectories for a free particle on a cylinder and a particle in a constant magnetic field. We compare the semi-classical spectrum, obtained by Bohr–Sommerfeld quantization with the exact quantum spectrum for low values of S. For the free particle, the semi-classical spectrum is exact. For the particle in a constant magnetic field, the semi-classical spectrum reproduces all the qualitative features of the exact quantum spectrum at all S. The quantitative fit for S = 1/2 is reasonably good.

For Klein–Gordon and Dirac waves representing massive quantum particles, the local group velocity v (weak value of the velocity operator) can exceed c. If the waves consist of superpositions of many plane waves, with different (but subluminal) group velocities u, the superluminal probability P_super, i.e. that |v| > c for a randomly selected state, can be calculated explicitly. P_super depends on two parameters describing the distribution (power spectrum) of u in the superpositions, and lies between 0 and 1/2 for Klein–Gordon waves and 1–1/$\sqrt{2}$ and 1/2 for Dirac waves. Numerical simulations display the superluminal intervals in space and regions in spacetime, and support the theoretical predictions for P_super.