Table of contents

It is shown that the celebrated Heun operator ${H}_{e}=-({a}_{0}{x}^{3}+{a}_{1}{x}^{2}+{a}_{2}x)\tfrac{{{\rm{d}}}^{2}}{{{\rm{d}}x}^{2}}$ + $({b}_{0}{x}^{2}+{b}_{1}x+{b}_{2})\tfrac{{\rm{d}}}{{\rm{d}}x}+{c}_{0}x$ is the Hamiltonian of the ${sl}(2,R)$-quantum Euler–Arnold top of spin ν in a constant magnetic field. For ${a}_{0}\ne 0$ it is canonically equivalent to ${{BC}}_{1}({A}_{1})-$ Calogero–Moser–Sutherland quantum models; if ${a}_{0}=0$, ten known one-dimensional quasi-exactly-solvable problems are reproduced, and if, in addition, ${b}_{0}={c}_{0}=0$, then four well-known one-dimensional quantal exactly-solvable problems are reproduced. If spin ν of the top takes a (half)-integer value the Hamiltonian possesses a finite-dimensional invariant subspace and a number of polynomial eigenfunctions occur. Discrete systems on uniform and exponential lattices are introduced which are canonically equivalent to the one described by the Heun operator.

We study diffusion on comb lattices of arbitrary dimension. Relying on the loopless structure of these lattices and using first-passage properties, we obtain exact and explicit formulae for the Laplace transforms of the propagators associated to nearest-neighbour random walks in both cases where either the first or the last point of the random walk is on the backbone of the lattice, and where the two extremities are arbitrarily chosen. As an application, we compute the mean-square displacement of a random walker on a comb of arbitrary dimension. We also propose an alternative and consistent approach of the problem using a master equation description, and obtain simple and generic expressions of the propagators. This method is more general and is extended to study the propagators of random walks on more complex comb-like structures. In particular, we study the case of a two-dimensional comb lattice with teeth of finite length.

We derive the exact n-point current expectation values in the Landauer–Büttiker non-equilibrium steady state of a multi terminal system with star graph geometry and a point-like defect localised in the vertex. The current cumulants are extracted from the connected correlation functions and the cumulant generating function is established. We determine the moments, show that the associated moment problem has a unique solution and reconstruct explicitly the corresponding probability distribution. The basic building blocks of this distribution are the probabilities of particle emission and absorption from the heat reservoirs, driving the system away from equilibrium. We derive and analyse in detail these probabilities, showing that they fully describe the quantum transport problem in the system.

The condition for distinguishability of a countably infinite number of pure states by a single measurement is given. Distinguishability is to be understood as the possibility of an unambiguous measurement. For a finite number of states, it is known that the necessary and sufficient condition of distinguishability is that the states are linearly independent. For an infinite number of states, several natural classes of distinguishability can be defined. We give a necessary and sufficient condition for a system of pure states to be distinguishable. It turns out that each level of distinguishability naturally corresponds to one of the generalizations of linear independence to families of infinite vectors. As an important example, we apply the general theory to von Neumann's lattice, a subsystem of coherent states which corresponds to a lattice in the classical phase space. We prove that the condition for distinguishability is that the area of the fundamental region of the lattice is greater than the Planck constant, and also find subtle behavior on the threshold. These facts reveal the measurement theoretical meaning of the Planck constant and give a justification for the interpretation that it is the smallest unit of area in the phase space. The cases of uncountably many states and of mixed states are also discussed.

We study a family of integrable systems of nonlinearly coupled harmonic oscillators on the classical and quantum levels. We show that the integrability of these systems follows from their symmetry characterized by algebras, here called Kummer shape algebras. The resolution of identity for a wide class of reproducing kernels is found. A number of examples, illustrating this theory, are also presented.

We discuss the use of a class of exact finite energy solutions to the vacuum source-free Maxwell equations as models for multi- and single cycle laser pulses in classical interaction with relativistic charged point particles. These compact solutions are classified in terms of their chiral content and their influence on particular charge configurations in space. The results of such classical interactions motivate a phenomenological quantum description of a propagating laser pulse in a medium in terms of an effective quantum Hamiltonian.

We consider a two-component ideal Fermi gas in an isotropic harmonic potential. Some eigenstates have a wavefunction that vanishes when two distinguishable fermions are at the same location, and would be unaffected by s-wave contact interactions between the two components. We determine the other, interaction-sensitive eigenstates, using a Faddeev ansatz. This problem is nontrivial, due to degeneracies and to the existence of unphysical Faddeev solutions. As an application we present a new conjecture for the fourth-order cluster or virial coefficient of the unitary Fermi gas, in good agreement with the numerical results of Blume and coworkers.