Reviews in Mathematical Physics

The basic concepts of factorizable problems in one-dimensional Quantum Mechanics, as well as the theory of Shape Invariant potentials are reviewed. The relation of this last theory with a generalization of the classical Factorization Method presented by Infeld and Hull is analyzed in detail. By the use of some properties of the Riccati equation the solutions of Infeld and Hull are generalized in a simple way.

This paper studies quantum systems with a finite number of degrees of freedom in the context of non-extensive thermodynamics. A trial density matrix, obtained by heuristic methods, is proved to be the equilibrium density matrix. If the entropic index q is larger than 1 then existence of the trial equilibrium density matrix requires that q is less than some critical value q_c which depends on the rate by which the eigenvalues of the Hamiltonian diverge. Existence of a unique equilibrium density matrix is proved if in addition q < 2 holds. For q between 0 and 1, such that 2 < q + q_c, the free energy has at least one minimum in the set of trial density matrices. If a unique equilibrium density matrix exists then it is necessarily one of the trial density matrices. Note that this is a finite rank operator, which means that in equilibrium high energy levels have zero probability of occupancy.

We develop a geometric representation for the ground state of the spin-1/2 quantum XXZ ferromagnetic chain in terms of suitably weighted random walks in a two-dimensional lattice. The path integral model so obtained admits a genuine classical statistical mechanics interpretation with a translation invariant Hamiltonian. This new representation is used to study the interface ground states of the XXZ model. We prove that the probability of having a number of down spins in the up phase decays exponentially with the sum of their distances to the interface plus the square of the number of down spins. As an application of this bound, we prove that the total third component of the spin in a large interval of even length centered on the interface does not fluctuate, i.e. has zero variance. We also show how to construct a path integral representation in higher dimensions and obtain a reduction formula for the partition functions in two dimensions in terms of the partition function of the one-dimensional model.

We investigate a family of Dirichlet Laplacians on randomly dented or bulged strips in ℝ²; for this random quantum waveguide model, dense point spectrum with exponentially localized eigenfunctions near its fluctuation boundary at the bottom of the spectrum and Lifshitz asymptotics of the integrated density of states are established. For this purpose, multi-scale analysis in the quite abstract form of [21] is applied, and domain perturbations of the Laplacian are studied.

We give a unifying description of the Dirac monopole on the 2-sphere S², of a graded monopole on a (2, 2)-supersphere S^{2, 2} and of the BPST instanton on the 4-sphere S⁴, by constructing a suitable global projector p via equivariant maps. This projector determines the projective modules of finite type of sections of the corresponding vector bundle. The canonical connection ∇ = p ◦ d is used to compute the topological charge which is found to be equal to -1 for the three cases. The transposed projector q = p^t gives the value +1 for the charges; this showing that transposition of projectors, although an isomorphism in K-theory, is not the identity map. We also study the invariance under the action of suitable Lie groups.

If u_is are periodic function on the line, the operator , acting on periodic functions, is called a Adler–Gelfand–Dikii (or AGD) operator. In this paper we consider a projective connection as defined by this nth order operator on the circle. In particular, projective connection as defined by a second order operator can be identified with the dual of Virasoro algebra, and it is well known that the KdV equation as a Euler–Arnold equation in the coadjoint orbit of the Bott–Virasoro group. In this paper we study (formally) the evolution equation of the Adler–Gelfand–Dikii operator, Δ⁽ⁿ⁾, (at least for n ≤ 4), under the action of Vect(S¹). This yields a single generating equation for periodic function u. We also establish a connection between the projective vector field, a vector field leaves fixed a given (extended) projective connection, and the C. Neumann system using the idea of Knörrer and Moser. We show that certain quadratic function of a projective field satisfies C. Neumann system.