Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.

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Some families of orthogonal matrix polynomials satisfying second order differential equations with coefficients independent of n have recently been introduced (see [DG1]). An important difference with the scalar classical families of Jacobi, Laguerre and Hermite, is that these matrix families do not satisfy scalar type Rodrigues ’ formulas of the type (Φ n W) (n) W −1, where Φ is a matrix polynomial of degree not bigger than 2. An example of a modified Rodrigues ’ formula, well suited to the matrix case, appears in [DG1]. In this note, we discuss some of the reasons why a second order differential equation with coefficients independent of n does not imply, in the matrix case, a scalar type Rodrigues ’ formula and show that scalar type Rodrigues ’ formulas are most likely not going to play in the matrix valued case the important role they played in the scalar valued case. We also mention the roles of a scalar type Pearson equation as well as that of a non-commutative version of it. 1

We study a two-dimensional charged particle interacting with a magnetic field, in general non-homogeneous, perpendicular to the plane, a confining potential, and a point interaction. If the latter moves adiabatically along a loop the state corresponding to an isolated eigenvalue acquires a Berry phase. We derive an expression for it and evaluate it in several examples such as a homogeneous field, a magnetic whisker, a particle confined at a ring or in quantum dots, a parabolic and a zero-range one. We also discuss the behavior of the lowest Landau level in this setting obtaining an explicit example of the Wilczek–Zee phase for an infinitely degenerated eigenvalue. 1

this article. Well-capped cones are conuclear. The main tool is Choquet's notion of conical measure, of which we present the necessary properties here.

Abstract. Based on a new analytical approach to the definition of additive free convolution on probability measures on the real line we prove free analogs of limit theorems for sums for non-identically distributed random variables in classical Probability Theory. 1.

this paper is to give a survey of results and questions concerning the problem of extending a continuous positive definite function from an open neighborhood of the neutral element of a Lie group to the whole group retaining positive definiteness. Whereas most positive results concern abelian groups, there are methods which carry over to non-commutative groups and which may prove useful to treat the extension problem in this case. 1. Definitions and Historical Remarks

this paper is to give a survey of results and questions concerning the problem of extending a continuous positive definite function from an open neighborhood of the neutral element of a Lie group to the whole group retaining positive definiteness. Whereas most positive results concern abelian groups, there are methods which carry over to non-commutative groups and which may prove useful to treat the extension problem in this case.

Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.

doi:10.3842/SIGMA.2009.018 Abstract. In this paper, the concept of generalized spectral function is introduced for finiteorder tridiagonal symmetric matrices (Jacobi matrices) with complex entries. The structure of the generalized spectral function is described in terms of spectral data consisting of the eigenvalues and normalizing numbers of the matrix. The inverse problems from generalized spectral function as well as from spectral data are investigated. In this way, a procedure for construction of complex tridiagonal matrices having real eigenvalues is obtained. Key words: Jacobi matrix; difference equation; generalized spectral function; spectral data