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35
The Ten Martini Problem
"... Abstract. We prove the conjecture (known as the \Ten Martini Problem " after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all nonzero values of the coupling and all irrational frequencies. 1. ..."
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Abstract. We prove the conjecture (known as the \Ten Martini Problem " after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all nonzero values of the coupling and all irrational frequencies. 1.
Limit theorems in free probability theory. I
 I ARXIV:MATH. OA/0602219 V
, 2006
"... 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 nonidentically distributed random variables in classical Probability Theory. ..."
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Cited by 22 (5 self)
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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 nonidentically distributed random variables in classical Probability Theory.
Stollmann: Eigenfunction expansion for Schrödinger operators on metric graphs (Preprint arXiv:0801.1376
"... 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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Cited by 13 (8 self)
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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.
Orthogonal matrix polynomials, scalar type Rodrigues’ formulas and Pearson equations
 J. Approx. Theory
"... 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 ..."
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Cited by 8 (1 self)
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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 noncommutative version of it. 1
Integral Representations in Conuclear Cones
, 1994
"... this article. Wellcapped cones are conuclear. The main tool is Choquet's notion of conical measure, of which we present the necessary properties here. ..."
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this article. Wellcapped cones are conuclear. The main tool is Choquet's notion of conical measure, of which we present the necessary properties here.
Geyler: Berry phase in magnetic systems with point perturbations
, 2000
"... We study a twodimensional charged particle interacting with a magnetic field, in general nonhomogeneous, 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 pha ..."
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We study a twodimensional charged particle interacting with a magnetic field, in general nonhomogeneous, 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 zerorange 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
Realization of the annihilation operator for generalized oscillatorlike system by a differential operator
, 2001
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Properties of matrix orthogonal polynomials via their RiemannHilbert characterization
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Inverse Spectral Problems for Tridiagonal N by N Complex Hamiltonians ⋆
, 2008
"... 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 o ..."
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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