L. Velázquez

Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle

It is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five-diagonal matrices depending on the Schur parameters. This result is achieved through the study of orthogonal Laurent polynomials on the unit circle. More precisely, it is a consequence of the five term recurrence relation obtained for these orthogonal Laurent polynomials, and the one to one correspondence established between them and the orthogonal polynomials on the unit circle. As an application, some results relating the behaviour of the zeros of orthogonal polynomials and the location of Schur parameters are obtained.

The CGMV method for quantum walks

We review the main aspects of a recent approach to quantum walks, the CGMV method. This method proceeds by reducing the unitary evolution to canonical form, given by the so-called CMV matrices, which act as a link to the theory of orthogonal polynomials on the unit circle. This connection allows one to obtain results for quantum walks which are hard to tackle with other methods. Behind the above connections lies the discovery of a new quantum dynamical interpretation for well known mathematical tools in complex analysis. Among the standard examples which will illustrate the CGMV method are the famous Hadamard and Grover models, but we will go further showing that CGMV can deal even with non-translation invariant quantum walks. CGMV is not only a useful technique to study quantum walks, but also a method to construct quantum walks à la carte. Following this idea, a few more examples illustrate the versatility of the method. In particular, a quantum walk based on a construction of a measure on the unit circle due to F. Riesz will point out possible non-standard behaviours in quantum walks.

Matrix orthogonal polynomials whose derivatives are also orthogonal

In this paper we prove some characterizations of the matrix orthogonal polynomials whose derivatives are also orthogonal, which generalize other known ones in the scalar case. In particular, we prove that the corresponding orthogonality matrix functional is characterized by a Pearson-type equation with two matrix polynomials of degree not greater than 2 and 1. The proofs are given for a general sequence of matrix orthogonal polynomials, not necessarily associated with a hermitian functional. We give several examples of non-diagonalizable positive definite weight matrices satisfying a Pearson-type equation, which show that the previous results are non-trivial even in the positive definite case. A detailed analysis is made for the class of matrix functionals which satisfy a Pearson-type equation whose polynomial of degree not greater than 2 is scalar. We characterize the Pearson-type equations of this kind that yield a sequence of matrix orthogonal polynomials, and we prove that these matrix orthogonal polynomials satisfy a second order differential equation even in the non-hermitian case. Finally, we prove and improve a conjecture of Duran and Grunbaum concerning the triviality of this class in the positive definite case, while some examples show the non-triviality for hermitian functionals which are not positive definite.

Bulk-edge correspondence of one-dimensional quantum walks

We outline a theory of symmetry protected topological phases of one-dimensional quantum walks. We assume spectral gaps around the symmetry-distinguished points +1 and -1, in which only discrete eigenvalues are allowed. The phase classification by integer or binary indices extends the classification known for translation invariant systems in terms of their band structure. However, our theory requires no translation invariance whatsoever, and the indices we define in this general setting are invariant under arbitrary symmetric local perturbations, even those that cannot be continuously contracted to the identity. More precisely we define two indices for every walk, characterizing the behavior far to the right and far to the left, respectively. Their sum is a lower bound on the number of eigenstates at +1 and -1. For a translation invariant system the indices add up to zero, so one of them already characterizes the phase. By joining two bulk phases with different indices we get a walk in which the right and left indices no longer cancel, so the theory predicts bound states at +1 or -1. This is a rigorous statement of bulk-edge correspondence. The results also apply to the Hamiltonian case with a single gap at zero.

A pseudoclassical model for the massive Dirac particle in d dimensions

Abstract A new pseudoclassical model is proposed for the description of the relativistic massive Dirac particle. It is P - and T -noninvariant in the case of odd space-time dimensions. The quantization of the model leads exactly to the corresponding d -dimensional Dirac equation for arbitrary d , conserving its P - and T -noninvariance at the quantum level for the case of odd d .

Electromagnetic interaction of anyons

Abstract A U(1) gauge theory of a particle with arbitrary spin in three spacetime dimensions is introduced. All the spin dependent effects are a consequence of a Chern-Simons field which is coupled to a conserved current with a piece involving the U(1) gauge field. In the case of a spin- 1 2 particle one reproduces all the results of the spinning particle in the presence of an electromagnetic field.

ELECTROMAGNETIC INTERACTION OF ANYONS IN NONRELATIVISTIC QUANTUM FIELD THEORY

The minimal (reduced) and extended canonical formulations for (2+1)-dimensional fractional spin particles are considered. We investigate the relationship between them, clearing up the meaning of the coordinates for such particles, and analyse the related question of correlation between spin and momentum. The classical lagrangian corresponding to the extended canonical formulation is constructed, and its gauge symmetries are identified.The nonrelativistic quantum-field-theoretic Lagrangian which describes an anyon system in the presence of an electromagnetic field is identified. A nonminimal magnetic coupling to the Chern–Simons statistical field as well as to the electromagnetic field together with a direct coupling between both fields are the nontrivial ingredients of the Lagrangian obtained from the nonrelativistic limit of the fermionic relativistic formulation. The results, an electromagnetic gyromagnetic ratio 2 for any spin together with a nontrivial dynamical spin-dependent contact interaction between anyons as well as the spin dependence of the electromagnetic effective action, agree with the quantum-mechanical formulation.A new formulation of fermions based on a second order action is proposed. An analysis of the

A Quantum Dynamical Approach to Matrix Khrushchev's Formulas

U(1)

Direct and inverse polynomial perturbations of hermitian linear functionals

anomaly allows us to test the validity of the formalism at the quantum level. This formulation gives a new perpective to the introduction of parity non-invariant interactions.

The Topological Classification of One-Dimensional Symmetric Quantum Walks

Khrushchevs formula is the cornerstone of the so called Khrushchev theory, a body of results which has revolutionized the theory of orthogonal polynomials on the unit circle. This formula can be understood as a factorization of the Schur function for an orthogonal polynomial modification of a measure on the unit circle. No such formula is known in the case of matrix-valued measures. This constitutes the main obstacle to generalize Khrushchev theory to the matrix-valued setting which we overcome in this paper. It was recently discovered that orthogonal polynomials on the unit circle and their matrix-valued versions play a significant role in the study of quantum walks, the quantum mechanical analogue of random walks. In particular, Schur functions turn out to be the mathematical tool which best codify the return properties of a discrete time quantum system, a topic in which Khrushchevs formula has profound and surprising implications. We will show that this connection between Schur functions and quantum walks is behind a simple proof of Khrushchevs formula via ‘quantum’ diagrammatic techniques for CMV matrices. This does not merely give a quantum meaning to a known mathematical result, since the diagrammatic proof also works for matrix-valued measures. Actually, this path-counting approach is so fruitful that it provides different matrix generalizations of Khrushchevs formula, some of them new even in the case of scalar measures. Furthermore, the path-counting approach allows us to identify the properties of CMV matrices which are responsible for Khrushchevs formula. On the one hand, this helps to formalize and unify the diagrammatic proofs using simple operator theory tools. On the other hand, this is the origin of our main result which extends Khrushchevs formula beyond the CMV case, as a factorization rule for Schur functions related to general unitary operators.