Generalized eigenfunctions for quantum walks via path counting approach

We consider the time-independent scattering theory for time evolution operators of one-dimensional two-state quantum walks. The scattering matrix associated with the position-dependent quantum walk naturally appears in the asymptotic behavior at the spatial infinity of generalized eigenfunctions. The asymptotic behavior of generalized eigenfunctions is a consequence of an explicit expression of the Green function associated with the free quantum walk. When the position-dependent quantum walk is a finite rank perturbation of the free quantum walk, we derive a kind of combinatorial construction of the scattering matrix by counting paths of quantum walkers. We also mention some remarks on the tunneling effect.

The Detectable Subspace for the Friedrichs Model: Applications of Toeplitz Operators and the Riesz–Nevanlinna Factorisation Theorem

We discuss how much information on a Friedrichs model operator (a finite rank perturbation of the operator of multiplication by the independent variable) can be detected from ‘measurements on the boundary’. The framework of boundary triples is used to introduce the generalised Titchmarsh–Weyl M-function and the detectable subspaces which are associated with the part of the operator which is ‘accessible from boundary measurements’. In this paper, we choose functions arising as parameters in the Friedrichs model in certain Hardy classes. This allows us to determine the detectable subspace by using the canonical Riesz–Nevanlinna factorisation of the symbol of a related Toeplitz operator.

The Detectable Subspace for the Friedrichs Model

This paper discusses how much information on a Friedrichs model operator can be detected from ‘measurements on the boundary’. We use the framework of boundary triples to introduce the generalised Titchmarsh–Weyl M-function and the detectable subspaces which are associated with the part of the operator which is ‘accessible from boundary measurements’. The Friedrichs model, a finite rank perturbation of the operator of multiplication by the independent variable, is a toy model that is used frequently in the study of perturbation problems. We view the Friedrichs model as a key example for the development of the theory of detectable subspaces, because it is sufficiently simple to allow a precise description of the structure of the detectable subspace in many cases, while still exhibiting a variety of behaviours. The results also demonstrate an interesting interplay between modern complex analysis, such as the theory of Hankel operators, and operator theory.

Canonical holomorphic sections of determinant line bundles

We investigate the analytic properties of torsion isomorphisms (determinants) of mapping cone triangles of Fredholm complexes. Our main tool is a generalization to Fredholm complexes of the perturbation isomorphisms constructed by R. Carey and J. Pincus for Fredholm operators. A perturbation isomorphism is a canonical isomorphism of determinants of homology groups associated to a finite rank perturbation of Fredholm complexes. The perturbation isomorphisms allow us to establish the invariance properties of the torsion isomorphisms under finite rank perturbations. We then show that the perturbation isomorphisms provide a holomorphic structure on the determinant lines over the space of Fredholm complexes. Finally, we establish that the torsion isomorphisms and the perturbation isomorphisms provide holomorphic sections of certain determinant line bundles.

Mesoscopic perturbations of large random matrices

We consider the eigenvalues and eigenvectors of small rank perturbations of random [Formula: see text] matrices. We allow the rank of perturbation [Formula: see text] to increase with [Formula: see text], and the only assumption is [Formula: see text]. The spiked population model, proposed by Johnstone [On the distribution of the largest eigenvalue in principal components analysis, Ann. Statist. 29(2) (2001) 295–327], is of this kind, in which all the population eigenvalues are 1’s except for a few fixed eigenvalues. Our model is more general since we allow the number of non-unit population eigenvalues to grow with the population size. In both additive and multiplicative perturbation models, we study the nonasymptotic relation between the extreme eigenvalues of the perturbed random matrix and those of the perturbation. As [Formula: see text] goes to infinity, we derive the empirical distribution of the extreme eigenvalues of the perturbed random matrix. We also compute the appropriate projection of eigenvectors corresponding to the extreme eigenvalues of the perturbed random matrix. We prove that they are approximate eigenvectors of the perturbation. Our results can be regarded as an extension of the finite rank perturbation case to the full generality up to [Formula: see text].

THE FINITE SUM OF THE PRODUCTS OF TWO TOEPLITZ OPERATORS

We consider in this paper the question of when the finite sum of products of two Toeplitz operators is a finite-rank perturbation of a single Toeplitz operator on the Hardy space over the unit disk. A necessary condition is found. As a consequence we obtain a necessary and sufficient condition for the product of three Toeplitz operators to be a finite-rank perturbation of a single Toeplitz operator.