Weyl Disk Research Articles

Square dA-integrable solutions of differential systems with measures on the semiaxis

This paper extends the Weyl-Titchmarsh theory to the case of generalized differential systems. It is that proved the characteristic matrix of the resolvent kernel of a differential system with measures belongs to a locus, which is a matrix analog of the Weyl disk. It is established that such matrix disks are nested and converge to the limiting disk or point depending on the limiting behavior of the radii. This limiting set plays an important role in discussing of the number of solutions to such system that are square dA-integrated on the semiaxis.

Holonomic quantum manipulation in the Weyl disk

It has been shown that a Weyl point in a superconducting nanostructure may give rise to a Weyl disk where two quantum states are almost degenerate in a two-dimensional manifold in the parametric space. This opens up the possibility of a holonomic quantum manipulation: A transformation of the wave function upon an adiabatic change of the parameters within the degenerate manifold. In this paper, we investigate in detail the opportunities for holonomic manipulation in Weyl disks. We compute the connection at the manifold in quasiclassical approximation to show it is Abelian and can be used for a phase gate. To provide a closed example of quantum manipulation that includes a state preparation and readout, we augment the holonomic gate with a change of parameters that brings the system out of the degenerate subspace. For numerical illustrations, we use a finite value of quasiclassical parameter and exact quantum dynamics. We investigate the fidelity of an example gate for different execution times. We evaluate the decoherence rate and show it can be made small to ensure a wide frequency range where an adiabatic manipulation remains coherent.

Titchmarsh–Weyl theory for vector-valued discrete Schrödinger operators

We develop the Titchmarsh–Weyl theory for vector-valued discrete Schrodinger operators. We show that the Weyl m functions associated with these operators are matrix valued Herglotz functions that map complex upper half plane to the Siegel upper half space. We discuss about the Weyl disk and Weyl circle corresponding to these operators by defining these functions on a bounded interval. We also discuss the geometric properties of Weyl disk and find the center and radius of the Weyl disk explicitly in terms of matrices.

Weyl disks: Theoretical prediction

A variety of quantum systems exhibits Weyl points in their spectra where two bands cross in a point of three-dimensional parameters space with conical dispersion in the vicinity of the point. We consider theoretically the soft constraint regime where the parameters are dynamical quantum variables. We have shown that in general the soft constraints, in quasi-classical limit, result in Weyl discs where two states are (almost) degenerate in a finite 2d region of the 3d parameter space. We provide concrete calculations for two setups: Weyl point in a four-terminal superconducting structure and a Weyl exciton, i.e., a bound state of Weyl electron and a massive hole.

Time scale symplectic systems with analytic dependence on spectral parameter

This paper is devoted to the study of time scale symplectic systems with polynomial and analytic dependence on the complex spectral parameter . We derive fundamental properties of these systems (including the Lagrange identity) and discuss their connection with systems known in the literature, in particular with linear Hamiltonian systems. In analogy with the linear dependence on , we present a construction of the Weyl disks and determine the number of linearly independent square integrable solutions. These results extend the discrete time theory considered recently by the authors. To our knowledge, in the continuous time case this concept is new. We also establish the invariance of the limit circle case for a special quadratic dependence on and its extension to two (generally nonsymplectic) time scale systems, which yields new results also in the discrete case. The theory is illustrated by several examples.

Limit point and limit circle classification for symplectic systems on time scales

In this paper we study the limit point and limit circle classification for symplectic systems on time scales, which depend linearly on the spectral parameter. In a broader context, we develop a unified Weyl–Titchmarsh theory for continuous and discrete linear Hamiltonian and symplectic systems. Both separated and coupled boundary conditions are allowed. Our results include the study of the Weyl disks and circles and their limiting behavior, as well as a precise analysis of the number of linearly independent square integrable solutions. We also prove an analogue of the famous Weyl alternative. We connect and unify many known results in the Weyl–Titchmarsh theory for continuous, discrete, and special time scales systems and explain the differences between them. Some of our statements, in particular those connected with coupled endpoints or the Weyl alternative, are new even in the continuous time setting.

Weyl disks and square summable solutions for discrete symplectic systems with jointly varying endpoints

In this paper we develop the spectral theory for discrete symplectic systems with general jointly varying endpoints. This theory includes a characterization of the eigenvalues, construction of the M-lambda function and Weyl disks, their matrix radii and centers, statements about the number of square summable solutions, and limit point or limit circle analysis. These results are new even in some particular cases, such as for the periodic and antiperiodic endpoints, or for discrete symplectic systems with special linear dependence on the spectral parameter. The method utilizes a new transformation to separated endpoints, which is simpler and more transparent than the one in the known literature.MSC: 39A12, 34B20, 34B05, 47B39.

Weyl‐Titchmarsh Theory for Time Scale Symplectic Systems on Half Line

We develop the Weyl‐Titchmarsh theory for time scale symplectic systems. We introduce the M(λ)‐function, study its properties, construct the corresponding Weyl disk and Weyl circle, and establish their geometric structure including the formulas for their center and matrix radii. Similar properties are then derived for the limiting Weyl disk. We discuss the notions of the system being in the limit point or limit circle case and prove several characterizations of the system in the limit point case and one condition for the limit circle case. We also define the Green function for the associated nonhomogeneous system and use its properties for deriving further results for the original system in the limit point or limit circle case. Our work directly generalizes the corresponding discrete time theory obtained recently by S. Clark and P. Zemánek (2010). It also unifies the results in many other papers on the Weyl‐Titchmarsh theory for linear Hamiltonian differential, difference, and dynamic systems when the spectral parameter appears in the second equation. Some of our results are new even in the case of the second‐order Sturm‐Liouville equations on time scales.

Rotordynamics prediction in engineering (second edition)

In this paper we study the limit point and limit circle classification for symplectic systems on time scales, which depend linearly on the spectral parameter. In a broader context, we develop a unified Weyl–Titchmarsh theory for continuous and discrete linear Hamiltonian and symplectic systems. Both separated and coupled boundary conditions are allowed. Our results include the study of the Weyl disks and circles and their limiting behavior, as well as a precise analysis of the number of linearly independent square integrable solutions. We also prove an analogue of the famous Weyl alternative. We connect and unify many known results in the Weyl–Titchmarsh theory for continuous, discrete, and special time scales systems and explain the differences between them. Some of our statements, in particular those connected with coupled endpoints or the Weyl alternative, are new even in the continuous time setting.

Geometric aspects of the Carathéodory extension problem

Both Euclidean geometric and non-Euclidean geometric descriptions for the regions of the solution set to the Carathéodory extension problem are discussed. In particular, it is shown that all the solution functions are located inside a Euclidean disk (the Weyl disk) and a hyperbolic disk (the Apollonian disk). The radius of each disk depends on the norm of the parametrizing Schur functions. Moreover, the so-called maximum entropy solution is exactly the hyperbolic center of the Apollonian disk, while the associated power spectral density function is the geometric mean center of the real region where the Weyl disk is projected to. Sensitivity analysis on the Weyl radius and Apollonian center due to the perturbation of the given finite moments is presented. The applications of these results to the spectrum estimation and inverse scattering problems are discussed.

Spectral Asymptotics for Sturm-Liouville Equations

The basis results of this paper giver asymptotic formulas and order of magnitude estimates for the socalled Titchmarsh-Weyl m-function of the equation (0.2) which are valid also for quite irregular coefficients. As a consequence asymptotic results are obtained for many other spectral quantities, e.g. the eigen-value distribution, Green's function, the spectral function and the integral kernels of the spectral projectors. A formula for the radius of the ‘Weyl disk’ is also given. The classical ‘right-definite’ case is considered as well as the ‘left-definite’ case.

NESTED MATRIX DISKS ANALYTICALLY DEPENDING PARAMETER, AND THEOREMS ON THE INVARIANCE RADII OF LIMITING DISKS

In this work there is an investigation of a family of invertible (i.e. ) analytic matrix-valued functions () which are -contractive and which have monotonically increasing -forms as . Invariance with respect to of the rank of the matrix is established, and conditions for convergence of are investigated. As a special case a theorem is obtained on the invariance of ranks of limiting radii of Weyl disks, which is fundamentally of significance in the theory of classical problems (the moment problem, the Nevanlinna-Pick problem, the Weyl problem on the number of square-integrable solutions of a system of differential equations, and so forth).Bibliography: 17 titles.