Application of the theory of linear operators to the waveguide discontinuity problems

In this chapter, we are going to study linear operators on finite-dimensional inner product spaces. In the last chapter, we introduced adjoints of linear maps between possibly different inner product spaces. Here we shall see how the adjoint can be used to understand linear operators on a fixed inner product space. The important operators we study here are the self-adjoint, skew-adjoint, normal, orthogonal, and unitary operators. We shall spend several sections on the existence of eigenvalues, diagonalizability, and canonical forms for these special but important linear operators. Having done that, we go back to the study of general linear maps and operators and establish the singular value and polar decompositions. We also show Schur’s theorem to the effect that complex linear operators have upper triangular matrix representations. This result does not depend on the spectral theorem. It is also possible to give a quick proof of the spectral theorem using only the material covered in Sect. 4.1. The chapter finishes with a section on quadratic forms and how they tie in with the theory of self-adjoint operators.

We give a generalization of a theorem concerning the change of basis for the matrix representation of unbounded operators defined in a scalar product space. We introduce for the proof a suitable structure which can be useful when one has to make operations with operators defined between different scalar product spaces.

A new geometric approach to the study of the partial stability of linear systems is proposed, which is based on the application of the geometric theory of linear operators. Using the theory of conjugate spaces and conjugate linear operators, bases are constructed in which the system under study takes the canonical form. A cyclic subspace with respect to the conjugate linear operator is considered. A basis is constructed for the dual space of a linear operator, in which its matrix takes the canonical form. This basis corresponds to the dual basis of the original linear space. Then, in a pair of bases of dual spaces the system under study takes the simplest form. The geometric properties of the system are realized using a non-singular linear transformation in the space of a part of the components of the system’s phase vector. This allows us to decompose the system under study in order to obtain necessary and sufficient conditions for the partial stability of the linear system. In an equivalent system, an independent subsystem is distinguished, whose nature of stability determines the behavior of the investigated component of the original system’s phase vector. The relationship between the partial stability of the system and the existence of an invariant subspace of a linear operator characterizing the dynamics of the system is established. The canonical form of the resulting subsystem makes it easy to exclude auxiliary variables and write an equation equivalent to this system. The application of the obtained results to the solution of the problem of partial stability for linear systems with constant coefficients belonging to the classes of ordinary differential equations, discrete systems and systems with deviating argument is shown. An example of a linear system of differential equations is given to illustrate the result obtained.

In NMR quantum computing, spin states of spin-1/2 nuclei are called qubits. Quantum logic gates are represented by unitary matrices. As a universal gate, controlled-NOT (CNOT) is a two-qubit gate. For the IS (I = 1/2 and S = 1/2) spin system, two-qubit CNOT gate is represented by a 4 × 4 matrix. SWAP logic gate, which exchanges two quantum states, is constructed by CNOT gates. In this study, first, four-qubit CNOT gates are constructed for the IS (I = 3/2, S = 3/2) spin system. Then, by using these CNOT gates, a four-qubit SWAP logic gate is found. As an application and verification, an obtained SWAP logic gate is applied to the matrix representation of product operators for the IS (I = 3/2, S = 3/2) spin system. SWAP logic gate can also be presented by an NMR pulse sequence. By using the product operator theory, the pulse sequence of the SWAP logic gate is applied to product operators of the IS (I = 3/2, S = 3/2) spin system. The expected exchange results are obtained for both the matrix representation and the pulse sequence of SWAP logic gate.

Dispersion properties of the substrate integrated rectangular waveguide (SIRW) are rigorously obtained using the BI-RME method combined with the Floquet's theorem. Our analysis shows that the SIRW basically has the same guided-wave characteristics as the conventional rectangular waveguide. Empirical equations are derived from the calculated dispersion curves in order to estimate the cutoff frequency of the first two dominant modes of the SIRW To validate the analysis results, an SIRW guide was designed and measured. Very good agreements between the experimental and theoretical results were obtained.

A finite difference method (FDM) with an analytical truncation boundary condition is developed for the analysis of cyclic H-plane waveguide three-port circulators with an arbitrary shape ferrite post. The method is based on scalar wave equations and the analytical field distribution in the rectangular waveguides. A general boundary field equation for an arbitrarily shaped ferrite or dielectric interface is derived. To verify the validity of the present method, the eigenvalue phases of the cyclic three-port empty junctions and the characteristics of H-plane waveguide Y-junction circulators with lossless, lossy cylindrical, composite and triangular ferrite post are calculated. Good agreements with previously published experimental and theoretical results have been achieved. The performance of a Y-junction circulator with a hexagonal ferrite post and the characteristics of a cyclic three-port circulator with a ferrite rod are also investigated.

The matrix representation of a linear operator is used in the conjugate gradient algorithm to solve electromagnetic boundary value problems. The use of this approach obviates the difficult task of finding the adjoint of the operator.

The substrate-integrated waveguide (SIW) is an artificial waveguide structure that is fabricated in planar dielectric substrate with periodic metallic cylinders, and can be analyzed as an equivalent rectangular waveguide (RW). In this letter, the formula of the conductor loss in SIW has been derived from the conductor loss formula of conventional RW. In addition, the equality condition at which the conductor losses in SIW and RW are exactly equal has been found. The conductor losses in SIW and RW have been investigated and compared, good agreement has been observed between the theoretical calculations and HFSS numerical simulations.

The bases used in quantum mechanics, in general, are orthonormal. The relations in orthonormal basis vectors are simpler than those in nonorthogonal basis vectors. Also, the operators that represent different physical quantities are Hermitian and the eigenvectors of these operators belonging to different eigenvalues are orthonormal to each other. But, in many cases, such as sets of screened hydrogenic wavefunctions, nonorthogonal sets of basis vectors occur. It is, therefore, necessary to consider the representation of vectors and operators in nonorthogonal bases. Making use of the set of vectors reciprocal to the given set of nonorthogonal base vectors, the expression for the norms of vectors and for the matrix elements has been found. Also, adjoint and Hermitian matrices in nonorthogonal bases have been worked out using a set of reciprocal basis vectors.

Spaces of σ(p)-nuclear linear and multilinear operators on Banach spaces and their duals

Certain bounded linear operators $T$ on a complex Hilbert space $\h$ which have 2-isometric liftings $S$ on another space $\ka \supset \h$ are being investigated. We refer also to a more special type of liftings, as well as to those which additionally meet the condition $S^*S\h \subset \h$. Furthermore we describe other types of dilations for $T$, which are close to 2-isometries. Among these we refer to expansive (concave) operators and also to Brownian unitary dilations. Different matrix representations for such operators are obtained, where matrix entries involve contractive operators.

A novel analysis for computing the admittance characteristics of edge slots with a salient feature in successfully considering the finite waveguide wall thickness is presented. The variational equation for the electric field in the slot region between the inner and outer waveguide walls is derived by applying the variational reaction theory and solved by the finite-element method. The division of the slot region into small elements of triangular cylinders and the basis function for each element are described in detail. In the limiting case of zero wall thickness, this finite-element analysis is reduced to a moment-method analysis using Galerkin's approach with triangular basis functions. The effects of waveguide wall thickness on the characteristics of the slot are investigated by comparing the numerical results with and without considering the wall thickness. Negligence of the wall thickness may result in a severe underestimation of the resonant length by as large as 14% for edge slots cut in typical X-band waveguides. The computed results with the finite wall thickness taken into account have been checked with the measured data published in previous literature and those obtained by a carefully devised experimental setup. Good agreement between theoretical and experimental results is obtained to demonstrate the validity of the analysis.

This paper focuses on investigating the problems of matrix representations of adjoint and anti-adjoint operators as well as computations for these matrices in multi-spin 1/2 systems. By introducing a multi-index transformation mapping, adjoint and anti-adjoint operators on tensor space as well as their matrix representations are defined to describe dynamics of multi-spin 1/2 systems. Formulas for computing these matrices of the adjoint and anti-adjoint operators in multi-spin 1/2 systems are given in terms of matrix representations of the adjoint and anti-adjoint operators in single-spin 1/2 systems.

In this paper we use some ideas from [12, 13] and consider the description of Hörmander type pseudo-differential operators on Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^d$$\\end{document} (d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 1$$\\end{document}), including the case of the magnetic pseudo-differential operators introduced in [15, 16], with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calderón-Vaillancourt theorem and Beals’ commutator criterion, and also establish local trace-class criteria.

Using the theory of linear operators, guided electromagnetic wave propagation in inhomogeneous (nonreciprocal) biisotropic planar structures is analyzed in terms of a 2*2 matrix differential operator. Based on the concept of adjoint waveguide, a new biorthogonality relation for the guided hybrid modes is derived. For the special case of reciprocal biisotropic media or chiral media, the linear-operator formalism leads to a self-adjoint problem. As an application example, a general analysis of the radiation modes of a grounded chiroslabguide is also presented. >