Jordan Blocks Research Articles

Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. I. Two-Dimensional Model

A shape invariant nonseparable and nondiagonalizable two-dimensional model with quadratic complex interaction, first studied by Cannata, Ioffe, and Nishnianidze, is re-examined with the purpose of exhibiting its hidden algebraic structure. The two operators $A^+$ and $A^-$, coming from the shape invariant supersymmetrical approach, where $A^+$ acts as a raising operator while $A^-$ annihilates all wavefunctions, are completed by introducing a novel pair of operators $B^+$ and $B^-$, where $B^-$ acts as the missing lowering operator. These four operators then serve as building blocks for constructing ${\mathfrak{gl}}(2)$ generators, acting within the set of associated functions belonging to the Jordan block corresponding to a given energy eigenvalue. This analysis is extended to the set of Jordan blocks by constructing two pairs of bosonic operators, finally yielding an ${\mathfrak{sp}}(4)$ algebra, as well as an ${\mathfrak{osp}}(1/4)$ superalgebra. Hence, the hidden algebraic structure of the model is very similar to that known for the two-dimensional real harmonic oscillator.

Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model

A shape invariant nonseparable and nondiagonalizable three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze. However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remained to be studied. We present a set of six operators $\{A^{\pm},B^{\pm},C^{\pm}\}$ that can be combined to build a ${\mathfrak{gl}}(3)$ hidden algebra. The latter can be embedded in an ${\mathfrak{sp}}(6)$ algebra, as well as in an ${\mathfrak{osp}}(1/6)$ superalgebra. The states associated with the eigenstates and making Jordan blocks are induced in different ways by combinations of operators acting on the ground state. We present the action of these operators and study the construction of an extended biorthogonal basis. These rely on establishing various nontrivial polynomial and commutator identities. We also make a connection between the hidden symmetry and the underlying superintegrability property of the model. Interestingly, the integrals generate a cubic algebra. This work demonstrates how various concepts that have been applied widely to Hermitian Hamiltonians, such as hidden symmetries, superintegrability, and ladder operators, extend to the pseudo-Hermitian case with many differences.

Real dimension of the Lie algebra of S-skew-Hermitian matrices

Let $M_n(\mathbb{C})$ be the set of $n\times n$ matrices over the complex numbers. Let $S \in M_n(\mathbb{C})$. A matrix $A\in M_n(\mathbb{C})$ is said to be $S$-skew-Hermitian if $SA^*=-AS$ where $A^*$ is the conjugate transpose of $A$. The set $\mathfrak{u}_S$ of all $S$-skew-Hermitian matrices is a Lie algebra. In this paper, we give a real dimension formula for $\mathfrak{u}_S$ using the Jordan block decomposition of the cosquare $S(S^*)^{-1}$ of $S$ when $S$ is nonsingular.

The solution of an infinite system of ternary differential equations

The present paper is devoted to an infinite system of differential equations. This system consists of ternary differential equations corresponding to 3×3 Jordan blocks. The system is considered in the Hilbert space l2. A theorem about the existence and uniqueness of solution of the system is proved.

On $m$-th roots of nilpotent matrices

A new necessary and sufficient condition for the existence of an $m$-th root of a nilpotent matrix in terms of the multiplicities of Jordan blocks is obtained and expressed as a system of linear equations with nonnegative integer entries which is suitable for computer programming. Thus, computation of the Jordan form of the $m$-th power of a nilpotent matrix is reduced to a single matrix multiplication; conversely, the existence of an $m$-th root of a nilpotent matrix is reduced to the existence of a nonnegative integer solution to the corresponding system of linear equations. Further, an erroneous result in the literature on the total number of Jordan blocks of a nilpotent matrix having an $m$-th root is corrected and generalized. Moreover, for a singular matrix having an $m$-th root with a pair of nilpotent Jordan blocks of sizes $s$ and $l$, a new $m$-th root is constructed by replacing that pair by another one of sizes $s+i$ and $l-i$, for special $s,l,i$. This method applies to solutions of a system of linear equations having a special matrix of coefficients. In addition, for a matrix $A$ over an arbitrary field that is a sum of two commuting matrices, several results for the existence of $m$-th roots of $A^k$ are obtained.

Deformed W-algebras in Type A for Rectangular Nilpotent

For any \(n,r \in {\mathbb {N}}\), we construct an algebra \({\mathcal {P}}_n^r\) via generators and quadratic relations, and show that it deforms the W–algebra of \({\mathfrak {gl}}_{nr}\) with respect to a nilpotent with Jordan block decomposition \(r+ \ldots +r\). We introduce a surjective morphism from half of the quantum toroidal algebra of \({\mathfrak {gl}}_n\) to \({\mathcal {P}}_n^r\), and show that the action of the quantum toroidal algebra on the K–theory groups of the moduli spaces of parabolic sheaves factors through \({\mathcal {P}}_n^r\).

Bifurcation analysis of the eigenstructure of the discrete single-curl operator in three-dimensional Maxwell’s equations with Pasteur media

Abstract This paper focuses on studying the bifurcation analysis of the eigenstructure of the $\gamma $-parameterized generalized eigenvalue problem ($\gamma $-GEP) arising in three-dimensional source-free Maxwell’s equations with Pasteur media, where $\gamma $ is the magnetoelectric chirality parameter. The weakly coupled case, namely, $\gamma < \gamma _{\ast } \equiv $ critical value, the $\gamma $-GEP is positive definite, has been well studied by Chern et al. (2015, Singular value decompositions for single-curl operators in three-dimensional Maxwell’s equations for complex media. SIAM J. Matrix Anal. Appl., 36, 203–224). For the strongly coupled case, namely, $\gamma> \gamma _{\ast }$, the $\gamma $-GEP is no longer positive definite, introducing a totally different and complicated structure. For the critical strongly coupled case, numerical computations for electromagnetic fields have been presented by Huang et al. (2018, Solving three-dimensional Maxwell eigenvalue problem with fourteen Bravais lattices. Technical Report). In this paper, we build several theoretical results on the eigenstructure behaviour of the $\gamma $-GEPs. We prove that the $\gamma $-GEP is regular for any $\gamma> 0$, and the $\gamma $-GEP has $2 \times 2$ Jordan blocks of infinite eigenvalues at the critical value $\gamma _{\ast }$. Then we show that the $2 \times 2$ Jordan block will split into a complex conjugate eigenvalue pair that rapidly goes down and up and then collides at some real point near the origin. Next it will bifurcate into two real eigenvalues, with one moving towards the left and the other to the right along the real axis as $\gamma $ increases. A newly formed state whose energy is smaller than the ground state can be created as $\gamma $ is larger than the critical value. This stunning feature of the physical phenomenon would be very helpful in practical applications. Therefore, the purpose of this paper is to clarify the corresponding theoretical eigenstructure of the three-dimensional Maxwell equations with Pasteur media.

Fukaya–Seidel categories of Hilbert schemes and parabolic category $\mathcal{O}$

We realise Stroppel’s extended arc algebra \[13, 51] in the Fukaya–Seidel category of a natural Lefschetz fibration on the generic fibre of the adjoint quotient map on a type $A$ nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation of certain parabolic two-block versions of Bernstein–Gel’fand–Gel’fand category $\\mathcal{O}$. As an application, we give a new geometric construction of the spectral sequence from annular to ordinary Khovanov homology. The heart of the paper is the development of a cylindrical model to compute Fukaya categories of (affine open subsets of) Hilbert schemes of quasi-projective surfaces, which may be of independent interest.

Homoclinic Renormalization Group Flows, or When Relevant Operators Become Irrelevant.

We study an N=1 supersymmetric quantum field theory with O(M)×O(N) symmetry. Working in 3-ε dimensions, we calculate the beta functions up to second loop order and analyze in detail the renormalization group (RG) flow and its fixed points. We allow N and M to assume general real values, which results in them functioning as bifurcation parameters. In studying the behavior of the model in the space of M and N, we demarcate the region where the RG flow is nonmonotonic and determine curves along which Hopf bifurcations take place. At a number of points in the space of M and N we find that the model exhibits an interesting phenomenon: at these points the RG flow possesses a fixed point located at real values of the coupling constants g_{i} but with a stability matrix (∂β_{i}/∂g_{j}) that is not diagonalizable and has a Jordan block of size two with zero eigenvalue. Such points correspond to logarithmic conformal field theories and represent Bogdanov-Takens bifurcations, a type of bifurcation known to give rise to a nearby homoclinic orbit-an RG flow that originates and terminates at the same fixed point. In the present example, we are able to employ analytic and numeric evidence to display the existence of the homoclinic RG flow.

Doubly commuting mixed invariant subspaces in the polydisc

We obtain a complete characterization for doubly commuting mixed invariant subspaces of the Hardy space over the unit polydisc. We say a closed subspace Q of H2(Dn) is mixed invariant if Mzj(Q)⊆Q for 1≤j≤k and Mzj⁎(Q)⊆Q, k+1≤j≤n for some integer k∈{1,2,…,n−1}. We prove that a mixed invariant subspace Q of H2(Dn) is doubly commuting if and only ifQ=ΘH2(Dk)⊗Qθ1⊗⋯⊗Qθn−k, where Θ∈H∞(Dk) is some inner function and Qθj is either a Jordan block H2(D)⊖θjH2(D) for some inner function θj or the Hardy space H2(D). Furthermore, an explicit representation for the commutant of an n-tuple of doubly commuting shifts as well as a representation for the commutant of a doubly commuting tuple of shifts and co-shifts are obtained. Finally, we discuss some concrete examples of mixed invariant subspaces.

Exceptional points of degeneracy in traveling wave tubes

The traveling wave tube (TWT) is a powerful vacuum electronic device used to amplify radio-frequency signals with numerous applications, including radar, television, and telephone satellite communications. TWT design in a nutshell comprises a pencil-like electron beam (e-beam) in vacuum interacting with a guiding slow-wave structure (SWS). In our studies here, the e-beam is represented by one-dimensional electron flow and SWS is represented by a transmission line (TL). The interaction between the e-beam and the TL is modeled by an analytic theory that generalizes the well-known Pierce model by taking into account the so-called space-charge effects, particularly electron-to-electron repulsion (debunching). Many important aspects of the analytic theory of TWTs have been already analyzed in our monograph on the subject. The focus of the studies here is on degeneracies of the TWT dispersion relations, particularly on exceptional points of degeneracy and their applications. The term exceptional point of degeneracy (EPD) refers to the property of the relevant matrix to have a nontrivial Jordan block structure. Using special parameterization, particularly suited to chosen EPD, we derive exact formulas for the relevant Jordan basis, including the eigenvectors and the so-called root vector associated with the Jordan block. Based on these studies, we develop a constructive approach to sensing of small signals.

Waveguides with Exceptional Points of Degeneracy of Order 2, 3, 4 and 6 without Loss and Gain

We show various examples of fully-planar two-way and three-way microstrip coupled waveguides that exhibit second, third, fourth and sixth order exceptional points of degeneracy (EPD). EPDs are shown in reciprocal waveguides that do not have gain or loss. The occurrence of EPDs is observed using three methods: (i) the mathematical description of modes and the similarity of the system matrix to a Jordan block; (ii) the dispersion diagram with prescribed flatness; (iii) the coalescence of eigenvectors that is observed analytically or experimentally. We have defined a coalescence parameter that is used in conjunction with experimental results to measure the separation of the coalescing eigenvectors (polarization states). The presented structures can serve various applications like leaky-wave antennas, distributed amplifiers, oscillators, delay lines, pulse generators, and sensors.

Anomalous-order exceptional point and non-Markovian Purcell effect at threshold in one-dimensional continuum systems

For a system consisting of a quantum emitter coupled near threshold (band edge) to a one-dimensional continuum with a van Hove singularity in the density of states, we demonstrate general conditions such that a characteristic triple level convergence occurs directly on the threshold as the coupling $g$ is shut off. For small $g$ values the eigenvalue and norm of each of these states can be expanded in a Puiseux expansion in terms of powers of $g^{2/3}$, which suggests the influence of a third-order exceptional point. However, in the actual $g \rightarrow 0$ limit, only two discrete states in fact coalesce as the system can be reduced to a $2 \times 2$ Jordan block; the third state instead merges with the continuum. Moreover, the decay width of the resonance state involved in this convergence is significantly enhanced compared to the usual Fermi golden rule, which is consistent with the Purcell effect. However, non-Markovian dynamics due to the branch-point effect are also enhanced near the threshold. Applying a perturbative analysis in terms of the Puiseux expansion that takes into account the threshold influence, we show that the combination of these effects results in quantum emitter decay of the unusual form $1 - C t^{3/2}$ on the key timescale during which most of the decay occurs. We then present two conditions that must be satisfied at the threshold for the anomalous exceptional point to occur: the density of states must contain an inverse square-root divergence and the potential must be non-singular. We further show that when the energy of the quantum emitter is detuned from threshold, the anomalous exceptional point splits into three ordinary exceptional points, two of which appear in the complex-extended parameter space. These results provide deeper insight into a well-known problem in spontaneous decay at a photonic band edge.

Exceptional Jordan matrix models, octonionic strings/branes and star product deformations

A brief review of the essentials behind the construction of a Chern-Simons-like brane action from the Large N limit of Exceptional Jordan Matrix Models paves the way to the construction of actions for membranes and p-branes moving in octonionic-spacetime backgrounds endowed with octonionic-valued metrics. It is found that the action of a membrane moving in spacetime backgrounds endowed with an octonionic-valued metric is not invariant under the usual diffeomorphisms of its world volume coordinates σa→σ′a(σb), but instead, it is invariant under the rigid E6(−26) transformations which preserve the volume (cubic) form. An extensive study of the bosonic octonionic string moving in flat backgrounds, and its quantization, follows. One of the most significant findings (without invoking supersymmetry) is that the number of degrees of freedom of three (massless) fermion generations is the same as the number of transverse dimensions of an octonionic string moving in D=26 octonionic dimensions. The star-product deformations of octonionic p-branes follow. In particular, we focus on the octonionic membrane along with the phase space quantization methods developed by [30] within the context of Nonassociative Quantum Mechanics. We finalize with an analysis of two octonionic gravitational actions and with some concluding remarks on Double and Exceptional Field theories, Nonassociative Gravity and A∞,L∞ algebras.

On finite GK-dimensional Nichols algebras over abelian groups

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GKdim \operatorname {GKdim} for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces over Z \mathbb {Z} with the generator acting as a single Jordan block and show that the corresponding Nichols algebra has finite GKdim \operatorname {GKdim} if and only if the size of the block is 2 and the eigenvalue is ± 1 \pm 1 ; when this is 1, we recover the quantum Jordan plane. We consider next a class of braided vector spaces that are direct sums of blocks and points that contains those of diagonal type. We conjecture that a Nichols algebra of diagonal type has finite GKdim \operatorname {GKdim} if and only if the corresponding generalized root system is finite. Assuming the validity of this conjecture, we classify all braided vector spaces in the mentioned class whose Nichols algebra has finite GKdim \operatorname {GKdim} . Consequently we present several new examples of Nichols algebras with finite GKdim \operatorname {GKdim} , including two not in the class alluded to above. We determine which among these Nichols algebras are domains.

Decomposition of exterior and symmetric squares in characteristic two

Let V be a finite-dimensional vector space over a field of characteristic two. As the main result of this paper, for every nilpotent element e∈sl(V), we describe the Jordan normal form of e on the sl(V)-modules ∧2(V) and S2(V). In the case where e is a regular nilpotent element, we are able to give a closed formula.We also consider the closely related problem of describing, for every unipotent element u∈SL(V), the Jordan normal form of u on ∧2(V) and S2(V). A recursive formula for the Jordan block sizes of u on ∧2(V) was given by Gow and Laffey [J. Group Theory 9 (2006), 659–672]. We show that their proof can be adapted to give a similar formula for the Jordan block sizes of u on S2(V).

Perturbations of circuit evolution matrices with Jordan blocks

In our prior studies, we synthesized special circuits with evolution matrices featuring degenerate eigenfrequencies and nontrivial Jordan blocks. The degeneracy of this type is sometimes referred to as exceptional point of degeneracy (EPD). Our focus here is on the simplest of our circuits featuring EPDs that are composed of only two LC-loops coupled by a gyrator. These circuits, when near an EPD state, can be used for enhanced sensitivity applications. With that in mind, we develop here a comprehensive perturbation theory for these simple circuits near an EPD. Using this theory, we propose an approach to sensing, allowing one to benefit from the proximity to an EPD on the one hand when providing for stable operation on the other hand. We also address a broader scope of problems related to perturbations of Jordan blocks and their numerical treatment that allow us to effectively detect proximity to Jordan blocks.

Titchmarsh–Weyl Formula for the Spectral Density of a Class of Jacobi Matrices in the Critical Case

We consider a class of Jacobi matrices with unbounded entries in the so-called critical (double root, Jordan block) case. We prove a formula which relates the spectral density of a matrix to the asymptotics of orthogonal polynomials associated with it.

Square roots of H-nonnegative matrices

Roots of matrices are well-studied. The conditions for their existence are understood: The block sizes of nilpotent Jordan blocks, arranged in pairs, have to satisfy some simple algebraic property.More interesting are structured roots of structured matrices. Probably the best known example is the existence and uniqueness of positive definite square roots of a positive definite matrix. If one drops the requirement of positive definiteness of the square root, it turns out that there exists an abundance of square roots. Here a description of all canonical forms of all square roots is possible and is straightforward.H-nonnegative matrices are H-selfadjoint and are nonnegative with respect to an indefinite inner product with Gramian H. An H-nonnegative matrix B allows a decomposition in a negative definite, a nilpotent H-nonnegative, and a positive definite matrix, B=B−⊕B0⊕B+. The interesting part is B0, as only Jordan blocks of size one and two occur. Determining a square root of B reduces to determining a square root of each of B−, B0, and B+. Here we investigate for an H-nonnegative matrix: its square roots without additional structure, as well as its structured square roots that are H-nonnegative or H-selfadjoint.For these three classes of square roots of H-nonnegative matrices we show a simple criterion for their existence and describe all possible canonical forms. This is based mainly on known results but an important new part is that in all three cases we describe all possible square roots of the nilpotent H-nonnegative matrix B0 explicitly. Moreover, we show how our results can be applied to the conditional and unconditional stability of H-nonnegative square roots of H-nonnegative matrices, where the explicit description of the square roots of B0 is used.

The integrable (hyper)eclectic spin chain

We refine the notion of eclectic spin chains introduced in [1] by including a maximal number of deformation parameters. These models are integrable, nearest-neighbor n-state spin chains with exceedingly simple non-hermitian Hamiltonians. They turn out to be non-diagonalizable in the multiparticle sector (n > 2), where their “spectrum” consists of an intricate collection of Jordan blocks of arbitrary size and multiplicity. We show how and why the quantum inverse scattering method, sought to be universally applicable to integrable nearest-neighbor spin chains, essentially fails to reproduce the details of this spectrum. We then provide, for n=3, detailed evidence by a variety of analytical and numerical techniques that the spectrum is not “random”, but instead shows surprisingly subtle and regular patterns that moreover exhibit universality for generic deformation parameters. We also introduce a new model, the hypereclectic spin chain, where all parameters are zero except for one. Despite the extreme simplicity of its Hamiltonian, it still seems to reproduce the above “generic” spectra as a subset of an even more intricate overall spectrum. Our models are inspired by parts of the one-loop dilatation operator of a strongly twisted, double-scaled deformation of mathcal{N} = 4 Super Yang-Mills Theory.