Non-real Eigenvalues Research Articles

A New Analysis of Block Preconditioners for Saddle Point Problems

We consider symmetric saddle point matrices. We analyze block preconditioners based on the knowledge of a good approximation for both the top left block and the Schur complement resulting from its elimination. We obtain bounds on the eigenvalues of the preconditioned matrix that depend only of the quality of these approximations, as measured by the related condition numbers. Our analysis applies to indefinite block diagonal preconditioners, block triangular preconditioners, inexact Uzawa preconditioners, block approximate factorization preconditioners, and a further enhancement of these preconditioners based on symmetric block Gauss--Seidel-type iterations. The analysis is unified and allows the comparison of these different approaches. In particular, it reveals that block triangular and inexact Uzawa preconditioners lead to identical eigenvalue distributions. These theoretical results are illustrated on the discrete Stokes problem. It turns out that the provided bounds allow one to localize accurately both real and nonreal eigenvalues. The relative quality of the different types of preconditioners is also as expected from the theory.

Some new inequalities on geometric multiplicities and Gersgorin discs

In this paper the authors continue their work on geometric multiplic- ities and Gersgorin discs done in a series of four recent papers. The new results involve principal submatrices, an upper bound on the absolute value of an eigenvalue, the rank of a matrix, non-real eigenvalues, and powers of matrices. Some consequences of the results and examples are provided.

Spectral Properties of the Iteration Matrix of the HSS Method for Saddle Point Problem

We discuss spectral properties of the iteration matrix of the HSS method for saddle point problems and derive estimates for the region containing both the nonreal and real eigenvalues of the iteration matrix of the HSS method for saddle point problems.

Non-real eigenvalues of indefinite Sturm–Liouville problems

The present paper deals with non-real eigenvalues of regular indefinite Sturm–Liouville problems. A priori bounds and sufficient conditions of the existence for non-real eigenvalues are obtained under mild integrable conditions of coefficients.

Eigenvalue Estimates for Non-Selfadjoint Dirac Operators on the Real Line

We show that the non-embedded eigenvalues of the Dirac operator on the real line with non-Hermitian potential $V$ lie in the disjoint union of two disks in the right and left half plane, respectively, provided that the $L^1-norm$ of $V$ is bounded from above by the speed of light times the reduced Planck constant. An analogous result for the Schr\"odinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on $V$ implies the absence of nonreal eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials.

On the theory of $ \mathcal{P}\mathcal{T} $ -symmetric operators

We develop a general theory of $ \mathcal{P}\mathcal{T} $ -symmetric operators. Special attention is given to $ \mathcal{P}\mathcal{T} $ -symmetric quasiself-adjoint extensions of symmetric operator with deficiency indices : For these extensions, the possibility of their interpretation as self-adjoint operators in Krein spaces is investigated and the description of nonreal eigenvalues is presented. These abstract results are applied to the Schr¨odinger operator with Coulomb potential on the real axis.

On the nonreal eigenvalues of elliptic differential operators with indefinite weights on Lipschitz domains

AbstractThe nonreal spectrum of a second order elliptic differential operator with an indefinite weight function on a Lipschitz domain is investigated with the help of Krein space techniques and perturbation methods. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Spectral properties of primal-based penalty preconditioners for saddle point problems

For large and sparse saddle point linear systems, this paper gives further spectral properties of the primal-based penalty preconditioners introduced in [C.R. Dohrmann, R.B. Lehoucq, A primal-based penalty preconditioner for elliptic saddle point systems, SIAM J. Numer. Anal. 44 (2006) 270–282]. The regions containing the real and non-real eigenvalues of the preconditioned matrix are obtained. The model of the Stokes problem is supplemented to illustrate the theoretical results and to test the quality of the primal-based penalty preconditioner.

Switching near a network of rotating nodes

We study the dynamics of a Z 2 ⊕ Z 2-equivariant vector field in the neighbourhood of a heteroclinic network with a periodic trajectory and symmetric equilibria. We assume that around each equilibrium the linearization of the vector field has non-real eigenvalues. Trajectories starting near each node of the network turn around in space either following the periodic trajectory or due to the complex eigenvalues near the equilibria. Thus, in a network with rotating nodes, the rotations combine with transverse intersections of two-dimensional invariant manifolds to create switching near the network; close to the network, there are trajectories that visit neighbourhoods of the saddles following all the heteroclinic connections of the network in any given order. Our results are motivated by an example where switching was observed numerically by forced symmetry breaking of an asymptotically stable network with O(2) symmetry.

Non-real eigenvalues of singular indefinite Sturm-Liouville operators

We study a Sturm-Liouville expression with indefinite weight of the form sgn (−d^2/dx^2+V ) on \mathbb{R} and the non-real eigenvalues of an associated selfadjoint operator in a Krein space. For real-valued potentials V with a certain behaviour at \pm \infty we prove that there are no real eigenvalues and the number of non-real eigenvalues (counting multiplicities) coincides with the number of negative eigenvalues of the selfadjoint operator associated to −d^2/dx^2 + V in L^2(\mathbb{R}). The general results are illustrated with examples.

Another dichotomy for surface diffeomorphisms

We prove that a C 1 generic orientation-preserving diffeomorphism of a closed orientable surface either has infinitely many periodic points with complex (nonreal) eigenvalues or is Axiom A without cycles. This improves Mane's dichotomy.

Random-matrix approach to RPA equations. I

We study the RPA equations in their most general form by taking the matrix elements appearing in the RPA equations as random. This yields either a unitary or an orthogonally invariant random-matrix model that does not appear in the Altland–Zirnbauer classification. The average spectrum of the model is studied with the help of a generalized Pastur equation. Two independent parameters govern the behaviour of the system: the strength α 2 of the coupling between positive- and negative-energy states and the distance between the origin and the centers of the two semicircles that describe the average spectrum for α 2 = 0, the latter measured in units of the equal radii of the two semicircles. With increasing α 2, positive- and negative-energy states become mixed and ever more of the spectral strength of the positive-energy states is transferred to those at negative energy, and vice versa. The two semicircles are deformed and pulled toward each other. As they begin to overlap, the RPA equations yield non-real eigenvalues: The system becomes unstable. We determine analytically the critical value of the strength for the instability to occur. Several features of the model are illustrated numerically.

Perturbing non-real eigenvalues of non-negative real matrices

Let σ = ( ρ , b + i c , b - i c , λ 4 , … , λ n ) be the spectrum of an entry non-negative matrix and t ⩾ 0 . Laffey [T. J. Laffey, Perturbing non-real eigenvalues of nonnegative real matrices, Electron. J. Linear Algebra 12 (2005) 73–76] has shown that σ = ( ρ + 2 t , b - t + i c , b - t - i c , λ 4 , … , λ n ) is also the spectrum of some nonnegative matrix. Laffey (2005) has used a rank one perturbation for small t and then used a compactness argument to extend the result to all nonnegative t. In this paper, a rank two perturbation is used to deduce an explicit and constructive proof for all t ⩾ 0 .

Minimality of the system of root functions of Sturm–Liouville problems with decreasing affine boundary conditions

We consider Sturm–Liouville problems with a boundary condition linearly dependent on the eigenparameter. We study the case of decreasing dependence where non-real and multiple eigenvalues are possible. By determining the explicit form of a biorthogo

Detecting broken -symmetry

A fundamental problem in the theory of PT-invariant quantum systems is to determine whether a given system `respects' this symmetry or not. If not, the system usually develops non-real eigenvalues. It is shown in this contribution how to algorithmically detect the existence of complex eigenvalues for a given PT-symmetric matrix. The procedure uses classical results from stability theory which qualitatively locate the zeros of real polynomials in the complex plane. The interest and value of the present approach lies in the fact that it avoids diagonalization of the Hamiltonian at hand.

Overlap Dirac Operator at Nonzero Chemical Potential and Random Matrix Theory

We show how to introduce a quark chemical potential in the overlap Dirac operator. The resulting operator satisfies a Ginsparg-Wilson relation and has exact zero modes. It is no longer gamma5 Hermitian, but its nonreal eigenvalues still occur in pairs. We compute the spectral density of the operator on the lattice and show that, for small eigenvalues, the data agree with analytical predictions of non-Hermitian chiral random matrix theory for both trivial and nontrivial topology. We also explain an observed change in the number of zero modes as a function of chemical potential.

Non-self-adjoint Jacobi matrices with a rank-one imaginary part

We develop direct and inverse spectral analysis for finite and semi-infinite non-self-adjoint Jacobi matrices with a rank-one imaginary part. It is shown that given a set of n not necessarily distinct nonreal numbers in the open upper (lower) half-plane uniquely determines an n × n Jacobi matrix with a rank-one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity. Algorithms of reconstruction for such finite Jacobi matrices are presented. A new model complementing the well-known Livsic triangular model for bounded linear operators with a rank-one imaginary part is obtained. It turns out that the model operator is a non-self-adjoint Jacobi matrix. We show that any bounded, prime, non-self-adjoint linear operator with a rank-one imaginary part acting on some finite-dimensional (respectively separable infinite-dimensional Hilbert space) is unitarily equivalent to a finite (respectively semi-infinite) non-self-adjoint Jacobi matrix. This obtained theorem strengthens a classical result of Stone established for self-adjoint operators with simple spectrum. We establish the non-self-adjoint analogs of the Hochstadt and Gesztesy–Simon uniqueness theorems for finite Jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non-self-adjoint tri-diagonal matrices to the case of mixed eigenvalues, real and nonreal. A unique Jacobi matrix, unitarily equivalent to the operator of integration ( F f ) ( x ) = 2 i ∫ x l f ( t ) d t in the Hilbert space L 2 [ 0 , l ] is found as well as spectral properties of its perturbations and connections with the well-known Bernoulli numbers. We also give the analytic characterization of the Weyl functions of dissipative Jacobi matrices with a rank-one imaginary part.

On the QR iterations of real matrices

We answer a question of D. Serre on the QR iterations of a real matrix with nonreal eigenvalues whose moduli are distinct except for the conjugate pairs. Numerical experiments by MATLAB are performed.

Some remarks on matrix stability with application to Turing instability

An example of a 4×4 matrix is given that provides a counterexample to a result on Turing (diffusion-driven) instability and also answers negatively a conjecture on strong stability. Such instability is shown to arise from nonreal eigenvalues. The relevance and the connection of our example to classes of matrix stability known in the literature are discussed.

Indefinite Sturm–Liouville problems

We study the spectrum of regular and singular Sturm–Liouville problems with real-valued coefficients and a weight function that changes sign. The self-adjoint boundary conditions may be regular or singular, separated or coupled. Sufficient conditions are found for (i) the spectrum to be real and unbounded below as well as above and (ii) the essential spectrum to be empty. Also found is an upper bound for the number of non-real eigenvalues. These results are achieved by studying the interplay between the indefinite problems (with weight function which changes sign) and the corresponding definite problems. Our approach relies heavily on operator theory of Krein space.