Non-real Eigenvalues Research Articles

Real diffusion with complex spectral gap

The low-lying eigenvalues of the generator of a Langevin process are known to satisfy the Eyring–Kramers law in the low temperature regime under suitable assumptions. These eigenvalues are generically real. We construct generators whose spectral gap is given by non-real eigenvalues or by a real eigenvalue having a Jordan block.

The lower bounds of non-real eigenvalues for singular indefinite Sturm–Liouville problems

The present paper deals with the non-real eigenvalues for singular indefinite Sturm–Liouville problems. The lower bounds on non-real eigenvalues for this singular problem associated with a special separated boundary condition are obtained.

4 × 4 Irreducible sign pattern matrices that require four distinct eigenvalues

A sign pattern matrix is a matrix whose entries are from the set {+,−,0}. For a sign pattern matrix A, the qualitative class of A, denoted Q(A), is the set of all real matrices whose entries have signs given by the corresponding entries of A. An n×n sign pattern matrix A requires all distinct eigenvalues if every real matrix in Q(A) has n distinct eigenvalues. Li and Harris (2002) [13] characterized the 2×2 and 3×3 irreducible sign pattern matrices that require all distinct eigenvalues, and established some useful general results on n×n sign patterns that require all distinct eigenvalues. In this paper, we characterize 4×4 irreducible sign patterns that require four distinct eigenvalues. This is done by characterizing 4×4 irreducible sign patterns that require four distinct real eigenvalues, that require four distinct nonreal real eigenvalues, or that require two distinct real eigenvalues and a pair of conjugate nonreal eigenvalues. The last case turns out to be much more involved. Some interesting open problems are presented. Three important tools that are used in the paper are the following: the discriminant of a polynomial; the fact that if a square sign pattern matrix A requires all distinct eigenvalues then A requires a fixed number of real eigenvalues; and the known result that if A is a “k-cycle” sign pattern then for each B∈Q(A), the k nonzero eigenvalues of B are evenly distributed on a circle in the complex plane centered at the origin.

Elementary proof for the existence of periodic points with real and simple spectrum for diffeomorphisms in any dimension

Using only elementary techniques we prove that for any $ C^r $ diffeomorphism, $ f $, of a compact manifold of dimension $ d>2 $, $ 1\leq r\leq \infty $, admitting a transverse homoclinic intersection, we can find a $ C^1 $-open neighborhood of $ f $ containing a $ C^1 $-open and $ C^r $-dense set of $ C^r $ diffeomorphisms which have a periodic point with real and simple spectrum. We use this fact to prove that $ C^r $-generically among $ C^r $ diffeomorphisms with horseshoes, we have density of periodic points with real and simple spectrum inside the horseshoe. As a corollary, we obtain that generically in the $ C^1 $-topology the unique obstruction to the existence of periodic points with real and simple spectrum are the Morse-Smale diffeomorphisms with all the periodic points admitting non-real eigenvalues.

Relatively Bounded Perturbations of J-Non-Negative Operators

We improve known perturbation results for self-adjoint operators in Hilbert spaces and prove spectral enclosures for diagonally dominant J-self-adjoint operator matrices. These are used in the proof of the central result, a perturbation theorem for J-non-negative operators. The results are applied to singular indefinite Sturm-Liouville operators with L^p-potentials. Known bounds on the non-real eigenvalues of such operators are improved.

A two-parameter shift-splitting preconditioner for saddle point problems

For saddle point problems, a two-parameter shift-splitting (TPSS) preconditioner is proposed. Estimated lower and upper bounds of the real eigenvalues and the real and imaginary parts of the non-real eigenvalues of the TPSS preconditioned matrix are obtained, respectively. As special cases, bounds obtained are tighter than the ones for the GSS, PGSS, IGSS and NESS preconditioners, respectively. Meanwhile, the corresponding TPSS iteration method is constructed and the convergence analysis is studied. Numerical experiments for two frequently used problems are given to illustrate the great robustness of the TPSS preconditioner.

Perturbing eigenvalues of nonnegative centrosymmetric matrices

An matrix C is said to be centrosymmetric if it satisfies the relation JCJ = C, where J is the counteridentity matrix. Centrosymmetric matrices have a rich eigenstructure that has been studied extensively in the literature. Many results for centrosymmetric matrices have been generalized to wider classes of matrices that arise in a wide variety of disciplines. In this paper, we obtain interesting spectral properties for nonnegative centrosymmetric matrices. We show how to change one single eigenvalue, two or three eigenvalues of an nonnegative centrosymmetric matrix without changing any of the remaining eigenvalues, the nonnegativity, or the centrosymmetric structure. Moreover, our results allow partially answer some known questions given by [Guo W. Eigenvalues of nonnegative matrices. Linear Algebra Appl. 266;1997:261–270] and by [Guo S, Guo W. Perturbing non-real eigenvalues of non-negative real matrices. Linear Algebra Appl. 426;2007:199–203]. Our proofs generate algorithmic procedures that allow one to compute a solution matrix.

Bounds on the Non-real Eigenvalues of Nonlocal Indefinite Sturm–Liouville Problems with Coupled Boundary Conditions

Bounds on the Non-real Eigenvalues of Nonlocal Indefinite Sturm–Liouville Problems with Coupled Boundary Conditions

Evaluation of user dynamics created by weak ties among divided communities

Flaming phenomena represent the divergence in the strength of user dynamics as created by user interactions in online social networks (OSNs). Although it has been known that flaming phenomena occur when the Laplacian matrix of the OSN has non-real eigenvalues, it was recently shown that flaming phenomena may occur even if all the eigenvalues are real numbers. This effect appears only in the situation that some eigenvalues are degenerate, and a special unitary transformation is applied to the equations representing user dynamics; whether actual OSNs satisfy this condition has not been fully discussed. In this paper, we clarify that the user dynamics caused by the degeneration of eigenvalue 0 is one specific example of the above condition. We also investigate the mechanism and characteristics of flaming phenomena generated by degenerated eigenvalues. Furthermore, we demonstrate through numerical simulations that the degeneration of eigenvalues can cause divergence.

Spectra of rank-one perturbations of self-adjoint operators

We characterize possible spectra of rank-one perturbations B of a self-adjoint operator A with discrete spectrum and, in particular, prove that the spectrum of B may include any number of real or non-real eigenvalues of arbitrary algebraic multiplicity

Second order asymptotics for Krein indefinite multipliers with multiplicity two

We consider linear Hamiltonian equations in R2n of the following typedγdt(t)=J2nA(t)γ(t),γ(0)∈Sp(2n,R), where J=J2n=def[0Idn−Idn0] and A:t↦A(t) is a C1 curve in the space of 2n×2n real matrices which are symmetric. Then, t↦γ(t) is a C2 curve in the space of 2n×2n (real) symplectic matrices. We obtain second order asymptotics for the eigenvalues bifurcated from non-real Krein indefinite eigenvalues with algebraic multiplicity two and geometric multiplicity one. As a corollary, we obtain a simple formula about the derivative of the sum of the bifurcated eigenvalues at time t=0. In the end, we discuss possible potential applications for the linear stability of the elliptic Lagrangian solutions of the planar three-body problem.

Maximal [formula omitted]-semi-definite invariant subspaces of unbounded [formula omitted]-selfadjoint operators in Krein spaces

In this paper we establish conditions for the existence of maximal J-semi-definite invariant subspaces of unbounded J-selfadjoint operators. Our results allow for operators where all entries of the formal matrix representation induced by the indefinite metric are unbounded and they do not require any definiteness or J-dissipativity assumptions. As a consequence of the existence of invariant subspaces, we obtain an unexpected result on the accumulation of non-real eigenvalues at the real axis which is of independent interest. An application to some dissipative two-channel Hamiltonians illustrates this new phenomenon.

Non-real eigenvalues of a class of fourth order discontinuous differential operators with indefinite weight function

The spectral theory of differential operators is one of the basic problems of differential operator theory. It has been an important tool in quantum mechanics and other fields. For the Sturm-Liouville Problems with weighted functions, the spectral theory of right-definite problems with self-adjoint boundary condition has been accomplished. But the spectral structure of indefinite problems, is quite different from and more complicated than that of right-definite problems. Compare with second order case, much less known for the fourth order indefinite differential operators. The present paper discusses a class of fourth order discontinuous differential operators. We obtained the estimate on the non-real eigenvalues, under separated boundary condition and the condition that weighted functions change signs for only once or any times.

Bounds of Non-real Eigenvalues for Indefinite p-Laplacian Problems

The present paper deals with the non-real eigenvalues of one dimensional p-Laplacian problem with an indefinite weight. The upper bounds on the imaginary parts and the absolute values of non-real eigenvalues for this indefinite p-Laplacian problem with Dirichlet boundary conditions are obtained through a new method, partly inspired by the estimates obtained in Kikonko and Mingarelli (J Differ Equ 261:6221–6232, 2016).

Non-real eigenvalues of the harmonic oscillator perturbed by an odd, two-point interaction

In this paper, we consider the perturbations of the Harmonic Oscillator Operator by an odd pair of point interactions: $z (\delta(x - b) - \delta(x + b))$. We study the spectrum by analyzing a convenient formula for the eigenvalue. We conclude that if $z = ir$, $r$ real, as $r \to \infty$, the number of non-real eigenvalues tends to infinity.

Fractional Sturm–Liouville eigenvalue problems, I

We introduce and present the general solution of three two-term fractional differential equations of mixed Caputo/Riemann–Liouville type. We then solve a Dirichlet type Sturm–Liouville eigenvalue problem for a fractional differential equation derived from a special composition of a Caputo and a Riemann–Liouville operator on a finite interval where the boundary conditions are induced by evaluating Riemann–Liouville integrals at those end-points. For each $$1/2<\alpha <1$$ it is shown that there is a finite number of real eigenvalues, an infinite number of non-real eigenvalues, that the number of such real eigenvalues grows without bound as $$\alpha \rightarrow 1^-$$, and that the fractional operator converges to an ordinary two term Sturm–Liouville operator as $$\alpha \rightarrow 1^-$$ with Dirichlet boundary conditions. Finally, two-sided estimates as to their location are provided as is their asymptotic behavior as a function of $$\alpha $$.

Non-real eigenvalues of nonlocal indefinite Sturm\u2013Liouville problems

The present paper deals with non-real eigenvalues of regular nonlocal indefinite Sturm–Liouville problems. The existence of non-real eigenvalues of indefinite Sturm–Liouville differential equation with nonlocal potential K(x,t) associated with self-adjoint boundary conditions is studied. Furthermore, a priori upper bounds of non-real eigenvalues for a class of indefinite differential equation involving nonlocal point interference potential function is obtained.

Spectra of a discrete Sturm-Liouville problem with eigenparameter-dependent boundary conditions in Pontryagin space

In this paper, we consider the spectra of a discrete Sturm-Liouville problem with two eigenparameter-dependent boundary conditions. Different from the previous results, the operator corresponding to the problem here is not self-adjoint in the corresponding Hilbert space, but J-self-adjoint in a new Pontryagin space under a new J-metric. No matter what, the existence, simplicity and interlacing properties of real eigenvalues, nonreal eigenvalues and the oscillation properties of eigenfunctions are obtained in this paper. Meanwhile, we also obtain the existence of a pair of nonreal eigenvalues and an unconditional basis of the Pontryagin space under the new J-metric.

A priori bounds and existence of non-real eigenvalues for singular indefinite Sturm–Liouville problems with limit-circle type endpoints

AbstractThe present paper deals with non-real eigenvalues of singular indefinite Sturm–Liouville problems with limit-circle type endpoints. A priori bounds and the existence of non-real eigenvalues of the problem associated with a special separated boundary condition are obtained.

Unphysical Energy Sheets and Resonances in the Friedrichs–Faddeev Model

We consider the Friedrichs-Faddeev model in the case where the kernel of the potential operator is holomorphic in both arguments on a certain domain of $\mathbb{C}$. For this model we, first, study the structure of the $T$- and $S$-matrices on unphysical energy sheet(s). To this end, we derive representations that explicitly express them in terms of these same operators considered exclusively on the physical sheet. Furthermore, we allow the Friedrichs-Faddeev Hamiltonian undergo a complex deformation (or even a complex scaling/rotation if the model is associated with an infinite interval). Isolated non-real eigenvalues of the deformed Hamiltonian are called the deformation resonances. For a class of perturbation potentials with analytic kernels, we prove that the deformation resonances do correspond to the scattering matrix resonances, that is, they represent the poles of the scattering matrix analytically continued to the respective unphysical sheet.