Non-real Eigenvalues Research Articles

ON THE SPECTRUM OF THE TRANSPORT OPERATOR FOR A BOUNDED CONVEX BODY

The eigenvalue problem of the transport operator for a general bounded convex body V is studied mathematically. Where either V or the eigenfunction of the operator is not necessarily of spherical symmetry. It is shown that the spectrum of the operator consists of pure eigenvalues, possibly plusthe point of negative infinity. There is a countable infinity of real eigenvalues accumulating at minus infinity. Each eigenvalue, especially one in the left half-plane, is of index one, There is no complex (non-real) eigenvalue in the right half-plane Re λ > ∑. The solution of the corresponding initial-value problem is also discussed.

The spectrum of some abstract kinetic operators in inhomogeneous media with applications

Abstract In this paper the spectrum of abstract kinetic operator A defined by on 0 < x < a with φ(0) = φ(a) is studied, where T and B are self adjoint operators on a Hilbert space. The location of non-real eigenvalues and the number of isolated real eigenvalues are estimated. In particular, we investigate the spectrum of the slab transport operator in an inhomogeneous medium.

Classification of eigentuples for uniformly elliptic multiparameter problems

Let T m , V mn be self-adjoint operators in Hilbert spaces H m the T 3 having compact resolvents and lower bounds τ m > 0, and the V mn satisfying “uniform ellipticity,” 1 ⩽ m , n ⩽ k . When τ m > 0 for each m these conditions are called “uniform left definiteness” and guarantee that all eigenvalues λ for the simultaneous equations W m ( λ ) x m = 0, x m ≠ 0, W m ( λ ) = T m − ∑ n = 1 k λ n V mn ; belong to R k and are “semi-simple” in an appropriate sense. When some τ m ⩽ 0, these conclusions may fail and we use an embedding technique to discuss, for example, algebraic multiplicities of eigenvalues, transitions between real and non-real eigenvalues, and bounds for the numbers of eigenvalues of various types.

Two parameter eigenvalue problems for matrices

The linked pair of two parameter eigenvalue problems ( T m − λ 1 V m1 − λ 2 V m2 ) x m = 0, x m ≠ 0, x m ∈ H m is considered for Hermitian matrices T m , V m1 , V m2 acting on on the finite dimensional Hilbert spaces H m , m = 1,2. We dispense with customary definiteness conditions, although for m = 1,2 we assume the existence of a linear combination of V m1 , V m2 which is definite. An embedding technique is used to study eigenvalues λ ∈ R 2 of specified variational index, the transition between real and nonreal eigenvalues, and algebraic multiplicity.

The complex scaling method for Dirac resonances

The method of complex scaling, usually used in atomic and molecular resonance calculations, is generalized to the Dirac equation. It is shown that Dirac resonances are associated with nonreal eigenvalues of the scaled Dirac Hamiltonian. The perturbation theory for the resonance parameters is also discussed.

On a class of matrices with real eigenvalues

It is easy to prove that if A is a real irreducible square matrix and if a real nonsingular diagonal matrix D exists such that AD is symmetric and positive semidefinite, then for any real diagonal matrix Y, AY has only real eigenvalues. This paper proves the converse result that if no such D exists, then for some Y, AY will possess some nonreal eigenvalues.

Optical guiding in a sheet-beam free-electron laser.

The influence of electron density gradients on optical guiding in free-electron lasers (FEL's) is considered. We study a model problem of a FEL which employs a sheet electron beam in the absence of a cavity or a waveguide. We calculate the gain and the rate of optical guiding by solving for the eigenvalues and the actual eigenmodes of the system. Similar to what has been found by Moore for a cylindrical beam (Nucl. Instrum. Methods A239, 19 (1985)), we find that two parameters characterize the interaction: a coupling parameter and a detuning parameter. We solve the problem for two density profiles, a uniform density profile and a triangular-shaped density profile. For large and small values of the coupling parameter we obtain the results analytically. For intermediate values of the coupling parameter we obtain numerical results. When the coupling parameter is small, diffraction is large. The gain and the wave profile are then similar for the two density profiles. When the coupling parameter is large, optical guiding is large. The gain for the triangular-shaped beam is then larger by 2/sup 1/3/ than the gain for the uniform beam. When the coupling parameter is large, the wave profile for the uniform densitymore » beam converges to a certain profile confined to the beam volume. For the triangular-shaped beam the wave profile becomes more and more concentrated near the midplane of the sheet beam. For the solution of the problem we derive an energy integral that determines the domain in the complex plane where nonreal eigenvalues are allowed. We also show the existence of an accumulation point of nonreal eigenvalues in certain cases.« less

The F test for model discrimination with exponential functions

The performance of the F test in identifying the correct number of components in exponential functions is investigated by mathematical and computational techniques. Uncoupled systems can be most easily identified when the relaxation times are widely separated but the exponential functions encountered in pharrnacokinetics and biochemistry are linked in that the amplitudes and eigenvalues are not independent. In such systems correct model discrimination occurs as the relaxation times become closer together. The possibility of nonreal eigenvalues is emphasized and an enzyme kinetic model with this property is investigated.

The role of nonreal eigenvalues in the identification of cycles in a compartmental system

The paper concerns the relationship between the cycles in the graph of a compartmental system and the modes of the impulse response function associated with an input-output experiment. Suppose that there is at least one oscillatory mode, e μt cos( vt - α). Let e ϱt be the slowest mode. The main result is that the system contains a cycle of length 3 or longer and that the length of the longest cycle is at least π/tan -1[¦ v¦/( ϱ- μ)]. The paper also deals with the problem of estimating the cycle length from discrete data.

Inverse Elementary Divisor Problem for Nonnegative Matrices

Given a diagonalizable positive matrix $A$, there exists a positive matrix with the same spectrum as $A$, and with arbitrarily prescribed elementary divisors, provided that elementary divisors corresponding to nonreal eigenvalues occur in conjugate pairs. It is also shown that a similar result holds for doubly stochastic matrices.

Inverse elementary divisor problem for nonnegative matrices

Given a diagonalizable positive matrix A A , there exists a positive matrix with the same spectrum as A A , and with arbitrarily prescribed elementary divisors, provided that elementary divisors corresponding to nonreal eigenvalues occur in conjugate pairs. It is also shown that a similar result holds for doubly stochastic matrices.

Existence of complex eigenvalues for the mono-energetic neutron transport equation

Abstract If the assumption is made that the neutron speed has a positive lower bound, the eigenvalue spectrum of the neutron transport equation in a finite body is discrete.1 Let λ∗ be the minimum collision rate. Then it can also be shown, under rather broad assumptions, the most relevant of which is the isotropy of scattering, that the eigenvalues belonging to the half-plane Reλ > −λ∗ must be real. Real eigenvalues also exist, as a rule, in the remaining half-plane Reλ ≤ −λ∗.2 An old problem is: does this half-plane contain complex (i.e., nonreal) eigenvalues? In this paper we study a particular one-speed problem (a thin absorbing and scattering spherical shell surrounding a purely absorbing core) and show that such complex eigenvalues actually exist.

DEFECT INDICES OF $ J_m$-MATRICES AND OF DIFFERENTIAL OPERATORS WITH POLYNOMIAL COEFFICIENTS

The problem of the defect indices of the symmetric operator which acts on the space and is generated by a regular Hermitian -matrix with increasing elements is investigated. The asymptotics, as , of the eigenvectors of the operator which correspond to the nonreal eigenvalues are obtained. The results are applied to ordinary differential operators with polynomial coefficients defined on the entire -axis. Figures: 2. Bibliography. 15 items.

Combinatorial approaches to Hopf bifurcations in systems of interacting elements

We describe combinatorial approaches to the question of whether families of real matrices admit pairs of nonreal eigenvalues passing through the imaginary axis. When the matrices arise as Jacobian matrices in the study of dynamical systems, these conditions provide necessary conditions for Hopf bifurcations to occur in parameterised families of such systems. The techniques depend on the spectral properties of additive compound matrices: in particular, we associate with a product of matrices a signed, labelled digraph termed a DSR^[2] graph, which encodes information about the second additive compound of this product. A condition on the cycle structure of this digraph is shown to rule out the possibility of nonreal eigenvalues with positive real part. The techniques developed are applied to systems of interacting elements termed “interaction networks”, of which networks of chemical reactions are a special case.