Negative-eigenvalue theorem

Application of Dean s Negative Eigenvalue Theorem to Aperiodic Polymers.—The... 

The total density of states is evaluated as the sum over all the r branches of the dispersion relation (Eq. (3-43)). Nmnerical methods such as the root sampling method have been proposed for simple one- or tridimensional lattices [51]. When the systems become more complex, as in the cases discussed in this chapter, the easiest numerical method is that proposed by Dean based on the application of the Negative Eigenvalues Theorem (NET) [79]. The method is extremely useful and has been extensively used in the laboratory of tlie writer for many polymeric cases. The details of the method are discussed in the next section. 

Dean developed a simple method (based on his so-called negative eigenvalue theorem) for determining the distribution of eigenvalues (density of states) in order to calculate the vibrational spectra of disordered systems. This method cannot be used — as we shall see — for simple topological reasons in the case of the electronic states of two- and three-dimensional solids (and therefore was generally not applied in... 

More detailed investigation requires us to analyze the spectra through more sophisticated methods. Some methods have been rqmrted to analyze the compUcaied infrared spectra of the copolymers, for example, normal mode calcubtion for m I chains with particular monomeric sequences (8,195, s negative eigenvalue theorem for disordered structure [197], s utilization of Omen function [198], and so on. Some of them will be mentioned here. 

Consider the space state model R deflned by Eq.(52), showing an equilibrium point such that the matrix of the linearized system at this point has a real negative eigenvalue A and a pair of complex eigenvalues a j/3, j = /—1) with positive real parts 0.. In this situation, the equilibrium point has onedimensional stable manifold and two-dimensional unstable manifold. If the condition A < a is verified, it is possible that an homoclinic orbit appears, which tends to the equilibrium point. This orbit is very singular, and then the Shilnikov theorem asserts that every neighborhood of the homoclinic orbit contains infinite number of unstable periodic orbits. 

The manifold M- a has locally stable and unstable manifolds that are close to the unperturbed locally stable and unstable manifolds and if these manifolds intersect transversely, then the Smale-Birkhoff theorem predicts the existence of horseshoes and their chaotic dynamics in the perturbed dissipative system. A 2D hyperbolic invariant torus Tja(Pi, P2) may be located on by averaging the perturbed dissipative vector field 7 > 0 and a > 0 restricted to M q, over the angular variables Qi and Q2- The averaged equations have a unique stable hyperbolic fixed point (Pi,P2) = (0,0) with two negative eigenvalues provided that the... 

Actually, if we could apply Koopmans theorem conditions also to the ion core levels, Eb would be the negative of the eigenvalue of the electron — e bn (obtained, e.g., by a one-electron Hartree-Fock calculation - see Chap. A) in the initial state. In reality, the existence of a hole within the ion core implies that Eb is given rather by the expression ... 

If the real parts of all eigenvalues e, Ree, <0 are negative, according to the Lyapunov theorem [14, 15] the stationary point is asymptotically stable... 

Remark 7 Theorem 2.2.8 provides the conditions for checking the convexity or concavity of a function /( ). These conditions correspond to positive semidefinite (P.S.D.) or negative semidef-inite (N.S.D.) Hessian of /(jc) for all x S, respectively. One test of PSD or NSD Hessian of f x) is based on the sign of eigenvalues of the Hessian. If all eigenvalues are greater than or equal to zero for all jc S, then the Hessian is PSD and hence the function /(jc) is convex. If all eigenvalues are less or equal than zero for all x S then the Hessian is NSD and therefore the function /(jc) is concave. 

The eigenvalues e are the well-known orbital energies, and represent (in absolute value) the amount of energy necessary to remove the electron from the respective orbital. According to Koop-mans theorem, e corresponds to the negative of the respective ionization potential (see Section IV). 

Proof. A sufficient condition for J, evaluated at to have eigenvalues with negative real parts is that if the oflF-diagonal elements are replaced by their absolute values, then the determinants of the principal minors alternate in sign (Theorem A.11). 

Proof. If El and E2 are unstable, then dissipativeness and uniform persistence (previous proposition) yield the existence of an interior rest point for -k(x, t) (Theorem D.3). If E exists then it is unique and has all eigenvalues negative (Lemma 5.2). Suppose that exists and that Ei is asymptotically stable. Then, since asymptotically stable, Theorem E.l contradicts the uniqueness of E. A similar argument applies if E2 is asymptotically stable. Note that the computations leading up to Lemma 5.1 explicitly determine the signs of the eigenvalues for linearization about Ei and E2. ... 

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