Negative energy densities in integrable quantum field theories at one-particle level

We study the phenomenon of negative energy densities in quantum field theories with self-interaction. Specifically, we consider a class of integrable models (including the sinh-Gordon model) in which we investigate the expectation value of the energy density in one-particle states. In this situation, we classify the possible form of the stress-energy tensor from first principles. We show that one-particle states with negative energy density generically exist in non-free situations, and we establish lower bounds for the energy density (quantum energy inequalities). Demanding that these inequalities hold reduces the ambiguity in the stress-energy tensor, in some situations fixing it uniquely. Numerical results for the lowest spectral value of the energy density allow us to demonstrate how negative energy densities depend on the coupling constant and on other model parameters

A new algorithm for computing the Geronimus transformations for large shifts

A monic Jacobi matrix is a tridiagonal matrix which contains the parameters of the three-term recurrence relation satisfied by the sequence of monic polynomials orthogonal with respect to a measure. The basic Geronimus transformation with shift ? transforms the monic Jacobi matrix associated with a measure d? into the monic Jacobi matrix associated with d?/(x????)?+?C?(x????), for some constant C. In this paper we examine the algorithms available to compute this transformation and we propose a more accurate algorithm, estimate its forward errors, and prove that it is forward stable. In particular, we show that for C?=?0 the problem is very ill-conditioned, and we present a new algorithm that uses extended precision

R esolution g eom etrique d'une equation de degr e 4 : Partie 1

nsi, > 0, alors l'hyperbole est definie sur ] 1 [#]x 2 ; +[. 2. Si d = 0. La racine de D(x) est double et est egale a . Par consequent, D(x) peut s'ecrire de la facon suivante (x x d ) . Dans ce cas la, D(x) est positif quelque soit x et les solutions de l'equation (2) sont . -(ax+b)-|a||x-x . -(ax+b)+|a||x-x . Les valeurs de y 1 (x) et y 2 (x) dependent du signe de a et de la position de x par rapport a celle de x d . Ainsi, . Si a > 0, . Si a < 0, = 0, independemment du signe de a, l'hyperbole est definie sur l'ensemble des reels tout entier. De plus, les deux branches de l'hyperbole sont reduites a deux droites qui se coupent au point critique x d . Dans les figures 1 et 2, nous avons mis sur un meme graphe, pour tout x, les valeurs de y 1 (x) en traits continus rouges et les valeurs de y 2 (x) en traits continus verts dans les cas ou a > 0 et a < 0. Ce cas est a noter car c'est le cas

d ¡ 0¢ (2)

En s’inspirant du livre de F. Klein [1], nous avons découvert une méthode utilisée par les anciens afin de résoudre géométriquement des problèmes dont la forme analytique est une équation du 3 ième degré ou de degré supérieur. Nous choisissons ici d’effectuer l’étude de la résolution géométrique d’une équation du 4 ième degré. 1 Description du procédé Nous considérons l’équation en x à coefficients réels de degré 4 suivante (1) Posons y ¡ x 2. L’équation (1) devient x 4   ax 3   bx 2   cx   d ¡ 0¢ y 2 axy by c

About Hölder Condition Numbers and the Stratification diagram for Defective Eigenvalues

In this paper, we look at a particular case of application which is Hölder continuous, namely the map from a matrix A to one of its eigenvalues , when it is multiple defective. Two asymptotic Hölder condition numbers are considered : one (resp. the other) is associated with a generalization of the Frechet (resp. Gateaux) derivative [9]. We illustrate on a fully developed 3 \Theta 3 example, why these asymptotic condition numbers may not be appropriate to analyze eigencomputations performed in finite precision. We present the complementary view point of differential geometry, developed by A. Ilahi in [12], which is based on the stratification diagram in C j j 3\Theta3 . The distance to the stratum indicates when this complementary viewpoint is required