Antieigenvalues and antisingularvalues of a matrix and applications to problems in statistics

Let A be p × p positive definite matrix. A p-vector x such that Ax = x is called an eigenvector with the associated with eigenvalue . Equivalent characterizations are: (i) cos = 1, where is the angle between x and Ax. (ii) (x0Ax)−1 = xA−1x. (iii) cos = 1, where is the angle between A1/2x and A−1/2x. We ask the question what is x such that cos as defined in (i) is a minimum or the angle of separation between x and Ax is a maximum. Such a vector is called an anti-eigenvector and cos an anti-eigenvalue of A. This is the basis of operator trigonometry developed by K. Gustafson and P.D.K.M. Rao (1997), Numerical Range: The Field of Values of Linear Operators and Matrices, Springer. We may define a measure of departure from condition (ii) as min[(x0Ax)(x0A−1x)]−1 which gives the same anti-eigenvalue. The same result holds if the maximum of the angle between A1/2x and A−1/2x as in condition (iii) is sought. We define a hierarchical series of anti-eigenvalues, and also consider optimization problems associated with measures of separation between an r(< p) dimensional subspace S and its transform AS. Similar problems are considered for a general matrix A and its singular values leading to anti-singular values. Other possible definitions of anti-eigen and anti-singular values, and applications to problems in statistics will be presented

Semiclassical analysis of a complex quartic Hamiltonian

It is necessary to calculate the C operator for the non-Hermitian PT-symmetric Hamiltonian H=\half p^2+\half\mu^2x^2-\lambda x^4 in order to demonstrate that H defines a consistent unitary theory of quantum mechanics. However, the C operator cannot be obtained by using perturbative methods. Including a small imaginary cubic term gives the Hamiltonian H=\half p^2+\half \mu^2x^2+igx^3-\lambda x^4, whose C operator can be obtained perturbatively. In the semiclassical limit all terms in the perturbation series can be calculated in closed form and the perturbation series can be summed exactly. The result is a closed-form expression for C having a nontrivial dependence on the dynamical variables x and p and on the parameter \lambda.Comment: 4 page

Refining Lagrange's four-square theorem

Lagrange's four-square theorem asserts that any $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as the sum of four squares. This can be further refined in various ways. We show that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb Z$ such that $x+y+z$ (or $x+2y$, $x+y+2z$) is a square (or a cube). We also prove that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb N$ such that $P(x,y,z)$ is a square, whenever $P(x,y,z)$ is among the polynomials \begin{gather*} x,\ 2x,\ x-y,\ 2x-2y,\ a(x^2-y^2)\ (a=1,2,3),\ x^2-3y^2,\ 3x^2-2y^2, \\x^2+ky^2\ (k=2,3,5,6,8,12),\ (x+4y+4z)^2+(9x+3y+3z)^2, \\x^2y^2+y^2z^2+z^2x^2,\ x^4+8y^3z+8yz^3, x^4+16y^3z+64yz^3. \end{gather*} We also pose some conjectures for further research; for example, our 1-3-5-Conjecture states that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb N$ such that $x+3y+5z$ is a square.Comment: 24 pages, final published versio

Bound States of Non-Hermitian Quantum Field Theories

The spectrum of the Hermitian Hamiltonian ${1\over2}p^2+{1\over2}m^2x^2+gx^4$ ($g>0$), which describes the quantum anharmonic oscillator, is real and positive. The non-Hermitian quantum-mechanical Hamiltonian $H={1\over2}p^2+{1 \over2}m^2x^2-gx^4$, where the coupling constant $g$ is real and positive, is ${\cal PT}$-symmetric. As a consequence, the spectrum of $H$ is known to be real and positive as well. Here, it is shown that there is a significant difference between these two theories: When $g$ is sufficiently small, the latter Hamiltonian exhibits a two-particle bound state while the former does not. The bound state persists in the corresponding non-Hermitian ${\cal PT}$-symmetric $-g\phi^4$ quantum field theory for all dimensions $0\leq D<3$ but is not present in the conventional Hermitian $g\phi^4$ field theory.Comment: 14 pages, 3figure

Phase diagram of quarter-filled band organic salts, [EDT-TTF-CONMe2]2X, X = AsF6 and Br

An investigation of the P/T phase diagram of the quarter-filled organic conductors, [EDT-TTF-CONMe2]2X, is reported on the basis of transport and NMR studies of two members, X=AsF6 and Br of the family. The strongly insulating character of these materials in the low pressure regime has been attributed to a remarkably stable charge ordered state confirmed by 13C NMR and the only existence of 1/4 Umklapp e-e scattering favoring a charge ordering instead of the 1D Mott localization seen in (TM)2X which are quarter-filled compounds with dimerization. A non magnetic insulating phase instead of the spin density wave state is stabilized in the deconfined regime of the phase diagram. This sequence of phases observed under pressure may be considered as a generic behavior for 1/4-filled conductors with correlations