Related papers: Sum rules and a second order Feynman-Hellman theor…

Square matrices often arise in microeconomics, particularly in network models addressing applications from opinion dynamics to platform regulation. Spectral theory provides powerful tools for analyzing their properties. We present an…

A representation of the perturbation series of a general functional measure is given in terms of generalized Feynman graphs and -rules. The graphical calculus is applied to certain functional measures of L\'evy type. A graphical notion of…

We study the eigenvalue problem for a superlinear convolution operator in the special case of bilinear constitutive laws and establish the existence and uniqueness of a one-parameter family of nonlinear eigenfunctions under a topological…

For odd primes p, we examine $\hat H^*(Aut(F_{2(p-1)}); \Z_{(p)})$, the Farrell cohomology of the group of automorphisms of a free group $F_{2(p-1)}$ on $2(p-1)$ generators, with coefficients in the integers localized at the prime $(p)…

In this paper, we discuss index theory for Toeplitz operators on a discrete quarter-plane of two-variable rational matrix function symbols. By using Gohberg-Krein theory for matrix factorizations, we extend the symbols defined originally on…

It is shown how the bilinear differential equations satisfied by Fredholm determinants of integral operators appearing as spectral distribution functions for random matrices may be deduced from the associated systems of nonautonomous…

We derive a sum rule which establishes a linear relation between a particle's anomalous magnetic moment and a quantity connected to the photoabsorption cross-section. This quantity cannot be measured directly. However, it can be computed…

We study the restriction of representations of Cayley-Hamilton algebras to subalgebras. This theory is applied to determine tensor products and branching rules for representations of quantum groups at roots of 1.

For correlators in $\mathcal{N}=4$ Super Yang-Mills preserving half the supersymmetry, we manifestly recast the gauge theory Feynman diagram expansion as a sum over dual closed strings. Each individual Feynman diagram maps on to a Riemann…

For discretisations of hyperbolic conservation laws, mimicking properties of operators or solutions at the continuous (differential equation) level discretely has resulted in several successful methods. While well-posedness for nonlinear…

In this study, we are concerned with spectral problems of second-order vector dynamic equations with two-point boundary value conditions and mixed derivatives, where the matrix-valued coefficient of the leading term may be singular, and the…

We study for subgroups $G\subseteq U(N)$ partial summations of the $\theta$-expanded perturbation theory. On diagrammatic level a summation procedure is established, which in the U(N) case delivers the full star-product induced rules.…

We prove that there is an isomorphism between the Hopf Algebra of Feynman diagrams and the Hopf algebra corresponding to the Homogenous Multiple Zeta Value ring H in C<<X,Y>> . In other words, Feynman diagrams evaluate to Multiple Zeta…

For a $n\times n$ unitary matrix $u=e^z$ with $z$ skew-Hermitian, the angles of $u$ are the arguments of its spectrum, i.e. the spectrum of $-iz$. For $1\le m\le n$, we show that $s_m(t)$, the sum of the first $m$ angles of the path…

The theory of abstract Friedrichs operators was introduced some fifteen years ago with the aim of providing a more comprehensive framework for the study of positive symmetric systems of first-order partial differential equations, nowadays…

We give first-order asymptotic expansions for the resolvent and Hadamard-type formulas for the eigenvalue curves of one-parameter families of canonically symplectic operators. We allow for parameter dependence in the boundary conditions,…

We present algorithms for the group independent reduction of group theory factors of Feynman diagrams. We also give formulas and values for a large number of group invariants in which the group theory factors are expressed. This includes…

Normality of the Dirac operator is shown to be necessary for chiral properties. From the global chiral Ward identity, which in the continuum limit gives the index theorem, a sum rule results which constrains the spectrum. The…

This article deals with the multidimensional Borg-Levinson theorem for perturbed bi-harmonic operator. More precisely, in a bounded smooth domain of $\R^n$, with $n \geq 2$, we prove the stability of the first and zero order coefficients of…

It is shown that the well known sum rules for oscillator strengths for Hydrogen atom can be generalised to a whole class of sum rules. The sum rules have contributions from the discrete and the continuum parts of the spectrum neither of…