Related papers: On the calculation of exact sum rules of rational …
We study the Helmholtz equation for a heterogeneous system in $d$ dimensions and show that it is possible to calculate exactly the sum rules of rational order using perturbation theory by relating the sum rules to suitable traces. The…
We have derived explicit expressions for the sum rules of order one of the eigenvalues of the negative Laplacian on two dimensional domains of arbitrary shape. Taking into account the leading asymptotic behavior of these eigenvalues, as…
We have obtained explicit integral expressions for the sums of inverse powers of the eigenvalues of the Laplacian on a unit sphere, in presence of an arbitrary variable density. The exact expressions for the sum rules are obtained by…
We study the sum rules of the form $Z(s) = \sum_n E_n^{-s}$, where $E_n$ are the eigenvalues of the time--independent Schr\"odinger equation (in one or more dimensions) and $s$ is a rational number for which the series converges. We have…
We show that the formulas for the sum rules for the eigenvalues of inhomogeneous systems that we have obtained in two recent papers are incomplete when the system contains a zero mode. We prove that there are finite contributions of the…
The sum-of-squares method can give rigorous lower bounds on the energy of quantum Hamiltonians. Unfortunately, typically using this method requires solving a semidefinite program, which can be computationally expensive. Further, the…
We study the asymptotic behavior of the spectrum of a quantum system which is a perturbation of a spherically symmetric anharmonic oscillator in dimension 2. We prove that a large part of its eigenvalues can be obtained by Bohr-Sommerfeld…
We discuss how the renormalisation scheme ambiguities in QCD can be fixed, when two observables are related, by requiring the coefficients in the perturbative expansion relating the two observables to have their conformal limit values, i.e.…
We derive new expressions for the Rayleigh-Schr\"odinger seriesdescribing the perturbation of eigenvalues of quantumHamiltonians. The method, somehow close to the so-called dimensionalrenormalization in quantum field theory, involves the…
We derive spectral sum rules for inverse powers of the eigenvalues of the Helmholtz equation on a $d$-sphere in the presence of an arbitrary density. By adopting a rigorous renormalization scheme, we remove the divergent contributions of…
We provide a systematic formula, in terms of integer partitions, that generates perturbation theory explicitly at an arbitrary order. Our approach naturally includes an infinite number of perturbations and uses a single matrix equation that…
A $q$--deformed anharmonic oscillator is defined within the framework of $q$--deformed quantum mechanics. It is shown that the Rayleigh--Schr\"odinger perturbation series for the bounded spectrum converges to exact eigenstates and…
A perturbation scheme is discussed for the computation of the normalization constant of the large order behavior arising from an ultraviolet renormalon. In this scheme the normalization constant is expressed in a convergent series that can…
We use the BLM method to show that perturbatively-calculable observables in QCD can be related to each other without renormalization scale or scheme ambiguity. We define and study the commensurate scale relations. We show that the…
Thesis includes review on the large order behaviour of perturbation theory in quantum mechanical and field theory models; generalization of the Borel summability and strong asymptotic conditions to various (including horn-shaped) regions;…
Results of applying analytic perturbation theory (APT) to the Bjorken sum rule are presented. We study the third-order QCD correction within the analytic approach and investigate its renormalization scheme dependence. We demonstrate that,…
We discuss the role of the Feynman-Hellmann theorem for abstract one-parameter families of Hamiltonians in sum rules and trace identities of Harrell and the author and its application to spectral theory. In particular, we derive a sum rule…
The effects of quantum corrections to a conformally invariant scalar field theory on a curved manifold of positive constant curvature with boundary are considered in the context of a renormalisation procedure. The renormalisation of the…
A determined algorithm is presented for solving the rSUM problem for any natural r with a sub-quadratic assessment of time complexity in some cases. In terms of an amount of memory used the obtained algorithm is the nlog^3(n) order. The…
In the periodic orbit quantization of physical systems, usually only the leading-order hbar contribution to the density of states is considered. Therefore, by construction, the eigenvalues following from semiclassical trace formulae…