GENERALIZED HYPERVIRIAL AND RECURRENCE RELATIONS FOR RADIAL MATRIX ELEMENTS IN ARBITRARY DIMENSIONS

Asymptotic behaviours and general recurrence relations

CHAPTER 4 - Recurrence Relations and Algebraic Equations

Recursiveness in matrix rational interpolation problems

An algebraic approach to obtain ladder operators for hydrogenlike wave functions from the algebraic representation of the ${\mathit{L}}_{\mathit{n}}^{\mathit{k}}$ Laguerre associated polynomials directly involved is presented. The method leads to diverse ladder operators depending on the manner as the operational form of ${\mathit{L}}_{\mathit{n}}^{\mathit{k}}$ is used. As an example, this work deals with linear creation and annihilation operators of two kinds: some that influence the n energy quantum number and others acting on the l orbital number according to usual ladder properties. In addition, as a useful application of the raising and lowering operators thus obtained, generalized recurrence relations and closed formulas for the calculation of multipole matrix elements are algebraically deduced by means of a procedure that takes advantage of the relationship between the ladder operators associated with the bra and those related to the ket.

Some properties of generalized secant integrals: extended definition and recurrence relations

A closed formula for and interpretation of exact quantum mechanical off-diagonal matrix elements of powers of r is given in terms of orthogonal polynomials. The compact formulae for r-k-2 and rk-1 are in exact correspondence to the semiclassical (diagonal) expectation values and are most suitable for the practical evaluation of the matrix elements. The expressions for < nl | r-k-2 | l'n > and < nl | rk-1 | l'n > consist only of the minimum of n-l or k+1 terms. An off-diagonal generalization of the well known diagonal inversion formula is derived and the matrix elements can thus be evaluated for arbitrary (complex) powers k. Also, an analytic continuation to an arbitrary effective n* is possible. It is thus established that, complementary to the O(2,1) approach by Armstrong (1970, 1971), orthogonality properties for purely radial quantities can be obtained without the introduction of a fictitious angular variable. Using the analytical results obtained, certain deficiencies of the semiclassical Coulomb (or WKB) approximation are discussed. The normalization error involved in the WKB approximation is identified analytically and an improved version of the semiclassical limit, a correspondence limit, is derived. Furthermore, it is established that, like the exact results, the WKB matrix elements obey a three-term recursion relation and an inversion formula. For low k some explicit formulae for exact matrix elements are tabulated.

Analytical expressions for the matrix elements 〈v'J\ensuremath{\Vert}(r-${r}_{e}$${)}^{l}$exp[-ma(r-${\mathrm{r}}_{\mathrm{e}}$) ]\ensuremath{\Vert}vJ〉 of a rotating Morse oscillator are obtained, where l is a non-negative integer and m is any number. These matrix elements are also obtained by a recursive method that obviates the need for using explicit eigenfunctions. This procedure is based on the hypervirial theorem together with the second-quantization formalism. The results permit the diagonal (v=v',J=J') and off-diagonal (v\ensuremath{\ne}v',J=J') matrix elements of the operator (r-${r}_{e}$${)}^{l}$ to be calculated.

Several imaging algorithms including patch-based image denoising, image time series recovery, and convolutional neural networks can be thought of as methods that exploit the manifold structure of signals. While the empirical performance of these algorithms is impressive, the understanding of recovery of the signals and functions that live on manifold is less understood. In this paper, we focus on the recovery of signals that live on a union of surfaces. In particular, we consider signals living on a union of smooth band-limited surfaces in high dimensions. We show that an exponential mapping transforms the data to a union of low-dimensional subspaces. Using this relation, we introduce a sampling theoretical framework for the recovery of smooth surfaces from few samples and the learning of functions living on smooth surfaces. The low-rank property of the features is used to determine the number of measurements needed to recover the surface. Moreover, the low-rank property of the features also provides an efficient approach, which resembles a neural network, for the local representation of multidimensional functions on the surface. The direct representation of such a function in high dimensions often suffers from the curse of dimensionality; the large number of parameters would translate to the need for extensive training data. The low-rank property of the features can significantly reduce the number of parameters, which makes the computational structure attractive for learning and inference from limited labeled training data.

The underlying stochastic nature of the requirements for the Solvency II regulations has introduced significant challenges if the required calculations are to be performed correctly, without resorting to excessive approximations, within practical timescales. It is generally acknowledged by practising actuaries within UK life offices that it is currently impossible to correctly fulfil the requirements imposed by Solvency II using existing computational techniques based on commercially available valuation packages. Our work has already shown that it is possible to perform profitability calculations at a far higher rate than is achievable using commercial packages. One of the key factors in achieving these gains is to calculate reserves using recurrence relations that scale linearly with the number of time steps. Here, we present a general vector recurrence relation which can be used for a wide range of non-unit linked policies that are covered by Solvency II; such contracts include annuities, term assurances, and endowments. Our results suggest that by using an optimised parallel implementation of this algorithm, on an affordable hardware platform, it is possible to perform the ‘brute force’ approach to demonstrating solvency in a realistic timescale (of the order of a few hours).

Over 100 years ago Harald Bohr identified a deep problem about the convergence of Dirichlet series, and introduced an ingenious idea relating Dirichlet series and holomorphic functions in high dimensions. Elaborating on this work, almost twnety years later Bohnenblust and Hille solved the problem posed by Bohr. In recent years there has been a substantial revival of interest in the research area opened up by these early contributions. This involves the intertwining of the classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. New challenging research problems have crystallized and been solved in recent decades. The goal of this book is to describe in detail some of the key elements of this new research area to a wide audience. The approach is based on three pillars: Dirichlet series, infinite dimensional holomorphy and harmonic analysis.

We consider the relaxation and transfer in a symmetric trimer coupled to a phonon bath. The energy transfer within the trimer occurs via resonance interactions and coupling between the trimer and the bath occurs via modulation of the monomer energies by phonons. We compare the model introduced by Čápek with the Redfield model used commonly with secular approximation. We discuss two initial conditions: a) The highest eigenstate of the trimer is initially excited, and b) the initial excitation is local. We find that the occupation probabilities in the representation of eigenstates are the same in both models. But it is not so for off-diagonal density matrix elements. Only if the off-diagonal elements vanish initially (initial conidtion a), they vanish at arbitrary time in both models. If the initial excitation is local, off-diagonal elements differ essentially even in the long time limit.

In this paper, we investigate the existence of random absolutely continuous invariant measures (ACIP) for random expanding on average Saussol maps in higher dimensions. This is done by the establishment of a random Lasota–Yorke inequality for the transfer operators on the space of bounded oscillation. We prove that the number of ergodic skew product ACIPs is finite and will provide an upper bound for the number of these ergodic ACIPs. This work can be seen as a generalization of the work in [F. Batayneh and C. González-Tokman, On the number of invariant measures for random expanding maps in higher dimensions, Discrete Contin. Dyn. Syst. 41 (2021) 5887–5914] on admissible random Jabłoński maps to a more general class of higher-dimensional random maps.

Errata to "Analysis and Applications of Holomorphic Functions in Higher Dimensions"

In this paper formulas and recursion relations for the expectation values of the operators {1-exp[-a(r-${r}_{e}$${)]\mathrm{}}}^{n}$, (r-${r}_{e}$${)}^{n}$, exp[-a(r-${r}_{e}$${)]}^{n}$, and (r-${r}_{e}$){exp[-a(r-${r}_{e}$${)]\mathrm{}}}^{n}$ are derived for a rotating Morse oscillator. These equations can be used to calculate the diagonal (v=v', J=J') and off-diagonal (v\ensuremath{\ne}v', J\ensuremath{\ne}J') matrix elements. Asymptotic approximations for the diagonal elements of the (r-${r}_{e}$${)}^{n}$ operator, 〈vJ\ensuremath{\Vert}(r-${r}_{e}$)\ensuremath{\Vert}vJ〉 and 〈vJ\ensuremath{\Vert}(r-${r}_{e}$${)}^{2}$\ensuremath{\Vert}vJ〉, are also obtained.

Calculation of spectral wings involves computation of far off-diagonal radiative matrix elements, and therefore suffers from the problem of phase cancellation due to rapid oscillation of the matrix element integrands. We review two methods which have been used to address this difficulty: the complex plane method and the propagator method.