Perturbation Theory For Operators Research Articles

Effective Berry–Esseen and concentration bounds for Markov chains with a spectral gap

Applying quantitative perturbation theory for linear operators, we prove nonasymptotic bounds for Markov chains whose transition kernel has a spectral gap in an arbitrary Banach algebra of functions $\mathscr{X}$. The main results are concentration inequalities and Berry–Esseen bounds, obtained assuming neither reversibility nor “warm start” hypothesis: the law of the first term of the chain can be arbitrary. The spectral gap hypothesis is basically a uniform $\mathscr{X}$-ergodicity hypothesis, and when $\mathscr{X}$ consist in regular functions this is weaker than uniform ergodicity. We show on a few examples how the flexibility in the choice of function space can be used. The constants are completely explicit and reasonable enough to make the results usable in practice, notably in MCMC methods.

Supersymmetric QCD: Renormalization and mixing of composite operators

We study $4$-dimensional SQCD with gauge group $SU(N_c)$ and $N_f$ flavors of chiral super-multiplets on the lattice. We perform extensive calculations of matrix elements and renormalization factors of composite operators in Perturbation Theory. In particular, we compute the renormalization factors of quark and squark bilinears, as well as their mixing at the quantum level with gluino and gluon bilinear operators. From these results we construct correctly renormalized composite operators, which are free of mixing effects and may be employed in non-perturbative studies of Supersymmetry. All our calculations have been performed with massive matter fields, in order to regulate the infrared singularities which are inherent in renormalizing squark bilinears. Furthermore, the quark and squark propagators are computed in momentum space with nonzero masses. This work is a feasibility study for lattice computations relevant to a number of observables, such as spectra and distribution functions of hadrons, but in the context of supersymmetric QCD, as a forerunner to lattice investigations of SUSY extensions of the Standard Model.

One method for the boundary value problem eigenvalues calcuating for a second-order differential equation with a fractional derivative

In this paper, we give a formula for computing the eigenvalues of the Dirichlet problem for a differential equation of second-order with fractional derivatives in the lower terms. We obtained this formula using the perturbation theory for linear operators. Using this formula we can write out the system of eigenvalues for the problem under consideration.

Spectral Indicator Method for a Non-selfadjoint Steklov Eigenvalue Problem

We propose an efficient numerical method for a non-selfadjoint Steklov eigenvalue problem. The Lagrange finite element is used for discretization and the convergence is proved using the spectral perturbation theory for compact operators. The non-selfadjointness of the problem leads to non-Hermitian matrix eigenvalue problem. Due to the existence of complex eigenvalues and lack of a priori spectral information, we employ the recently developed spectral indicator method to compute eigenvalues in a given region on the complex plane. Numerical examples are presented to validate the effectiveness of the proposed method.

СПЕКТРОСКОПІЯ БАГАТОЕЛЕКТРОННОГО АТОМА В DC ЕЛЕКТРИЧНОМУ ПОЛІ: МОДИФІКОВАНИЙ МЕТОД ОПЕРАТОРНОЇ ТЕОРІЇ ЗБУРЕНЬ ДЛЯ ОПИСУ ШТАРКIВСЬКИХ РЕЗОНАНСІВ

It is presented a new modified method to calculation of the Stark resonances energies characteristics (energies and widths) for the multielectron atomic systems in a DC electric field. The method is based on the modified operator perturbation theory. The latter allows an accurate, consistent treatment of a strong field DC Stark effect and includes the physically reasonable distorted-waves approximation in the frame of the formally exact quantum-mechanical procedure. As illustration, some test data for the Stark resonances energies and widths in the lithium atom spectrum are presented and compared with results of calculations within the alternative consistent sophisticated methods

OPTIMIZED RELATIVISTIC OPERATOR PERTURBATION THEORY IN SPECTROSCOPY OF MULTIELECTRON ATOM IN A DC ELECTRIC FIELD: SENSING SPECTRAL PARAMETERS

It is developed the optimized version of relativistic operator perturbation theory approach to calculation of the Stark resonances energies characteristics (energies and widths) for the multielectron atomic systems in an electromagnetic field. A new approach allows to perform an accurate, consistent treatment of a strong field DC(AC) Stark effect and includes the physically reasonable distorted-waves approximation in the frame of the formally exact relativistic quantum-mechanical procedure. As illustration, some test data for the Stark resonances energies and widths in the heavy multielectron atoms (caesium, francium) are presented and compared with results of calculations within the alternative consistent sophisticated methods etc.

Modified operator perturbation theory computational approach to atomic systems in a DC electric field: Stark resonances

New computational realization of the modified operator perturbation theory method to computing the Stark resonances energies and widths for the non-hydrogenic (non-H) atomic systems in a DC electric field is presented. The method is based on the operator form of the perturbation theory of the Schrödinger equation for the non-H atomic systems and includes the physically reasonable distorted-waves approximation in the frame of the formally exact quantum-mechanical procedure. The Stark resonances energies and widths in the lithium atom are calculated and compared with results of calculations on the basis of the method of complex eigenvalue Schrödinger equation by Themelis-Nicolaides, the complex absorbing potential method by Sahoo-Ho and the B-spline-based coordinate rotation method approach by Hui-Yan Meng et al.

Spectral analysis of photonic crystals made of thin rods

In this paper we address the question how to design photonic crystals that have photonic band gaps around a finite number of given frequencies. In such materials electromagnetic waves with these frequencies can not propagate; this makes them interesting for a large number of applications. We focus on crystals made of periodically ordered thin rods with high contrast dielectric properties. We show that the material parameters can be chosen in such a way that transverse magnetic modes with given frequencies can not propagate in the crystal. At the same time, for any frequency belonging to a predefined range there exists a transverse electric mode that can propagate in the medium. These results are related to the spectral properties of a weighted Laplacian and of an elliptic operator of divergence type both acting in $L^2(\mathbb{R}^2)$. The proofs rely on perturbation theory of linear operators, Floquet-Bloch analysis, and properties of Schroedinger operators with point interactions.

On the approximate discrete KLT of fractional Brownian motion and applications

This paper establishes connection between discrete cosine transform (DCT) and the discrete-time fractional Brownian motion process (dfBm). It is proved that the eigenvectors of the auto-covariance matrix of a dfBm can be approximated by DCT basis vectors in the asymptotic sense. This shows that DCT basis acts as discrete Karhunen–Loève transform (DKLT) for these processes in the approximate sense. Analytic perturbation theory of linear operators is used to prove this result. This result will be of great practical significance in applications where one is looking for an appropriate basis to work with signals that can perhaps be modeled as belonging to fBm processes. The utility of the proposed work has been illustrated with two real-life data (a) on compressive sampling based reconstruction of financial time-series and (b) in denoising gravitational wave event GW150914 data obtained from a binary black hole merger.

An exact solution method for a class of nonlinear loaded difference equations with multipoint boundary conditions

ABSTRACTSolvability criteria are established and closed-form solutions are obtained for a class of general boundary value problems consisting of an nth-order nonlinear loaded difference equation and nonlocal multipoint boundary conditions. The approach is based on the extension and perturbation theory of operators.

Approximation result for non-autonomous and non-local rock fracture models

The increasing number of earthquakes in the world and the unusual behavior observed in rock fracture processes across the globe have recently raised number of concerns. The way rocks undergo fractures in nature (sometime caused by human activities) is not always the idealized one as wished by geologists. This makes their mathematical modeling and analysis quite complex. In this paper a non-local and non-autonomous rock fracture model where sizes of clusters are discrete and rock fracture rate is time, position and size dependent is investigated by means of forward propagators. Such a model, where in addition, new particles are spatially randomly distributed according to some probabilistic law, has not yet been analyzed in the same work. Our analysis consists of approximating the solution of a discrete, non-local and non-autonomous rock fracture model by a sequence of solutions of cut-off problems of a similar form. Then, we make use of the classical argument of Dini (Lemma 4, Kato in Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften. Band, vol 132. Springer, New York, 1984) to show existence of strong solutions of the model in the class of Banach spaces (of functions with finite higher moments) $$\mathcal {X}_{r}:=L_{1}(\mathfrak {I},X_{r})$$ , with $$X_{r}=\{g:\mathbb {R}^{3}\times \mathbb {N}\ni (x,n)\rightarrow g(x,n), \Vert g\Vert _{r}:=\int _{\mathbb {R}^{3}}\sum _{n=1}^{\infty }n^{r}|g(x,n)|dx<\infty \}.$$ Numerical implementations of the approximation scheme show the fracture evolution of the shear stress resulting from rock fracture as a function of slip. A comparison with the standard well-know results reveals, as expected, that the approximation scheme maintains a sharp weakening of the shear stress in the real fracture case compared to the idealized (homogenous) case. This great observation together with this approximation scheme may contribute to the proof of uniqueness of a strong solution to the discrete, non-local and non-autonomous rock fracture model which remains open and unsolved.

Effective high-temperature estimates for intermittent maps

Using quantitative perturbation theory for linear operators, we prove a spectral gap for transfer operators of various families of intermittent maps with almost constant potentials (‘high-temperature’ regime). Hölder and bounded $p$-variation potentials are treated, in each case under a suitable assumption on the map, but the method should apply more generally. It is notably proved that for any Pommeau–Manneville map, any potential with Lipschitz constant less than 0.0014 has a transfer operator acting on $\operatorname{Lip}([0,1])$ with a spectral gap; and that for any two-to-one unimodal map, any potential with total variation less than 0.0069 has a transfer operator acting on $\operatorname{BV}([0,1])$ with a spectral gap. We also prove under quite general hypotheses that the classical definition of spectral gap coincides with the formally stronger one used in Giulietti et al [The calculus of thermodynamical formalism. J. Eur. Math. Soc., to appear. Preprint, 2015, arXiv:1508.01297], allowing all results there to be applied under the high-temperature bounds proved here: analyticity of pressure and equilibrium states, central limit theorem, etc.

Chiral perturbation theory for neutron\u2013antineutron oscillations

We construct the Chiral Perturbation Theory operators for neutron–antineutron oscillations and use these to estimate chiral and finite volume corrections at one-loop order.

Anharmonic vibrational analysis of s-trans and s-cis conformers of acryloyl fluoride using numerical-analytic Van Vleck operator perturbation theory

A new gas-phase infrared (IR) spectrum of acryloyl fluoride (ACRF, CH2CHCFO) with a resolution of 0.1cm−1 in the range 4000–450cm−1 was measured. Theoretical ab initio molecular structures, full quartic potential energy surfaces (PES), and cubic surfaces of dipole moments and polarizability tensor components (electro-optical properties, EOP) of the s-trans and s-cis conformers of the ACRF were calculated by the second-order Møller-Plesset electronic perturbation theory with a correlation consistent Dunning triple-ζ basis set. The numerical-analytic implementation of the second-order operator canonical Van Vleck perturbation theory was employed for predicting anharmonic IR and Raman scattering (RS) spectra of ACRF. To improve the anharmonic predictions, harmonic frequencies were replaced by their counterparts evaluated with the higher-level CCSD(T)/cc-pVTZ model, to form a “hybrid” PES. The original operator representation of the Hamiltonian is analytically reduced to a quasi-diagonal form, integrated in the harmonic oscillator basis and diagonalized to account for strong resonance couplings. Double canonical transformations of EOP expansions enabled prediction of integral intensities of both fundamental and multi-quanta transitions in IR/RS spectra. Enhanced band shape analysis reinforced the assignments. A thorough interpretation of the new IR experimental spectra and existing matrix-isolation literature data for the mixture of two conformers of ACRF was accomplished, and a number of assignments clarified.

An a posteriori estimator of eigenvalue/eigenvector error for penalty-type discontinuous Galerkin methods

We provide an abstract framework for analyzing discretization error for eigenvalue problems discretized by discontinuous Galerkin methods such as the local discontinuous Galerkin method and symmetric interior penalty discontinuous Galerkin method. The analysis applies to clusters of eigenvalues that may include degenerate eigenvalues. We use asymptotic perturbation theory for linear operators to analyze the dependence of eigenvalues and eigenspaces on the penalty parameter. We first formulate the DG method in the framework of quadratic forms and construct a companion infinite dimensional eigenvalue problem. With the use of the companion problem, the eigenvalue/vector error is estimated as a sum of two components. The first component can be viewed as a “non-conformity” error that we argue can be neglected in practical estimates by properly choosing the penalty parameter. The second component is estimated a posteriori using auxiliary subspace techniques, and this constitutes the practical estimate.

Existence and construction of resonances for atoms coupled to the quantized radiation field

The processes of emission and absorption of photons by atoms can be rigorously understood in the low-energy limit if we neglect the creation and annihilation of electrons. They are related to eigenvalues of the atomic Hamiltonian that are embedded in the continuous spectrum of the free Hamiltonian of the electromagnetic field. The mathematical analysis of the perturbation of these eigenvalues due to the electromagnetic interaction relies on a complex deformation technique relating the original Hamiltonian to a non-selfadjoint operator. We develop a new technique to analyze the spectrum of operators used in non-relativistic quantum electrodynamics. Our method can be applied to prove most of the results that previously required an involved renormalization group construction. We use analytic perturbation theory of operators in Hilbert spaces instead. More precisely, we extend the multi-scale analysis introduced by one of us in 2003 [15], which was used so far only for the study of selfadjoint operators, to non-selfadjoint operators. Compared to the selfadjoint case (see, for example, [3]) the analysis of these non-selfadjoint operators is more difficult, because we cannot make use of the functional calculus (spectral theorem) and the min–max principle in some crucial estimates.

Experimental-Computational Study of Carbon Nanotube Effects on Mitochondrial Respiration: In Silico Nano-QSPR Machine Learning Models Based on New Raman Spectra Transform with Markov-Shannon Entropy Invariants.

The study of selective toxicity of carbon nanotubes (CNTs) on mitochondria (CNT-mitotoxicity) is of major interest for future biomedical applications. In the current work, the mitochondrial oxygen consumption (E3) is measured under three experimental conditions by exposure to pristine and oxidized CNTs (hydroxylated and carboxylated). Respiratory functional assays showed that the information on the CNT Raman spectroscopy could be useful to predict structural parameters of mitotoxicity induced by CNTs. The in vitro functional assays show that the mitochondrial oxidative phosphorylation by ATP-synthase (or state V3 of respiration) was not perturbed in isolated rat-liver mitochondria. For the first time a star graph (SG) transform of the CNT Raman spectra is proposed in order to obtain the raw information for a nano-QSPR model. Box-Jenkins and perturbation theory operators are used for the SG Shannon entropies. A modified RRegrs methodology is employed to test four regression methods such as multiple linear regression (LM), partial least squares regression (PLS), neural networks regression (NN), and random forest (RF). RF provides the best models to predict the mitochondrial oxygen consumption in the presence of specific CNTs with R2 of 0.998-0.999 and RMSE of 0.0068-0.0133 (training and test subsets). This work is aimed at demonstrating that the SG transform of Raman spectra is useful to encode CNT information, similarly to the SG transform of the blood proteome spectra in cancer or electroencephalograms in epilepsy and also as a prospective chemoinformatics tool for nanorisk assessment. All data files and R object models are available at https://dx.doi.org/10.6084/m9.figshare.3472349 .

Spectrum analysis of the linear Fokker–Planck equation

In this work, we show the spectrum structure of the linear Fokker–Planck equation by using the semigroup theory and the linear operator perturbation theory. As an application, we show the large time behavior of the solutions to the linear Fokker–Planck equation.

ADVANCED DATA FOR HYDROGEN ATOM IN CROSSED ELECTRIC AND MAGNETIC FIELDS

Spectroscopy of atoms in the crossed external electric and magnetic fields is investigated on the basis of the operator perturbation theory. As a novel element within the operator perturbation theory, we use more flexible functions for model function, which imitates an electric field. In a case of the crossed electric and magnetic fields we develop more effective finite differences numerical scheme. As illustration, some advanced data for the hydrogen atom in the electric and crossed external electric and magnetic fields are listed. Advanced data for hydrogen atom are listed.

Long time asymptotics of unbounded additive functionals of Markov processes

Under hypercontractivity and $L_p$-integrability of transition density for some $p>1$, we use the perturbation theory of linear operators to obtain existence of long time asymptotics of exponentials of unbounded additive functionals of Markov processes and establish the moderate deviation principle for the functionals. For stochastic differential equations with multiplicative noise, we show the hypercontractivity and the integrability based on Wang’s Harnack inequality. As an application of our general results, we obtain the existence of these asymptotics and the moderate deviation principle of additive functionals with quadratic growth for the stochastic differential equations with multiplicative noise under some explicit conditions on the coefficients and prove that these asymptotics solve the related ergodic Hamilton-Jacobi-Bellman equation with nonsmooth and quadratic growth cost in viscosity sense.