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Classifying topological phases of matter with strong interactions is a notoriously challenging task and has attracted considerable attention in recent years. In this paper, we propose an unsupervised machine learning approach that can classify a wide range of symmetry-protected interacting topological phases directly from the experimental observables and without a priori knowledge. We analytically show that Green's functions, which can be derived from spectral functions that can be measured directly in an experiment, are suitable for serving as the input data for our learning proposal based on the diffusion map. As a concrete example, we consider a one-dimensional interacting topological insulators model and show that, through extensive numerical simulations, our diffusion map approach works as desired. In addition, we put forward a generic scheme to measure the spectral functions in ultracold atomic systems through momentum-resolved Raman spectroscopy. Our work circumvents the costly diagonalization of the system Hamiltonian, and provides a versatile protocol for the straightforward and autonomous identification of interacting topological phases from experimental observables in an unsupervised manner. [ABSTRACT FROM AUTHOR]

A Fan-Theobald-von Neumann system [7] is a triple (V , W , λ) , where V and W are real inner product spaces and λ : V → W is a norm-preserving map satisfying a Fan-Theobald-von Neumann type inequality together with a condition for equality. Examples include Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decomposition systems (Eaton triples). The present article is a continuation of [9] where the concepts of commutativity, automorphisms, majorization, and reduction were introduced and elaborated. Here, we describe some transfer principles and present Fenchel conjugate and subdifferential formulas. [ABSTRACT FROM AUTHOR]

In this study, the resolvent operator of the impulsive dynamic singular Sturm–Liouville problem is considered. The spectral function was first established. The integral representation of the resolvent operator is obtained using this function. [ABSTRACT FROM AUTHOR]

In this study, a singular impulsive Hahn–Dirac system is studied. A spectral function has been established for this type of system. With the help of this function, the Parseval equation and eigenfunction expansion were obtained. [ABSTRACT FROM AUTHOR]

In this article, we proved that a Schatten $ p $-(quasi)norm for $ 0 < p < 1 $, defined on Euclidean Jordan algebras, satisfied a relaxed triangle inequality with an optimal constant $ 2^{\frac{1}{p} - 1} $; hence, it indeed induced a quasinorm. This confirmed the validity of a conjecture raised by Huang, Chen, and Hu.

We study the question as to when the closed convex hull of the graph of a K-convex map equals its K-epigraph. In particular, we shed light onto the smallest cone K such that a given map has convex and closed K-epigraph, respectively. We apply our findings to several examples in matrix space as well as to convex composite functions. [ABSTRACT FROM AUTHOR]

We introduce an algorithm for estimating the trace of a matrix function f(A) using implicit products with a symmetric matrix A. Existing methods for implicit trace estimation of a matrix function tend to treat matrix-vector products with f(A) as a black box to be computed by a Krylov subspace method. Like other recent algorithms for implicit trace estimation, our approach is based on a combination of deflation and stochastic trace estimation. However, we take a closer look at how products with f(A) are integrated into these approaches which enables several efficiencies not present in previously studied methods. In particular, we describe a Krylov subspace method for computing a low-rank approximation of a matrix function by a computationally efficient projection onto Krylov subspace. [ABSTRACT FROM AUTHOR]

This study is based upon investigations of the existence of a singular Hahn difference equation of q,ω$$ q,\omega $$‐Sturm–Liouville problem with transmission conditions. Moreover, the Parseval identity and expansion formula in eigenfunctions are obtained. [ABSTRACT FROM AUTHOR]

We report multipronged progress on the stochastic averaging approach to numerical analytic continuation of imaginary-time correlation functions computed by quantum Monte Carlo simulations. After reviewing the conventional maximum-entropy approach and established stochastic analytic continuation methods, we present several new developments in which the configurational entropy of the sampled spectrum plays a key role. Parametrizing the spectrum as a large number of δ -functions in continuous frequency space, an exact calculation of the entropy lends support to a simple goodness-of-fit criterion for the optimal sampling temperature. We also compare spectra sampled in continuous frequency with those from amplitudes sampled on a fixed frequency grid. Insights into the functional form of the entropy in different cases allow us to demonstrate equivalence in a generalized thermodynamic limit (large number of degrees of freedom) of the average spectrum and the maximum-entropy solution, with different parametrizations corresponding to different forms of the entropy in the prior probability. These results revise prevailing notions of the maximum-entropy method and its relationship to stochastic analytic continuation. In further developments of the sampling approach, we explore various adjustable (optimized) constraints that allow sharp low-temperature spectral features to be resolved, in particular at the lower frequency edge. The constraints, e.g., the location of the edge or the spectral weight of a quasi-particle peak, are optimized using a statistical criterion based on entropy minimization under the condition of optimal fit. We show with several examples that this method can correctly reproduce both narrow and broad quasi-particle peaks. We next introduce a parametrization for more intricate spectral functions with sharp edges, e.g., power-law singularities. We present tests with synthetic data as well as with real simulation data for the spin-1/2 Heisenberg chain, where a divergent edge of the dynamic structure factor is due to deconfined spinon excitations. Our results demonstrate that distortions of sharp edges or quasi-particle peaks, which arise with other analytic continuation methods, propagate and cause artificial spectral features also at higher energies. The constrained sampling methods overcome this problem and allow analytic continuation of spectra with sharp edge features at unprecedented fidelity. We present results for S = 1 / 2 Heisenberg 2-leg and 3-leg ladders to illustrate the ability of the methods to resolve spectral features arising from both elementary and composite excitations. Finally, we also propose how the methods developed here could be used as "pre processors" for analytic continuation by machine learning. Edge singularities and narrow quasi-particle peaks being ubiquitous in quantum many-body systems, we expect the new methods to be broadly useful and take numerical analytic continuation to a new quantitative level in many applications. [ABSTRACT FROM AUTHOR]

Focusing towards ternary metal hydrides has recently been regarded as a new avenue for research in pressure-dependent high-temperature superconductors, thanks highly to a fairly large number of permutations of alloying metals, even metalloids, with hydrogen. Herein, new phases of Mg − B − H ternary hydrides are predicted from the first-principles evolutionary techniques, as a result of which the corresponding phonon and electronic calculations for the three candidate phases are performed successively to confirm their dynamic stability and the possibility to become conductors. The metallic MgBH 9 undergoes a superconducting phase with a maximum T c of 64 K at 110 GPa, with its spectral function predominantly active around optical modes. The significant increase in cumulative electron-phonon coupling constant is associated with a relatively low cutoff frequency according to the bandwidth function. As for the non-metallic candidate, hydrogen-vacancy diffusion kinetics of the MgB 2 H 8 phase are determined by means of total energy calculations. Stable pathways at varying pressure are reported, suggesting that elevated pressure lowers the activation energy which is presumably due to an optimal level of average nearest H − H(B) inter-fragment distances. [Display omitted] • New stable phases of Mg–B–H under pressure are determined across the ternary convex hull. • MgBH 9 adopts an orthorhombic structure and becomes metallic at pressures up to 110 GPa. • Frequencies around 178–212 K entail significant increase of the spectral function profile of MgBH 9. • Hydrogen-vacancy diffusion kinetics within MgB 2 H 8 are investigated through the context of total energy curves. [ABSTRACT FROM AUTHOR]

The Sturm‐Liouville equation SL−1p(x)ddxp(x)dudx=λu,x∈(0,∞),λ∈ℂ,$$ -\frac{1}{p(x)}\frac{d}{dx}\left(p(x)\frac{du}{dx}\right)=\lambda u,\kern0.30em x\in \left(0,\infty \right),\lambda \in \mathbb{C}, $$is studied. The characterization of the spectral data for (SL) with the boundary condition u′(0)−hu(0)=0,h∈ℝ,$$ {u}^{\prime }(0)- hu(0)=0,\kern0.30em h\in \mathbb{R}, $$is given. A series representation for the Jost solution is obtained in the form of a power series with respect to the parameter z=12+iλ12−iλ$$ z=\frac{\frac{1}{2}+i\sqrt{\lambda }}{\frac{1}{2}-i\sqrt{\lambda }} $$. The series converges in the unit disk |z|<1$$ \mid z\mid <1 $$ and leads to an explicit representation for spectral data and analytic method for their computation. [ABSTRACT FROM AUTHOR]

This paper aims to study the diffusion and localization of cold atoms in three-dimensional (3D) optical speckles by using the theory of self-energy for N diffusers. In this paper, the Self-consistent Born approximation (SCBA) is used to study the diffusion and localization of cold atoms in an optical disordered speckles, while the spectral function is adopted to study the effect of the matter waves energy and the disorder amplitude on the behaviour of cold atoms. For this purpose, the models of SCBA and spectral function are computed by a numerical algorithm. The calculation of the latter quantity involves the application of Simpson's integration methods. The diffusion and localization of a Bose–Einstein Condensate for First-order Born Approximation (FBA) and SCBA models are presented. In addition, the diffusion time and the mean free path between two diffusers are illustrated. Moreover, the effect of disorder amplitude and the scattering of matter waves in 3D disordered potentials are highlighted. In this study, the results show that the time response of cold atoms localization and metal–insulator transition in SCBA is faster and yields lower energy than the FBA approximation. [ABSTRACT FROM AUTHOR]

In this article, we investigate the resolvent of dynamic Dirac operators on unbounded time scales. For the resolvent of these operators, integral representations are obtained. Finally, a formula for the Titchmarsh-Weyl function is given. [ABSTRACT FROM AUTHOR]

The role of electron-phonon interaction in band gap shrinkage for some II-VI bulk and low-dimensional semiconductors is investigated in this work. The variation of the energy band gap is studied as a function of temperature using Varshni's, Vina's, and Passler's relations. It is observed that the change in the energy band gap is affected due to the quantum confinement as the dimensionality of these semiconductors is decreased. • The role of electron-phonon interaction in band gap shrinkage is studied. • The effect of quantum confinement on band gap shrinkage is investigated. • A plateau behavior in the band gap variation is observed. • The goodness of fitting is studied. [ABSTRACT FROM AUTHOR]

The G W method is a standard method to calculate the electronic band structure from first principles. It has been applied to a large variety of semiconductors and insulators but less often to metallic systems, in particular, with respect to a self-consistent employment of the method. In this work, we take a look at all-electron quasiparticle self-consistent G W (QS G W ) calculations for simple metals (alkali and alkaline earth metals) based on the full-potential linearized augmented-plane-wave approach and compare the results to single-shot (i.e., non-selfconsistent) G 0 W 0 calculations, density-functional theory (DFT) calculations in the local-density approximation, and experimental measurements. We show that, while DFT overestimates the bandwidth of most of the materials, the G W quasiparticle renormalization corrects the bandwidths in the right direction, but a full self-consistent calculation is needed to consistently achieve good agreement with photoemission data. The results mainly confirm the common belief that simple metals can be regarded as nearly free electron gases with weak electronic correlation. The finding is particularly important in light of a recent debate in which this seemingly established view has been contested. [ABSTRACT FROM AUTHOR]

We study pseudospectral and spectral functions for Hamiltonian system Jy′−B(t)=λΔ(t)y$Jy^{\prime }-B(t)=\lambda \Delta (t)y$ and differential equation l[y]=λΔ(t)y$l[y]=\lambda \Delta (t)y$ with matrix‐valued coefficients defined on an interval I=[a,b)$\mathcal {I}=[a,b)$ with the regular endpoint a. It is not assumed that the matrix weight Δ(t)≥0$\Delta (t)\ge 0$ is invertible a.e. on I$\mathcal {I}$. In this case a pseudospectral function always exists, but the set of spectral functions may be empty. We obtain a parametrization σ=στ$\sigma =\sigma _\tau$ of all pseudospectral and spectral functions σ by means of a Nevanlinna parameter τ and single out in terms of τ and boundary conditions the class of functions y for which the inverse Fourier transform y(t)=∫Rφ(t,s)dσ(s)ŷ(s)$y(t)=\int _\mathbb {R}\varphi (t,s)\, d\sigma (s) \widehat{y}(s)$ converges uniformly. We also show that for scalar equation l[y]=λΔ(t)y$l[y]=\lambda \Delta (t)y$ the set of spectral functions is not empty. This enables us to extend the Kats–Krein and Atkinson results for scalar Sturm–Liouville equation −(p(t)y′)′+q(t)y=λΔ(t)y$-(p(t)y^{\prime })^{\prime }+q(t)y=\lambda \Delta (t) y$ to such equations with arbitrary coefficients p(t)$p(t)$ and q(t)$q(t)$ and arbitrary non trivial weight Δ(t)≥0$\Delta (t)\ge 0$. [ABSTRACT FROM AUTHOR]

In this paper Carath'eodoery theorem for matrix-valued Schur functions and its two basic corollaries, due to Arov, are supplemented with several important results especially, under some conditions, about the uniqueness of the spectral function sup-porting maximal jump of full rank at a given boundary point. An essential step is to derive an effective coupling identity between the jumps of every spectral function and some canonical one at that boundary point. A by-product of this identity is simplified proofs of a number of remarkable properties for canonical spectral functions. These extend the region of their applications. As examples, we elucidate simultaneously corresponding known extremal feature for various matrix moment problems and interpolation problems of the Nevanlinna-Pick type in the completely indeterminate case, and a number of additions to them. [ABSTRACT FROM AUTHOR]

Subject and Purpose. Smoke, fog, avalanches, debris of collapsed structures and other optically opaque obstacles in both natural and man-made disasters make optical sensors useless for detecting victims. Electromagnetic waves of the decimeter range penetrate well almost all obstacles, reflect from the trapped people and return to the radar receiver. Due to the breathing and heartbeat, the human-reflected sounding signals get the Doppler phase modulation, which is an information signal. These information signals and their properties provide the subject matter for the present research with the aim to create optimal methods and algorithms of random event processing for the prompt location of survivors by rescuers. Method and Methodology. The method of stochastic analysis of the fluctuation Doppler spectra of reflected sounding signals shows that the information signals have properties of conditional Markov processes. Results. The problem of optimal nonlinear filtering of conditional Markov processes entering the radar signal processing unit has been examined closely. An optimal adaptive filter has been proposed to reduce the masking effect of interferences caused by non-stationary noises and sounding signal reflections from stationary objects. The optimality criterion is the minimum mean square error function whose current value is evaluated in real time during the filtering process as the statistics is accumulated. The filter coefficients are calculated by the recurrent, steepest descent algorithm. The real-time work is carried out through the use of fast Fourier transform algorithms. Conclusion. The structure of the optimal adaptive filter to be built into the radar signal processing unit has been developed. Real radar signals have shown that the optimal filtering during the signal processing in systems designed for detecting live people by their breathing and heartbeat facilitates the interpretation of the observed signals. Some spectra of real signals generated by human breathing and heartbeat are presented.

We propose a device composed of a quantum dot (QD) connected to a normal metal lead to detect Majorana bound states (MBSs), which are formed at the ends of a topological superconductor nanowire (TSNW) and coupled to the lead with spin-dependent hybridization strengths. The information of the MBSs leaked into the lead is inferred from the spectral function of the QD serving as the tip of a scanning tunneling microscope (STM). It is found that lead–MBSs interaction induces a bound state characterized by an infinitely high peak in the dot’s zero-energy spectral function. The overlap between the two modes of the MBSs turns this bound state into a resonant one, and thus the zero-energy peak is split into three with the height of the central one equaling that in the absence of lead–MBSs coupling. We also find that the MBSs have lower impacts on the additional peak in the dot’s spectral function induced by intradot Coulomb interaction.

The temperature behavior of the pion width in the hadronic phase is investigated in the framework of the Nambu–Jona–Lasinio (NJL) model. The contribution to the width from the pion–pion collision is considered with a scalar sigma-meson as an intermediate state. It is shown that the pion width significantly broadens at T > 0. 1 GeV. Using the two-step iteration method, suggested by Kadanoff and Baym, the pion spectral function in a hot pion gas is calculated at different temperatures. [ABSTRACT FROM AUTHOR]

Abstract In the present study, we investigate the existence of spectral functions and obtain the Parseval identity and expansion formula in eigenfunctions for the singular q-Sturm–Liouville problem with transmission conditions.

We aim to study the effects of chemical potential and angular velocity on the critical endpoint of quantum chromodynamics (QCD). We used several probes (drag force, jet quenching parameter, heavy vector meson spectral function) to characterize the phase transition and studied gravitational waves from the holographic QCD phase transition in the early universe. We used different holographic QCD models to discuss the QCD phase transition, energy loss, spectral function, and gravitational waves. We found that the chemical potential and angular velocity changed the location of the critical endpoint, and the drag force and jet quenching parameter were temperature dependent and enhanced near the phase transition temperature. The magnetic field had a nontrivial effect on the spectral function. We conclude that the chemical potential decreases ωc, and the angular velocity decreases µc and the phase transition temperature. The jet quenching parameter and drag force can characterize the phase transition, and the magnetic field promotes the dissociation of heavy vector mesons. Moreover, the energy density of gravitational waves decreases as the gluon condensate increases, and the peak frequency shifts downward with increasing gluon condensate.Exploring the phase structure of QCD is an important task in high-energy heavy ion collision physics, and recently, there has been considerable interest in the QCD phase transition for rotating backgrounds.

This article presents a randomized matrix-free method for approximating the trace of f (A) , where A is a large symmetric matrix and f is a function analytic in a closed interval containing the eigenvalues of A. Our method uses a combination of stochastic trace estimation (i.e., Hutchinson's method), Chebyshev approximation, and multilevel Monte Carlo techniques. We establish general bounds on the approximation error of this method by extending an existing error bound for Hutchinson's method to multilevel trace estimators. Numerical experiments are conducted for common applications such as estimating the log-determinant, nuclear norm, and Estrada index. We find that using multilevel techniques can substantially reduce the variance of existing single-level estimators. [ABSTRACT FROM AUTHOR]

In this work, we consider one dimensional Dirac operators on unbounded time scales. We construct a spectral function of such an operator. Using this function, we establish a Parseval equality and an expansion formula in eigenfunctions for the Dirac operator on unbounded time scales. [ABSTRACT FROM AUTHOR]

In this paper we focus on the spectrum of Toeplitz operators Tφ on the Hardy space H² of the unit disk D when φ is not continuous and especially when Tφ is a complex symmetric operator. Moreover, we show a necessary condition for Tφ to be complex symmetric, for φ belongs to subalgebra H∞ +C(T) of L∞ and we present a case of the continuity of the spectral function restricted to the space of Toeplitz operators. [ABSTRACT FROM AUTHOR]

We develop a method to obtain fermion spectral functions non-perturbatively in a non-Abelian gauge theory with high occupation numbers of gauge fields. After recovering the free field case, we extract the spectral function of fermions in a highly occupied non-Abelian plasma close to its non-thermal fixed point, i.e., in a self-similar regime of the non-equilibrium dynamics. We find good agreement with hard loop perturbation theory for medium-induced masses, dispersion relations and quasiparticle residues. We also extract the full momentum dependence of the damping rate of the collective excitations.

Subject and Purpose. The subject of the research is a system of coherent digital processing of radar signals such that includes a number of spatially spaced receiving points and is designed to detect moving objects hidden behind optically opaque obstacles. The aim of the work is to synthesize an optimal, in the sense of a minimum Bayesian risk average, signal structure in the processing system. The system performance is evaluated and recommendations as to the choice of signal preprocessing system parameters are developed. Methods and Methodology of the work are based on the analysis of probabilistic characteristics of the useful and interference signals and seek to synthesize an algorithm for processing observed signal implementations according to a single quality criterion of a minimum average loss from incorrect decisions. The task of detection and estimation of moving objects in terms of their number and angular coordinates is formulated as a statistical problem. The angular resolution algorithm for point targets is built on the principle of statistical hypothesis testing. The a priori information about the characteristics of useful and interference signals during the target movement and, also, about the target probabilistic properties allows a procedure of signal processing to be built through the spatial synthesis of the observation system aperture, offering a resolution exceeding the energy limiting resolution. Results. Analytical relations have been obtained for the joint procedure of signal detection and arrival angle measurement. The estimated angular positions of the targets are used in the aperture synthesis algorithm. It has been shown that the real-time aperture synthesis is possible in the case of relatively high signal-to-noise ratios and linear and unidirectional movement of targets. Conclusions. A two-channel algorithm for monitoring moving targets has been developed. Based on the a priori information about the nature of target movement and on the estimates of angular positions and numbers of the targets, a passive synthesis procedure for the observation system aperture has been constructed. Recommendations have been developed for algorithm implementations in real time.

In this article, we investigate the resolvent operator of a singular q-Dirac system. We obtain an integral representations for the resolvent of this system, in terms of the spectral function. Furthermore, we give a formula for the Titchmarsh-Weyl function of q-Dirac system using the integral representation of the resolvent.

A key aspect of ultracold bosonic quantum gases in deep optical lattice potential wells is the realization of the strongly interacting Mott insulating phase. Many characteristics of this phase are well understood, however little is known about the effects of a random external potential on its gapped quasiparticle and quasihole low-energy excitations. In the present study we investigate the effect of disorder upon the excitations of the Mott insulating state at zero temperature described by the Bose–Hubbard model. Using a field-theoretical approach we obtain a resummed expression for the disorder ensemble average of the spectral function. Its analysis shows that disorder leads to an increase of the effective mass of both quasiparticle and quasihole excitations. Furthermore, it yields the emergence of damped states, which exponentially decay during propagation in space and dominate the whole band when disorder becomes comparable to interactions. We argue that such damped-localized states correspond to single-particle excitations of the Bose-glass phase.

Abstract This paper is devoted to studying a q-analog of the singular Dirac problem. First, we investigate some spectral properties of the problem. Then we prove the existence of a spectral function and establish a Parseval’s equality, for the singular q-Dirac system in a Hilbert space. Although there were given some results for this type of problem, we think that Parseval’s equality has not been studied yet.

Subject and Purpose. The subject of the research is a digital signal processing system of a radar, consisting of an input filter, an analog-to-digital converter and a processor that implements an algorithm for detecting of information signal in accordance with some criterion given in advance. The aim of the work is to estimate losses in the signal-to-noise ratio during the transition from optimal to quasi-optimal signal processing taking into account the spectral characteristics of signals and interference. The methods and methodology of work are based on the analysis of the spectral functions of useful signals and noise and the calculation of errors of their approximation using lattice functions at real sample durations. It is shown that the approximation errors of the spectrum of the useful signal and the noise spectrum at the output of the smoothing filter have a different effect on the signal processing result. To accumulate the signal and reduce the noise level in real time, it was proposed by minimizing the signal-to-noise losses during quasi-optimal signal processing by choosing the frequency, duration of the lattice function samples and the width of the filter frequency response. Results of work. Analytical relations have been obtained for estimating losses in the signal-to-noise ratio for quasi-optimal signal processing with respect to the optimal one. It is shown that at limiting frequencies an additional loss reduction can be obtained by changing the sampling duration. Conclusion Discrete quasi-optimal signal processing at fixed sample durations and frequency bands of matching filters can lead to significant signal-to-noise losses. To reduce losses, it is necessary to find a compromise between the requirements for the speed of the processing system, the accuracy of the approximation of the spectral characteristics of signals and noise, and the passband of the matching filter.

In this paper, we provide a full conjugacy and subdifferential calculus for convex convex-composite functions in finite-dimensional space. Our approach, based on infimal convolution and cone convexity, is straightforward. The results are established under a verifiable Slater-type condition, with relaxed monotonicity and without lower semicontinuity assumptions on the functions in play. The versatility of our findings is illustrated by a series of applications in optimization and matrix analysis, including conic programming, matrix-fractional, variational Gram, and spectral functions. [ABSTRACT FROM AUTHOR]

The GW method is a standard method to calculate the electronic band structure from first principles. It has been applied to a large variety of semiconductors and insulators but less often to metallic systems, in particular, with respect to a self-consistent employment of the method. In this work, we take a look at all-electron quasiparticle self-consistent GW (QSGW) calculations for simple metals (alkali and alkaline earth metals) based on the full-potential linearized augmented-plane-wave approach and compare the results to single-shot (i.e., non-selfconsistent) G0W0 calculations, density-functional theory (DFT) calculations in the local-density approximation, and experimental measurements. We show that, while DFT overestimates the bandwidth of most of the materials, the GW quasiparticle renormalization corrects the bandwidths in the right direction, but a full self-consistent calculation is needed to consistently achieve good agreement with photoemission data. The results mainly confirm the common belief that simple metals can be regarded as nearly free electron gases with weak electronic correlation. The finding is particularly important in light of a recent debate in which this seemingly established view has been contested.

n this paper, we investigate the resolvent operator of the singular q-Sturm-Liouville problem defined as − ( 1 / q ) D q ⁻ ¹ [D q y ( x )] + [r ( x ) -λ ]y ( x )=0, with the boundary condition y ( 0, λ ) c o s β + D q ⁻ ¹ y ( 0, λ ) s i n β = 0, where λ ∈ C, r is a real function defined on [ 0, ∞ ), continuous at zero and r ∈ L q, l o c ¹ ( 0, ∞ ) . We give an integral representation for the resolvent operator and investigate some properties of this operator. Furthermore, we obtain a formula for the Titchmarsh-Weyl function of the singular q -Sturm-Liouville problem. [ABSTRACT FROM AUTHOR]

The discrete Schrodinger equation on a half-line lattice with the Dirichlet boundary condition is considered when the potential is real valued, is summable, and has a finite first moment. The Darboux transformation formulas are derived from first principles showing how the potential and the wave function change when a bound state is added to or removed from the discrete spectrum of the corresponding Schrodinger operator without changing the continuous spectrum. This is done by explicitly evaluating the change in the spectral density when a bound state is added or removed and also by determining how the continuous part of the spectral density changes. The theory presented is illustrated with some explicit examples.

In the present study, we investigate the existence of spectral functions and obtain the Parseval identity and expansion formula in eigenfunctions for the singular q-Sturm–Liouville problem with transmission conditions. [ABSTRACT FROM AUTHOR]

In this work, we prove the existence of a spectral function for singular conformable fractional Dirac system. Further, we establish a Parseval equality and spectral expansion formula by terms of the spectral function. [ABSTRACT FROM AUTHOR]

I review the application of self-consistent Green's function methods to study the properties of infinite nuclear systems. Improvements over the last decade, including the consistent treatment of three-nucleon forces and the development of extrapolation methods from finite to zero temperature, have allowed for realistic predictions of the equation of state of infinite symmetric, asymmetric and neutron matter based on chiral interactions. Microscopic properties, like momentum distributions or spectral functions, are also accessible. Using an indicative set of results based on a subset of chiral interactions, I summarize here the first-principles description of infinite nuclear systems provided by Green's function techniques, in the context of several issues of relevance for nuclear theory including, but not limited to, the role of short-range correlations in nuclear systems, nuclear phase transitions and the isospin dependence of nuclear observables.

A new stochastic approach for the approximation of (nonlinear) Lipschitz operators in normed spaces by their eigenvectors is shown. Different ways of providing integral representations for these approximations are proposed, depending on the properties of the operators themselves whether they are locally constant, (almost) linear, or convex. We use the recently introduced notion of eigenmeasure and focus attention on procedures for extending a function for which the eigenvectors are known, to the whole space. We provide information on natural error bounds, thus giving some tools to measure to what extent the map can be considered diagonal with few errors. In particular, we show an approximate spectral theorem for Lipschitz operators that verify certain convexity properties.