Mittag-Leffler Expansion Research Articles

Convergence of resonant state expansions for transmission in double delta resonators

We investigate the transmission features of symmetric double Dirac delta resonators using a non-Hermitian quantum approach, based on resonant state expansions involving complex energy eigenvalues. We focus on the convergence properties of these expansions, which involve a sum of resonance and anti-resonance terms We demonstrate that systems with transmission profiles featuring sharp and isolated resonances converge more rapidly, whereas those with overlapping resonances exhibit slower convergence. We also show that for the former systems, by taking into account only the resonance terms, we can accurately describe the transmission coefficient as a sum of Breit-Wigner resonances, each distinctly characterized by its energy and width. We demonstrate that there is a one-to-one correlation between transmission peaks and the resonance energies of the system. We also compare the convergence properties of transmission using Mittag-Leffler expansions, noting their slower rates compared to resonant state expansions. These findings emphasize the advantage of using non-Hermitian resonant expansions for analyzing quantum mechanical systems, providing a clearer understanding of systems characterized by resonant features.

Time-varying equivalent circuit for dynamically modulated dielectric slabs and its implications for stability analysis.

Equivalent circuits (ECs) have played an instrumental role in analyzing and modeling distributed electromagnetic structures for many decades, owing to their intrinsic simplicity and the valuable intuition and physical insights they provide. Here, we envision what we believe to be a novel class of ECs featuring linear time-varying (LTV) circuit elements to harness the power of EC analysis for studying LTV structures, particularly LTV dielectric slabs. Our time-varying equivalent circuit (TVEC)-consisting of infinitely many time-varying LC resonators interconnected in either series or parallel-is rigorously derived and exact, with closed-form expressions for the circuit elements. Its convenient form, enabled by a judicious use of Mittag-Leffler expansion, simplifies the analysis of LTV slabs experiencing various resonant and harmonic phenomena. We demonstrate this point by applying our TVEC to the problem of parametric instability in linear time-periodic slabs. Despite its immense importance, instability has been largely neglected in the literature on periodically modulated structures, mainly due to the inherent complexity of problems involving partial differential equations with time-periodic coefficients. Time-periodic circuits, on the other hand, enjoy an abundance of mathematical tools and methods in the form of the theory of Hill's equation. We utilize this rich theory to investigate the nature of parametric instability in linear time-periodic slabs. Numerical results are then presented to verify these findings. Our TVEC is simple, applicable to any form of temporal modulation (not just periodic), and potentially extensible to other geometries, making it likely to have a significant impact on a multitude of hotly researched LTV topics.

Quantum mechanical-like approach with non-Hermitian effective Hamiltonians in spin-orbit coupled optical cavities

We present a comprehensive analytical model of resonant states in birefringent microcavities with permeable mirrors. We derive an effective, non-Hermitian photonic Hamiltonian describing cavity mode dispersion and modal lifetimes by applying the Green's function technique based on the Mittag-Leffler expansion with respect to the resonant states and the k·p perturbation theory known from semiconductor physics. Using this formalism we obtained results which can be interpreted as effective cavity mode spin-orbit coupling. We use this method to derive the two-mode Hamiltonian to describe the properties of light in the cavity, which significantly reduces the computational effort and properly captures the polarization of the eigenmodes. This is done by introducing the necessary corrections to the Hamiltonian matrix and modifying the basis modes resulting from the coupling with other optical modes present in the system. This simplified Hamiltonian allows us to determine the positions of exceptional points in momentum space. These points are connected by Fermi arcs and appear due to non-Hermiticity and PT symmetry. Published by the American Physical Society 2024

Mittag–Leffler expansions for inverse spectral problems with mixed data

We consider the inverse spectral problem for the Sturm–Liouville problem in the interval with the Dirichlet boundary conditions at both ends. We provide a method for the unique reconstruction of the potential that is known a priori on the subinterval , where m and r being positive integers and satisfying , by using the spectral data set consisting of all the eigenvalues and a subset of the norming constants. The method is based on the Mittag–Leffler decomposition which can help us to decompose an entire function of exponential type into two functions of smaller types. This decomposition allows the use of Lagrange interpolation in order to reconstruct the potential in the Sturm–Liouville problem. We also provide necessary and sufficient conditions related to the existence issues.

Ramanujan’s formula for odd zeta values: a proof by Mittag-Leffler expansion and applications

Using Mittag-Leffler expansion a novel proof of Ramanujan’s famous formula for zeta (2m-1) is presented. Here, the formula can be derived by Taylor series of a generating function and its Mittag-Leffler expansion. Furthermore, generalized formulas for the fast-converging series of Plouffe are offered. But Ramanujan’s equation not only produces series for odd zeta values, surprisingly it is shown that in the limit to its singularity it provides the identities for even zeta values, namely Euler’s classical formula. Finally, a new triangle identity of Ramanujan’s formula is presented, hinting that properties and symmetries of the equation are far from all uncovered. For this work, a new representation of the equations is introduced, that significantly simplifies analyses and applications.

Analytic Map of Three-Channel S Matrix: Generalized Uniformization and Mittag-Leffler Expansion.

We explore the analytic structure of the three-channel S matrix by generalizing uniformization and making a single-valued map for the three-channel S matrix. First, by means of the inverse Jacobi's elliptic function we construct a transformation from eight Riemann sheets of the center-of-mass energy complex plane onto a torus, on which the three-channel S matrix is represented single-valued. Second, we show that the Mittag-Leffler expansion, a pole expansion, of the three-channel scattering amplitude includes not only topologically trivial but also nontrivial contributions and is given by the Weierstrass zeta function. Finally, taking a simple nonrelativistic effective field theory with contact interaction for the S=-2, I=0, J^{P}=0^{+}, ΛΛ-NΞ-ΣΣ coupled-channel scattering, we demonstrate that as a function of the uniformization variable the scattering amplitude is, in fact, given by the Mittag-Leffler expansion and is dominated by contributions from neighboring poles.

Near-threshold spectrum from a uniformized Mittag-Leffler expansion: Pole structure of the Z(3900)

We demonstrate how S-matrix poles manifest themselves as the physical spectrum near the upper threshold in the context of the two-channel uniformized Mittag-Leffler expansion, an expression written as a sum of pole terms under an appropriate variable where the S-matrix is made single-valued (uniformization). We show that the transition of the spectrum is continuous as a S-matrix pole moves across the boundaries of the complex energy Riemann sheets and that the physical spectrum peaks at or near the upper threshold when the S-matrix pole is positioned sufficiently close to it on the uniformized plane. There is no essential difference on which sheet the pole is positioned. What is important is the existence of a pole near the upper threshold and the distance between the pole and the physical region, not on which complex energy sheet the pole is positioned. We also point out that when the pole is close to the upper threshold, the complex pole does not have the usual meaning of the resonance. Neither the real part represents the peak energy, nor the imaginary part represents the half width. Subsequently, we try to understand the current status of $Z(3900)$ from the viewpoint of the uniformized Mittag-Leffler expansion reflecting in particular, Phys.Rev.Lett.117, 242001 (2016) in which they concluded that $Z(3900)$ is not a conventional resonance but a threshold cusp. We point out that their results turn out to indicate the existence of S-matrix poles near the $\bar D D^*$ threshold, which is most likely the origin of the peak found in their calculation of the near-threshold spectrum. In order to support our argument, we set up a separable potential model which shares common behavior of poles near the $\bar D D^*$ threshold to the above-mentioned reference and show in our model that the structures near the $\bar D D^*$ threshold are indeed caused by these near-threshold poles.

The radii of starlikeness and convexity of the functions including derivatives of Bessel functions

Let $J_\nu(z)$ denote the Bessel function of the first kind of order $\nu.$ In this paper, our aim is to determine the radii of starlikeness and convexity for three kind of normalization of the function $N_\nu(z)=az^{2}J_{\nu }^{\prime \prime }(z)+bzJ_{\nu }^{\prime }(z)+cJ_{\nu }(z)$ in the case where zeros are all real except for a single pair, which are conjugate purely imaginary. The key tools in the proof of our main results are the Mittag-Leffler expansion for function $N_\nu(z)$ and properties of real and complex zeros of it.

Radii of starlikeness and convexity of Bessel function derivatives

UDC 517.5 In this paper, our aim is to find the radii of starlikeness andconvexity of Bessel function derivatives for three different kind of normalization. The key tools in the proof of our main results are the Mittag-Leffler expansion for th derivative of Bessel function andproperties of real zeros of it. In addition, by using the Euler–Rayleigh inequalities we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the normalized th derivative of Bessel function. The main results of the paper are natural extensions of some known results on classical Bessel functions of the first kind.

Application of the uniformized Mittag-Leffler expansion to Λ(1405)

We study the pole properties of $\Lambda(1405)$ in a model-independent manner by applying the Uniformized Mittag-Leffler expansion proposed in our previous paper. The resonant energy, width and residues are determined by expanding the observable as a sum of resonant-pole pairs under an appropriate parameterization which expresses the observable to be single-valued, and fitting it to experimental data of the invariant-mass distribution of $\pi^+\Sigma^-$, $\pi^-\Sigma^+$, $\pi^0\Sigma^0$ final states in the reaction, $\gamma p \rightarrow K^+ \pi \Sigma$, and the elastic and inelastic cross section, $K^-p\to K^-p$, $\bar{K}^0n$, $\pi^+\Sigma^-$, $\pi^-\Sigma^+$. As we gradually increase the number of pairs from one to three, the first pair converges while the second and third pairs emerge further and further away from the first pair, implying that the Uniformized Mittag-Leffler expansion with three pairs is almost convergent in the vicinity of the $\Lambda(1405)$. The broad peak structure between the $\pi\Sigma$ and $\bar{K} N$ thresholds regarded to be $\Lambda(1405)$ is explained by a single pair with a resonant energy of 1420 $\pm$ 1 MeV, and a half width of 48 $\pm$ 2 MeV, which is consistent with the single-pole picture of $\Lambda(1405)$. We conclude that the Uniformized Mittag-Leffler expansion turns out to be a very powerful method to obtain resonance energy, width and residues from the near-threshold spectrum.

Spectral asymptotic and positivity for singular Dirichlet-to-Neumann operators

In the framework of Hilbert spaces we shall give necessary and sufficient conditions to define a Dirichlet-to-Neumann operator via Dirichlet principle. Analyzing singular Dirichlet-to-Neumann operators, we will establish Laurent expansion near singularities as well as Mittag–Leffler expansion for the related quadratic forms. The established results will be exploited to solve definitively the problem of positivity of the related semigroup in the framework of Lebesgue spaces. The obtained results are supported by some examples where we analyze properties of singular Dirichlet-to-Neumann operators related to Neumann and Robin Laplacian on Lipschitz domains. Among other results, we shall demonstrate that regularity of the boundary may affect positivity and derive Mittag-Leffler expansion for the eigenvalues of singular Dirichlet-to-Neumann operators.

Geometric and monotonic properties of Ramanujan type entire functions

In this paper, our aim is to find the radii of starlikeness and convexity of the Ramanujan type function for three different kinds of normalization by using their Mittag–Leffler expansion in such a way that the resulting functions are analytic in the unit disk of the complex plane. A result of Zhang (Proc Am Math Soc 145:241–250, 2017) on the reality of the zeros of Ramanujan type entire functions play important roles in this paper. Moreover, the interlacing properties of the zeros of Ramanujan type functions and its derivative are also useful in the proof of the main results. In addition, by using the Euler–Rayleigh inequalities, we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the Ramanujan type entire functions. Finally, we give monotonicity and Redheffer-type inequalities for this function.

Unique reconstruction of the potential for the interior transmission eigenvalue problem for spherically stratified media

This work deals with the interior transmission eigenvalue problem for a spherically stratified medium supported in {x: |x| = r = b}, which can be formulated as −y″ + q(x)y = λ2y with boundary conditions y(0) = 0 = y′(1) cos(aλ) − y(1) sin(aλ)/λ, where a = b/B with B being the size of the wave speed, measured by an integral. We provide a necessary and sufficient condition for the existence issue by giving a new method that allows unique reconstruction of a potential of this Sturm–Liouville problem from the spectrum of the problem and the set of the norming constants corresponding to the real eigenvalues when 0 < a < 1; and from the spectrum together with one additional piece of information when a = 1. The method is based on Mittag-Leffler expansions which can help us to decompose an entire function of exponential type into two functions of more smaller exponential types. This decomposition provides us a well-suited situation for utilizing Levin–Lyubarski interpolation formula to reconstruct the potential for our problem.

An Efficient Algorithm for Locating TE and TM Poles for a Class of Multiscale Inhomogeneous Media Problems

The Mittag-Leffler (ML) expansion method has been minimally studied in the electromagnetic theory community. Therefore, in this article, the ML expansion functions are effectively utilized to locate initial poles for practical dielectric materials backed by a ground plane. The initial poles from the ML expansion and other existing methods are, then, checked for convergence, and found that the proposed method is highly efficient for electrically thick substrates (ETSs). To understand why, a fundamental and intrinsic convergence (in terms of relative or absolute errors) analysis of ML expansion and Taylor series expansion functions has been carried out and it was found out that the ML expansion converges significantly faster. In addition, Pade approximant has been successfully applied at no computational expense to improve the accuracy of initial pole locations. Moreover, proper and improper surface wave (SW) modes have been clearly distinguished from the poles obtained using the ML expansion method. Finally, the spectral region between leaky and SW poles for ETSs is identified using the initial pole locations obtained from using the ML expansion function on the transverse electric characteristic mode equation.

Different manifestations of S-matrix poles

Making use of the analytical properties of the S-matrix and a theorem of Mittag-Leffler, model independent non-relativistic expressions for cross sections in single channel elastic scattering, scattering phase shifts and survival probabilities of resonances are derived. Provided certain conditions are satisfied by the poles, the residues can also be estimated analytically. Considerations of the low energy behaviour of the S-matrix and cross sections reveal additional conditions on the residues of the poles appearing in the Mittag-Leffler expansions. The exact expressions for the resonant cross section and phase shift are shown to reduce to the commonly used Breit–Wigner formula plus corrections. The latter is shown to approach the exact result with the example of a meson and a baryon resonance. Finally, a comparison of the exact expressions with some realistic examples is presented. The calculations of survival probabilities in particular reveal the reason behind the non-observability of non-exponential decay.

Radii of $$\varvec{\alpha }$$ α -Convexity of Some Normalized Bessel Functions of the First Kind

In this paper our aim is to determine the radii of \(\alpha -\)convexity of the normalized Bessel functions for two different kinds of normalization in the case when the order is between \(-2\) and \(-1\). The key tools in the proof of our main results are the Mittag-Leffler expansion for Bessel functions, properties of zeros of the Bessel functions and their derivatives and some inequalities for complex and real numbers.

Turán type inequalities for Struve functions

Some Turán type inequalities for Struve functions of the first kind are deduced by using various methods developed in the case of Bessel functions of the first and second kind. New formulas, like Mittag–Leffler expansion, infinite product representation for Struve functions of the first kind, are obtained, which may be of independent interest. Moreover, some complete monotonicity results and functional inequalities are deduced for Struve functions of the second kind. These results complement naturally the known results for a particular case of Lommel functions of the first kind, and for modified Struve functions of the first and second kind.

Modified Dini functions: monotonicity patterns and functional inequalities

We deduce some new functional inequalities, like Tur\'an type inequalities, Redheffer type inequalities, and a Mittag-Leffler expansion for a special combination of modified Bessel functions of the first kind, called modified Dini functions. Moreover, we show the complete monotonicity of a quotient of modified Dini functions by introducing a new continuous infinitely divisible probability distribution. The key tool in our proofs is a recently developed infinite product representation for a special combination of Bessel functions of the first, which was very useful in determining the radius of convexity of some normalized Bessel functions of the first kind.

Close‐to‐convexity of normalized Dini functions

In this paper necessary and sufficient conditions are deduced for the close‐to‐convexity of some special combinations of Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives and some newly discovered Mittag–Leffler expansions for Bessel functions of the first kind.

Starlikeness of Bessel functions and their derivatives

In this paper necessary and sufficient conditions are obtained for the starlikeness of Bessel functions of the first kind and their derivatives of the second and third order by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives and Mittag-Leffler expansions for the derivatives of Bessel functions of the first kind, as well as some results on the zeros of these functions. Mathematics subject classification (2010): 33C10, 30C45.