Sturm-Liouville Research Articles

Mathieu–Hill Equation Stability Analysis for Trapped Ions: Anharmonic Corrections for Nonlinear Electrodynamic Traps

The stability properties of the Hill equation are discussed, especially those of the Mathieu equation that characterize ion motion in electrodynamic traps. The solutions of the Mathieu-Hill equation for a trapped ion are characterized by employing the Floquet theory and Hill’s method solution, which yields an infinite system of linear and homogeneous equations whose coefficients are recursively determined. Stability is discussed for parameters a and q that are real. Characteristic curves are introduced naturally by the Sturm–Liouville problem for the well-known even and odd Mathieu equations cem(z,q) and sem(z,q). In the case of a Paul trap, the stable solution corresponds to a superposition of harmonic motions. The maximum amplitude of stable oscillations for ideal conditions (taken into consideration) is derived. We illustrate the stability diagram for a combined (Paul and Penning) trap and represent the frontiers of the stability domains for both axial and radial motion, where the former is described by the canonical Mathieu equation. Anharmonic corrections for nonlinear Paul traps are discussed within the frame of perturbation theory, while the frontiers of the modified stability domains are determined as a function of the chosen perturbation parameter and we demonstrate they are shifted towards negative values of the a parameter. The applications of the results include but are not restricted to 2D and 3D ion traps used for different applications such as mass spectrometry (including nanoparticles), high resolution atomic spectroscopy and quantum engineering applications, among which we mention optical atomic clocks and quantum frequency metrology.

Spectral properties of Sturm–Liouville operators on infinite metric graphs

Spectral properties of Sturm–Liouville operators on infinite metric graphs

How the twofold light cone arises from nonlinear electrodynamics

We analyze the propagation of light signals in the context of nonlinear electrodynamics. As a general feature of the nonlinear theories, the superposition principle is no longer satisfied. In the electromagnetic theory, this is due to the self-interactions of the field and light propagation is governed by an effective or optical metric. We present a simple derivation of the two light cones that arise if the Lagrangian depends on the electromagnetic invariants in a nonlinear way. Using the algebraic properties of the electromagnetic tensor fμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_{\\mu \ u }$$\\end{document}, we determine the dispersion relations from the eigenvalues of a Sturm–Liouville equation. It turns out that in the presence of a background field, light propagation can be slower or faster than the one in vacuum. We also derive the corresponding transport vector fields.

Existence of Solutions to the Non-Self-Adjoint Sturm–Liouville Problem with Discontinuous Nonlinearity

Existence of Solutions to the Non-Self-Adjoint Sturm–Liouville Problem with Discontinuous Nonlinearity

Recovering of the Stiffness Coefficients of the Sturm–Liouville Operator on a Star Graph from a Finite Set of Its Eigenvalues

Recovering of the Stiffness Coefficients of the Sturm–Liouville Operator on a Star Graph from a Finite Set of Its Eigenvalues

Quantum mechanics on sharply bent wires via two-interval Sturm-Liouville theory

We study quantum mechanics on the limiting case of a highly curved wire by approximating the physics around the curved region by three boundary condition parameters coming from the two-interval Sturm-Liouville theory. Since the geometric potential in this case is strong and non-integrable, these parameters depend crucially on the regularization of the curve. Hence, unless we know precisely the shape of the curve, any physical prediction is meaningless. In this context, the method presented in this paper becomes not only a useful approximation, but also a necessary scheme to deal with quantum mechanics on highly curved wires and (possibly) corners.

Nucleons and vector mesons in a confining holographic QCD model

We present a simple holographic QCD model that provides a unified description of vector mesons and nucleons in a confining background based on Einstein-dilaton gravity. For the confining background, we consider analytical solutions of the Einstein-dilaton equations where the dilaton is a quadratic function of the radial coordinate far from the boundary. We build actions for the 5D gauge field and the 5D Dirac field dual to the 4D flavor current and the 4D nucleon interpolator, respectively. In order to obtain asymptotically linear Regge trajectories, we impose for each sector the condition that the effective Schrödinger equation has a potential that grows quadratically in the radial coordinate far from the boundary. For the vector mesons, we show that this condition is automatically satisfied by a 5D Yang-Mills action minimally coupled to the metric and the dilaton. For the nucleons, we find that the mass term of the 5D Dirac action needs to be generalized to include couplings to the metric and the dilaton. Using Sturm-Liouville theory, we obtain a spectral decomposition for the hadronic correlators consistent with large Nc QCD. Our setup contains only three parameters: the mass scale associated with confinement, the 5D gauge coupling, and the 5D Dirac coupling. The last two are completely fixed by matching the correlators at high energies to perturbative QCD. We calculate masses and decay constants and compare our results against available experimental data. Our model can be thought of as a consistent embedding of soft wall models in Einstein-dilaton gravity. Published by the American Physical Society 2024

Integration of the Korteweg-de Vries equation with time-dependent coefficients in the case of moving eigenvalues of the Sturm–Liouville operator

The inverse scattering method is used to integrate the Korteweg-de Vries equation with time-dependent coefficients. We derive the evolution of the scattering data of the Sturm–Liouville operator whose coefficient is a solution of the Korteweg-de Vries equation with time-dependent coefficients. An algorithm for constructing exact solutions of the Korteweg-de Vries equation with time-dependent coefficients is also proposed; we reduce it to the inverse problem of scattering theory for the Sturm–Liouville operator. Examples illustrating the stated algorithm are given.

Gravity-matter sum rules in models with a single extra-dimension

We prove a set of sum rules needed for KK-graviton pair production from matter in orbifolded extra-dimensional models. The sum rules can be found in full generality by considering the properties of solutions to the Sturm-Liouville problem, which describes the wave functions and the masses of the KK-gravitons in four dimensions. They ensure cancellations in the amplitudes of the processes mentioned above which considerably reduce their growth with s in the high-energy limit. This protects extra-dimensional theories from the low-scale unitarity problems that plague other theories with massive spin-2 particles. We argue that such relations are valid for a broader category of models thus generalizing our previous results that were limited to the large μ limit of the Randall-Sundrum model.

The Schwarzian Approach in Sturm–Liouville Problems

A novel method for finding the eigenvalues of a Sturm–Liouville problem is developed. Following the minimalist approach, the problem is transformed to a single first-order differential equation with appropriate boundary conditions. Although the resulting equation is nonlinear, its form allows us to find the general solution by adding a second part to a particular solution. This splitting of the general solution into two parts involves the Schwarzian derivative: hence, the name of the approach. The eigenvalues that correspond to acceptable solutions can be found by requiring the second part to correct the asymptotically diverging behavior of the particular solution. The method can be applied to many different areas of physics, such as the Schrödinger equation in quantum mechanics and stability problems in fluid dynamics. Examples are presented.

On recovering Sturm–Liouville operators with two delays

Abstract We study the inverse spectral problems of recovering Sturm–Liouville differential operator with two constant delays a 1 {a_{1}} and a 2 {a_{2}} greater than one third of the interval. It has been proved that the operator can be recovered uniquely from four spectra under the condition 2 ⁢ a 1 + a 2 2 ≥ π {2a_{1}+\frac{a_{2}}{2}\geq\pi} , while it is not possible otherwise.

Countably Many Positive Symmetric Solutions For Sturm Liouville Type Boundary Conditions Of Second Order Iterative System

In this paper we consider second order iterative boundary value problem with Sturm Liouville type boundary conditions and establish the existence of countably many positive symmetric solutions by using Krasnoselskii’s fixed point theorem for operator on a cone.

Optimizing the noise control in a two-layer conduit

In the modern world, noise pollution is a major concern due to its prevalence. This work focuses on optimizing noise control in a two-layer conduit. A conduit comprises an inlet and double-outlet zones (DOZ). The upper walls of the DOZ are lined with perforated absorbent material while the lower walls are layered with fibrous. Mathematically, the physical problem is formulated with a field wave equation together with rigid and impedance boundary conditions in the respective zones. Such governing boundary value problem (BVP) leads to the Sturm-Liouville (SL) category in which standard orthogonality relations (OR) are indispensable. The system of the linear equations of the BVP is acquired with semi-analytical Mode-Matching (MM) approach by implementing the continuity conditions of sound pressures and velocities at the matching junction with the aid of OR. These systems are truncated and solved numerically with computation learning to obtain the reflected and transmitted modal amplitudes in respective zones. Due to the lining's perforated upper walls, fibrous lower walls, and reversal in the DOZ, the analysis of the reflected and transmitted powers versus frequencies is significantly observed and is shown in graphical findings. The algebraic derivation is validated by satisfying the conservation law of power flux and matching continuity conditions for impedance, perforated, and fibrous lined boundaries.

Inverse problems for discrete Sturm–Liouville operator having Bessel-type potential

In this paper, inverse problems are studied for Sturm–Liouville difference operators having p 2 − 1 / 4 m 2 type singularity. First, we define the Green's function and the corresponding Weyl function, some of their properties are investigated, and we give a method to find the potential function. Then we give the solution of the inverse nodal problem and present a constructive procedure for the solution of the inverse problem by using nodes.

An analytical fast field solver for surface permanent magnet co‐axial magnetic gears

AbstractA mathematical model of co‐axial magnetic gears is given for the prediction of magnetic field distribution and the magnetic forces. The modelling approach is based on the separation of variables method (SVM), Sturm–Liouville problem, and spectral analysis. This method is computationally more efficient and has the ability to account for finite permeability of magnetic saliency. An finite element method (FEM) validation is performed for a case study, and the results are observed to be almost matched. The computational time of the analytical model has been almost in the same order of that of the FEM for a specified rotor position (both in a matter of a second or less); however, when 100 number of rotor positions were accounted in the simulation, the computational time of FEM was increased to 82 s, while the computational complexity of SVM was almost unaffected.

A Study of the scattering analysis of the multiplicative Sturm-Liouville problem

A Study of the scattering analysis of the multiplicative Sturm-Liouville problem

A High-Resolution Exponential Spline Method and Its Convergence Analysis for Two-Point Mixed Boundary Value Problems

In this paper, we present a new numerical method for effectively solving general second-order ordinary differential equations with mixed boundary conditions. Our approach utilizes a quasi-variable mesh enabling us the flexibility to adapt the mesh density according to different boundary layer problems. By employing a discretization technique that incorporates the construction of exponential spline, we achieve third-order accuracy at internal grid points, while boundary points exhibit fourth-order accuracy. Since the method is based on off-step grid points, it eliminates the need to modify the method while employing it to singular problems. We provide computational results to various numerical problems which arise in different physical phenomena like Burger’s equation and Sturm–Liouville equation. A comparison with recent findings underscores the superior performance of our method.

Unraveling the Complexity of Inverting the Sturm–Liouville Boundary Value Problem to Its Canonical Form

The Sturm–Liouville boundary value problem (SLBVP) stands as a fundamental cornerstone in the realm of mathematical analysis and physical modeling. Also known as the Sturm–Liouville problem (SLP), this paper explores the intricacies of this classical problem, particularly the relationship between its canonical and Liouville normal (Schrödinger) forms. While the conversion from the canonical to Schrödinger form using Liouville’s transformation is well known in the literature, the inverse transformation seems to be neglected. Our study attempts to fill this gap by investigating the inverse of Liouville’s transformation, that is, given any SLP in the Schrödinger form with some invariant function, we seek the SLP in its canonical form. By closely examining the second Paine–de Hoog–Anderson (PdHA) problem, we argue that retrieving the SLP in its canonical form can be notoriously difficult and can even be impossible to achieve in its exact form. Finding the inverse relationship between the two independent variables seems to be the main obstacle. We confirm this claim by considering four different scenarios, depending on the potential and density functions that appear in the corresponding invariant function. In the second PdHA problem, this invariant function takes a reciprocal quadratic binomial form. In some cases, the inverse Liouville transformation produces an exact expression for the SLP in its canonical form. In other situations, however, while an exact canonical form is not possible to obtain, we successfully derived the SLP in its canonical form asymptotically.

Analysis of a Sturm–Liouville Problem Arising in Atmosphere

We present recent results in study of a mathematical model of the sea-breeze flow, arising from a general model of the ’morning glory’ phenomena. Based on analysis of the Dirichlet spectrum of the corresponding Sturm–Liouville problem and application of the Fredholm alternative, we establish conditions of existence/uniqueness of solutions to the given problem.

Numerical Analysis for Sturm–Liouville Problems with Nonlocal Generalized Boundary Conditions

For the generalized Sturm–Liouville problem (GSLP), a new formulation is undertaken to reduce the number of unknowns from two to one in the target equation for the determination of eigenvalue. The eigenparameter-dependent shape functions are derived for using in a variable transformation, such that the GSLP becomes an initial value problem for a new variable. For the uniqueness of eigenfunction an extra condition is imposed, which renders the right-end value of the new variable available; a derived implicit nonlinear equation is solved by an iterative method without using the differential; we can achieve highly precise eigenvalues. For the nonlocal Sturm–Liouville problem (NSLP), we consider two types of integral boundary conditions on the right end. For the first type of NSLP we can prove sufficient conditions for the positiveness of the eigenvalue. Negative eigenvalues and multiple solutions may exist for the second type of NSLP. We propose a boundary shape function method, a two-dimensional fixed-quasi-Newton method and a combination of them to solve the NSLP with fast convergence and high accuracy. From the aspect of numerical analysis the initial value problem of ordinary differential equations and scalar nonlinear equations are more easily treated than the original GSLP and NSLP, which is the main novelty of the paper to provide the mathematically equivalent and simpler mediums to determine the eigenvalues and eigenfunctions.