Related papers: The Sturm-Liouville problem and the Polar Represen…

A central result of Sturm-Liouville theory (also called the Sturm-Hurwitz Theorem) states that if $\phi_k$ is a sequence of eigenfunctions of a second order differential operator on the interval $I \subset \mathbb{R}$, then any linear…

There is $c_{} > 0$ such that for all $f \in C[0,\pi]$ with at most $d-1$ roots inside $(0,\pi)$ $$ \sum_{1 \leq n \leq d}{ \left| \left\langle f, \sin\left( n x\right) \right\rangle \right|} \geq \kappa^{-\kappa^2 \log{\kappa}}\|f\|_{L^2}…

Sturm-Liouville oscillation theory for periodic Jacobi operators with matrix entries is discussed and illustrated. The proof simplifies and clarifies the use of intersection theory of Bott, Maslov and Conley-Zehnder. It is shown that the…

We establish two theorems that illustrate the uniqueness of inverse q-Sturm-Liouville problems based on a specified set of spectral data. The first uniqueness theorem employs the method of transformation operators to provide a q-analog of…

The paper contains integral representations for certain classes of exponentially growing solutions of second order periodic elliptic equations. These representations are the analogs of those previously obtained by S. Agmon, S. Helgason, and…

In this study, we consider a boundary value problem generated by the Sturm-Liouville problem with a frozen argument and with non-separated boundary conditions on a time scale. Firstly, we present some solutions and characteristic function…

In this paper, we formulate a regular $q$-fractional Sturm--Liouville problem (qFSLP) which includes the left-sided Riemann--Liouville and the right-sided Caputo q-fractional derivatives of the same order $\alpha$, $\alpha\in (0,1)$. The…

The non-commutative harmonic oscillator (NCHO) was introduced as a specific Hamiltonian operator on $L^2(\mathbb{R})\otimes\mathbb{C}^2$ by Parmeggiani and Wakayama. Then it was proved by Ochiai and Wakayama that the eigenvalue problem for…

In this paper, Sturm-Liouville problem for difference equations is considered with potential function q(n). The representations of solutions are obtained by variation of parameters method. These solutions are proved, using summation by…

The one-dimensional harmonic oscillator wave functions are solutions to a Sturm-Liouville problem posed on the whole real line. This problem generates the Hermite polynomials. However, no other set of orthogonal polynomials can be obtained…

We consider the nonlinear equation $$-u'' = f(u) + h , \quad \text{on} \quad (-1,1),$$ where $f : {\mathbb R} \to {\mathbb R}$ and $h : [-1,1] \to {\mathbb R}$ are continuous, together with general Sturm-Liouville type, multi-point boundary…

The relativistic kinetic equations describing time evolution and space dependence of the density matrices of polarized photons and electrons interacting via Compton scattering are deduced from the quantum Liouville equation. The induced…

We present some general results for the time-dependent mass Hamiltonian problem with H=-{1/2}e^{-2\nu}\partial_{xx} +h^{(2)}(t)e^{2\nu}x^2. This Hamiltonian corresponds to a time-dependent mass (TM) Schr\"odinger equation with the…

This research was devoted to investigate the inverse spectral problem of Sturm-Liouville operator with many frozen arguments. Under some assumptions, the authors obtained uniqueness theorems. At the end, a numerical simulation for the…

We obtain asymptotic representations as $\lambda \to \infty$ in the upper and lower half-planes for the solutions of the Sturm--Liouville equation $$ -y"+p(x)y'+q(x)y= \lambda ^2 \rho(x)y, \qquad x\in [a,b] \subset \mathbb{R}, $$ under the…

Ambarzumyan's theorem for quadratic Sturm-Liouville problem is extended to second order differential systems of dimension d. It is shown that if the spectrum is the same as the spectrum belonging to the zero potential, then the matrix…

The first part of this thesis studies the notion of a "quantum representation", introduced by J.-M. Souriau in order to provide a polarization-free characterization of the Lie group representations attached to coadjoint orbits. When the…

We study a second-order differential equation involving a quasi-derivative, leading to a non-self-adjoint Sturm--Liouville-type problem with four coefficient functions. To analyze this equation, we develop a generalized Pr\"ufer…

We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem $$ \Delta^2 u=|u|^{p-1}u\ \{in} \ \R^n,$$ where $ p>1$ and $n\ge1$. We give a complete classification of stable and finite Morse index…

We study spectral properties of energy-dependent Sturm-Liouville equations, introduce the notion of norming constants and establish their interrelation with the spectra. One of the main tools is the linearization of the problem in a…