arXiv subjects

Víctor A. Vicente-Benítez

Publications and source records attributed to Víctor A. Vicente-Benítez.

9 recordsLinked to original sources

arXiv2026

Neumann series of Bessel functions for the solutions of the Sturm-Liouville equation in impedance form and related boundary value problems

We present a Neumann series of spherical Bessel functions representation for solutions of the Sturm--Liouville equation in impedance form \[ (κ(x)u')' + λκ(x)u = 0,\quad 0 < x < L, \] in the case where $κ\in W^{1,2}(0,L)$ and has no zeros on the interval of interest. The $x$-dependent coefficients of this representation can be constructed explicitly by means of a simple recursive integration procedure. Moreover, we derive bounds for the truncation error, which are uniform whenever the spectral parameter $ρ=\sqrtλ$ satisfies a condition of the form $|\operatorname{Im}ρ|\leq C$. Based on these representations, we develop a numerical method for solving spectral problems that enables the computation of eigenvalues with non-deteriorating accuracy.

math.CA↗

arXiv2026

Transmutation operators for Schrödinger equations with distributional potentials and the associated impedance equation

We present the construction of an integral transmutation operator for the Schrödinger equation \[ -y'' + q(x)y = λy, \quad x \in J, \ λ\in \mathbb{C}, \] in the case where $q$ is the distributional derivative of an $L^2$ function on a bounded interval $J \subset \mathbb{R}$. Such a transmutation operator transforms solutions of $ v'' + λv = 0 $ into solutions of the Schrödinger equation. The construction of the integral transmutation operator relies on a new regularization of the distributional Schrödinger equation based on the Polya factorization in terms of a solution $f$ that does not vanish on the closure of $J$. The existence of such a function $f$ is established, together with a constructive method for its computation. As a consequence of the Polya factorization, we obtain an integro-differential transmutation operator for the associated Sturm--Liouville operator in impedance form related to $f$, along with smoothness conditions for the transmutation kernel. Furthermore, we introduce the Darboux transform for both the Schrödinger and impedance operators and describe their relationships with the corresponding transmutation operators. Finally, we develop several series representations for the solutions, including the spectral parameter power series and the Neumann series of spherical Bessel functions.

math.CA↗

arXiv2025

Complete systems of solutions, transmutations and Darboux transform for Sturm-Liouville equations in impedance form

We present the construction of a complete system of functions associated with the Sturm-Liouville equation in impedance form on a finite interval $I$, given an impedance function $a\in L^2(I)$. The system, known as the formal powers, is generated through recursive integration of the impedance function $a$ and its reciprocal. We establish the completeness of this system in the space $L^p$ with the weight function $a^2$. Under additional conditions on $a$, we extend the completeness of this completeness to Sobolev spaces $W^{1,p}(I)$, along with a generalized Taylor formula. We show that the completeness of the formal powers implies key analytic properties for a transmutation operator associated with the Sturm-Liouville equation in impedance form, including the existence of a continuous inverse. Finally, we introduce a formulation of the Darboux-transformed equation and establish a relation between the transmutation operators to the original and transformed equations.

math.CA↗

arXiv2025

Entire solutions to a quasilinear purely critical competitive system

We establish the existence of a fully nontrivial solution with nonnegative components for a weakly coupled competitive system for the $p$-Laplacian in $\mathbb{R}^N$ whose nonlinear terms are purely critical. We also show that the purely critical equation for the $p$-Laplacian in $\mathbb{R}^N$ has infinitely many nodal solutions.

math.AP↗

arXiv2024

Bergman spaces and reproducing kernel for the biquaternionic Vekua equation

We analyze the main properties of the Bergman spaces of weak $L_p$- solutions for a biquaternionic Vekua equation of the form \[ \mathbf{D}w(x)-\mathbf{Q}_Aw(x)=0 \] on bounded domains of $\mathbb{R}^3$, where the operator $\mathbf{Q}_A$ involves quaternionic conjugation and multiplications, both left and right, by essentially bounded functions. Properties such as completeness, separability, and reflexivity are shown. It is demonstrated that the solutions belonging to the Bergman spaces are locally Hölder continuous and that the evaluation maps are bounded in the $L_p$-norm. Consequently, for the case $p=2$, we obtain a reproducing integral kernel and an explicit formula for the orthogonal projection onto the Bergman space. For $1<p<\infty$, the explicit form for the annihilator of the Bergman space in the dual $L_{p'}$ is presented, along with an orthogonal decomposition for $L_2$.

math.AP↗

arXiv2024

Schrödinger equation with finitely many $δ$-interactions: closed form, integral and series representations for solutions

A closed form solution for the one-dimensional Schrödinger equation with a finite number of $δ$-interactions \[ \mathbf{L}_{q,\mathfrak{I}_{N}}y:=-y^{\prime\prime}+\left( q(x)+\sum _{k=1}^{N}α_{k}δ(x-x_{k})\right) y=λy,\quad0<x<b,\;λ\in\mathbb{C}% \] is presented in terms of the solution of the unperturbed equation \[ \mathbf{L}_{q}y:=-y^{\prime\prime}+q(x)y=λy,\quad0<x<b,\;λ\in\mathbb{C}% \] and a corresponding transmutation operator $\mathbf{T}_{\mathfrak{I}_{N}}^{f}$ is obtained in the form of a Volterra integral operator. With the aid of the spectral parameter power series method, a practical construction of the image of the transmutation operator on a dense set is presented, and it is proved that the operator $\mathbf{T}_{\mathfrak{I}_{N}}^{f}$ transmutes the second derivative into the Schrödinger operator $\mathbf{L}_{q,\mathfrak{I}_{N}}$ on a Sobolev space $H^{2}$. A Fourier-Legendre series representation for the integral transmutation kernel is developed, from which a new representation for the solutions and their derivatives, in the form of a Neumann series of Bessel functions, is derived.

math.CA↗

arXiv2024

Optimal pinwheel partitions and pinwheel solutions to a nonlinear Schrödinger system

We establish the existence of a solution to a nonlinear competitive Schrödinger system whose scalar potential tends to a positive constant at infinity with an appropriate rate. This solution has the property that all components are invariant under the action of a group of linear isometries and each component is obtained from the previous one by composing it with some fixed linear isometry. We call it a pinwheel solution. We describe the asymptotic behavior of the least energy pinwheel solutions when the competing parameter tends to zero and to minus infinity. In the latter case the components are segregated and give rise to an optimal pinwheel partition for the Schrödinger equation, that is, a partition formed by invariant sets that are mutually isometric through a fixed isometry.

math.AP↗

arXiv2024

Bergman spaces for the bicomplex Vekua equation with bounded coefficients

We develop the theory for the Bergman spaces of generalized $L_p$-solutions of the bicomplex-Vekua equation $\overline{\boldsymbol{\partial}}W=aW+b\overline{W}$ on bounded domains, where the coefficients $a$ and $b$ are bounded bicomplex-valued functions. We study the completeness of the Bergman space, the regularity of the solutions, and the boundedness of the evaluation functional. For the case $p=2$, the existence of a reproducing kernel is established, along with a representation of the orthogonal projection onto the Bergman space in terms of the obtained reproducing kernel, and an explicit expression for the orthogonal complement. Additionally, we analyze the main Vekua equation ($a=0$, $b = \frac{\overline{\boldsymbol{\partial}}f}{f}$ with $f$ being a non-vanishing complex-valued function). Results concerning its relationship with a pair of conductivity equations, the construction of metaharmonic conjugates, and the Runge property are presented.

math.AP↗

arXiv2023

Transmutation operators and complete systems of solutions for the radial Bicomplex Vekua equation

The construction of a pair of transmutation operators for the radial main Vekua equation with a Bicomplex-valued coefficient is presented. The pair of operators transform the Bicomplex analytic functions into the solutions of the main Vekua equation. The analytical properties of the operators in the space of classical solutions and the pseudoanalytic Bergman space are established. With the aid of such operators, a complete system of solutions for the main Vekua equation, called radial formal powers, is obtained. The completeness of the radial formal powers is proven with respect to the uniform convergence on compact subsets and in the Bicomplex pseudoanalytic Bergman space with the $L_2$-norm. For the pseudoanalytic Bergman space on a disk, we show that the radial formal powers are an orthogonal basis, and a representation for the Bergman kernel in terms of such basis is obtained.

math.CV↗