Inverse Square Potential Research Articles

The cubic-quintic nonlinear Schrödinger equation with inverse-square potential

We consider the nonlinear Schrödinger equation in three space dimensions with a focusing cubic nonlinearity and defocusing quintic nonlinearity and in the presence of an external inverse-square potential. We establish scattering in the region of the mass-energy plane where the virial functional is guaranteed to be positive. Our result parallels the scattering result of [11] in the setting of the standard cubic-quintic NLS.

Well-posedness and scattering for a 2D inhomogeneous NLS with Aharonov-Bohm magnetic potential

We consider the magnetic nonlinear inhomogeneous Schrödinger equationi∂tu−(−i∇+α|x|2(−x2,x1))2u=±|x|−ϱ|u|p−1u,(t,x)∈R×R2, where α∈R∖Z,ϱ>0,p>1. We establish a dichotomy between global existence and scattering versus finite-time blow-up for energy solutions below the ground state threshold in the inter-critical regime. The scattering result is obtained by employing the novel approach developed by Dodson and Murphy (A new proof of scattering below the ground state for the 3D radial focusing cubic NLS, Proc. Am. Math. Soc. (2017)). We employ Tao's scattering criteria and Morawetz estimates as the foundation of this method. The novelty of our approach lies in two aspects: firstly, we explore the case of nonzero ϱα, and secondly, we consider general energy initial data that need not be radially symmetric. The specific instance of α=0, known as INLS, has garnered significant attention in recent years. Additionally, X. Gao and C. Xu recently investigated the case of ϱ=0 in the homogeneous regime, establishing scattering theory for spherically symmetric data in their work (Scattering theory for NLS with inverse-square potential in 2D, J. Math. Anal. Appl. (2020)). In the radial framework, the aforementioned problem can be translated to the INLS with an inverse square potential, which has been extensively studied in higher-dimensional spaces. The Hardy inequality ‖|x|−1f‖L2(RN)≤2N−2‖∇f‖L2(RN), which establishes the norm equivalence ‖f‖H1≃‖f‖H1+‖|x|−1f‖L2, does not hold in two space dimensions. Consequently, it remains unclear how to handle the NLS with an inverse square potential in the H1 framework for two-dimensional space. This article stands out as the first one to address the NLS with Aharonov-Bohm magnetic potential within the inhomogeneous regime, specifically when ϱ≠0.

Carleman estimate and applications for a heat equation with multiple singularities

The goal of this article is the analyses of inverse source problem and null controllability for a heat equation with multipolar inverse-square potentials: ∑ j = 1 m μ j | x − x j | 2 , where x 1 , x 2 , … , x m are different points, defined in a bounded C 2 domain Ω ⊂ R n , and μ 1 , … , μ m are real constants. More precisely, we show that for any μ j ≤ ( n − 2 ) 2 4 , j = 1 , … , m , we can obtain a Lipschitz stability result in inverse source problems and we will show that the system is null controllable. Our proofs are mainly based on Carleman estimates, and due to multiple singularities, we give a new Carleman estimate by choosing an appropriate weight function.

Duality between the quantum inverted harmonic oscillator and inverse square potentials

In this paper we show how the quantum mechanics of the inverted harmonic oscillator (IHO) can be mapped to the quantum mechanics of a particle in a super-critical inverse square potential (ISP). We demonstrate this by relating both of these systems to the Berry–Keating system with Hamiltonian H=(xp+px)/2 . It has long been appreciated that the quantum mechanics of the ISP has an ambiguity in choosing a boundary condition near the origin and we show how this ambiguity is mapped to the IHO system. Imposing a boundary condition requires specifying a distance scale where it is applied and changes to this scale come with a renormalisation group (RG) evolution of the boundary condition that ensures observables do not directly depend on the scale (which is arbitrary). Physical scales instead emerge as RG invariants of this evolution. The RG flow for the ISP is known to follow limit cycles describing the discrete breaking of classical scale invariance in a simple example of a quantum anomaly, and we find that limit cycles also occur for the IHO. However, unlike the ISP where the continuous scaling symmetry is explicit, in the case of the IHO it is hidden and occurs because the Hamiltonian is part of a larger su(1,1) spectrum generating algebra. Our map does not require the boundary condition to be self-adjoint, as can be appropriate for systems that involve the absorption or emission of particles.

Modified Attractive Inverse-Square Potential in the Induced Electric Dipole System

Modified Attractive Inverse-Square Potential in the Induced Electric Dipole System

On Hermite-Hadamard-type inequalities for second order differential inequalities with inverse-square potential

<abstract><p>We consider the class of functions $ u\in C^2((0, \infty)) $ satisfying second-order differential inequalities in the form $ u''(x)+\frac{k}{x^2}u(x)\geq 0 $ for all $ x > 0 $. For this class of functions, we establish Hermite-Hadamard-type inequalities in both cases ($ k=\frac{1}{4} $ and $ 0 < k < \frac{1}{4} $). We next extend our obtained results to the two-dimensional case. In the limit case $ k\rightarrow 0^+ $ we deriver some existing results from the literature that are related to convex functions and convex functions on the coordinates. In our approach, we make use of some tools from ordinary differential equations.</p></abstract>

Lorentz Symmetry Violation Effects Caused by the Coupling between the Field fμγ5 and the Derivative of the Fermionic Field on One-Dimensional Potentials

In search of physics beyond the standard model, new phenomena can be relevant in low energies. In view of the Standard Model Extension is an effective field theory, in this study we explore the fermionic sector by showing that the properties of nonrelativistic quantum systems can be modified. We study one-dimensional nonrelativistic quantum systems under Lorentz symmetry violation effects caused by the coupling between the fixed vector field fμγ5 and the derivative of the fermionic field. We deal with the quantum bouncer, the attractive inverse-square potential, a modified attractive inverse-square potential, and a scalar exponential potential inside this scenario of the Lorentz symmetry violation. Then, we show that the spectra of energy are influenced by the Lorentz symmetry violation effects.

The Hardy inequality and large time behaviour of the heat equation on [formula omitted

In this paper we study the large time asymptotic behaviour of the heat equation with Hardy inverse-square potential on corner spaces RN−k×(0,∞)k, k≥0. We first show a new improved Hardy-Poincaré inequality for the quantum harmonic oscillator with Hardy potential. In view of that, we construct the appropriate functional setting in order to pose the Cauchy problem. Then we obtain optimal polynomial large time decay rates and subsequently the first term in the asymptotic expansion of the solutions in L2(RN−k×(0,∞)k). Particularly, we extend and improve the results obtained by Vázquez and Zuazua (J. Funct. Anal. 2000), which correspond to the case k=0, to any k≥0. We emphasize that the higher the value of k the better time decay rates are. We employ a different and simplified approach than Vázquez and Zuazua, managing to remove the usage of spherical harmonics decomposition in our analysis.

Rotational and inverse-square potential effects on harmonic oscillator confined by flux field in a space–time with screw dislocation

This research paper delves into the study of a nonrelativistic quantum system, considering the interplay of noninertial effects induced by a rotating frame and confinement by the Aharonov–Bohm (AB) flux field with potential in the backdrop of topological defects, specifically a screw dislocation. We first focus on the harmonic oscillator problem, incorporating an inverse-square repulsive potential. Notably, it becomes evident that the energy eigenvalues and wave functions are intricately influenced by multiple factors: the topological defect parameter [Formula: see text] (representing the screw dislocation), the presence of a rotating frame engaged in constant angular motion with speed [Formula: see text] and the external potential. Then we study the quantum behavior of nonrelativistic particles, engaging in interactions governed by an inverse-square potential, all while taking into account the effects of the rotating frame. In both scenarios, a significant observation is made: the quantum flux field’s existence brings about a shift in the energy spectrum. This phenomenon bears a resemblance to the electromagnetic Aharonov–Bohm effect.

Late-time asymptotics for geometric wave equations with inverse-square potentials

We introduce a new, physical-space-based method for deriving the precise leading-order late-time behaviour of solutions to geometric wave equations on asymptotically flat spacetime backgrounds and apply it to the setting of wave equations with asymptotically inverse-square potentials on Schwarzschild black holes. This provides a useful toy model setting for introducing methods that are applicable to more general linear and nonlinear geometric wave equations, such as wave equations for electromagnetically charged scalar fields, wave equations on extremal Kerr black holes and geometric wave equations in even space dimensions, where existing proofs for deriving precise late-time asymptotics might not apply. The method we introduce relies on exploiting the spatial decay properties of time integrals of solutions to derive the existence and precise genericity properties of asymptotic late-time tails and obtain sharp, uniform decay estimates in time.

A Perturbed Cauchy Viscoelastic Problem in an Exterior Domain

A Cauchy viscoelastic problem perturbed by an inverse-square potential, and posed in an exterior domain of RN, is considered under a Dirichlet boundary condition. Using nonlinear capacity estimates specifically adapted to the non-local nature of the problem, the potential function and the boundary condition, we establish sufficient conditions for the nonexistence of weak solutions.

A system of wave inequalities with inverse-square potentials in an exterior domain

ABSTRACT We study the existence and nonexistence of solutions to a system of wave inequalities with inverse-square potentials in an exterior domain of R N , under inhomogeneous Dirichlet-type boundary conditions. Our study yields naturally existence and nonexistence results for the corresponding stationary wave system.

Existence and asymptotic behavior of solutions for Choquard equations with singular potential under Berestycki–Lions type conditions

We prove the existence and asymptotic behavior of solutions to the following problem: − Δ u + V ( x ) u − g ( x ) u = ( I α ∗ F ( u ) ) f ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where g ( x ) : = μ | x | is called the Coulomb potential, g ( x ) : = β | x | 2 is called the Hardy potential (the inverse-square potential). μ , β > 0 are parameters, I α : R N ⟶ R is the Riesz potential. Moreover, the nonlinearity f satisfies Berestycki–Lions type conditions which are introduced by Moroz and Van Schaftingen (Trans. Amer. Math. Soc. 367 (2015) 6557–6579). When μ ∈ ( 0 , α ( N − 2 ) / 2 ( α + 1 ) ) and β ∈ ( 0 , α ( N − 2 ) 2 / 4 ( 2 + α ) ), under some mild assumptions on V, we establish the existence and asymptotic behavior of solutions. Particularly, our results extend some relate ones in the literature.

Point Charge Subject to an Attractive Inverse-Square-Type Potential and Anharmonic-Type Potentials

By applying the WKB (Wentzel, Kramers, Brillouin) approximation, we search for bound state solutions to the time-independent Schrödinger equation for an attractive inverse-square potential and anharmonic oscillators that stem from the interaction of a point charge with radial electric fields. We focus on the bound states associated with the s-waves. Further, we obtain the revival time associated with each case studied.

Stability and instability of radial standing waves to NLKG equation with an inverse-square potential

In this paper, we consider radial standing waves to a nonlinear Klein-Gordon equation with a repulsive inverse-square potential. It is known that there exist “radial” ground states to the stationary problem of the nonlinear Klein-Gordon equation. Here, “radial” ground states are non-trivial solutions having least energy among all non-trivial radial solutions to the stationary equation. We deal with the stability and instability of “radial” ground state standing waves.

Global existence, blow-up and mass concentration for the inhomogeneous nonlinear Schrödinger equation with inverse-square potential

<abstract><p>In the current paper, the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation including inverse-square potential is considered. First, some criteria of global existence and finite-time blow-up in the mass-critical and mass-supercritical settings with $ 0 < c\leq c^{*} $ are obtained. Then, by utilizing the potential well method and the sharp Sobolev constant, the sharp condition of blow-up is derived in the energy-critical case with $ 0 < c < \frac{N^{2}+4N}{(N+2)^{2}}c^{*} $. Finally, we establish the mass concentration property of explosive solutions, as well as the dynamic behaviors of the minimal-mass blow-up solutions in the $ L^{2} $-critical setting for $ 0 < c < c^{*} $, by means of the variational characterization of the ground-state solution to the elliptic equation, scaling techniques and a suitable refined compactness lemma. Our results generalize and supplement the ones of some previous works.</p></abstract>

Global existence and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential

<p style='text-indent:20px;'>In this paper, we study the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ iu_{t} +\Delta u-c|x|^{-2}u+|x|^{-b} |u|^{\sigma } u=0,\; u(0)=u_{0} \in H_{c}^{1},\;(t, x)\in \mathbb R\times\mathbb R^{d}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ d\ge3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ 0<b<2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \frac{4-2b}{d}<\sigma<\frac{4-2b}{d-2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ c>-c(d):=-\left(\frac{d-2}{2}\right)^{2} $\end{document}</tex-math></inline-formula>. We first establish the criteria for global existence and blow-up of general (not necessarily radial or finite variance) solutions to the equation. Using these criteria, we study the global existence and blow-up of solutions to the equation with general data lying below, at, and above the ground state threshold. Our results extend the global existence and blow-up results of Campos-Guzmán (Z. Angew. Math. Phys., 2021) and Dinh-Keraani (SIAM J. Math. Anal., 2021).</p>

Self-Bound vortex states in nonlinear Schrödinger equations with LHY correction

We study the cubic-quartic nonlinear Schrödinger equation (NLS) in two and three spatial dimension. This equation arises in the mean-field description of Bose-Einstein condensates with Lee-Huang-Yang correction. We first prove global existence of solutions in natural energy spaces which allow for the description of self-bound quantum droplets with vorticity. Existence of such droplets, described as central vortex states in 2D and 3D, is then proved using an approach via constrained energy minimizers. In 2D we also obtain a sharp lower bound on their mass. Even though these vortex states are known to be unstable in general, we prove that they are indeed orbitally stable (as a set) under the flow associated to the NLS with repulsive inverse-square potential in 2D.

The 𝑊^{𝑠,𝑝}-boundedness of stationary wave operators for the Schrödinger operator with inverse-square potential

In this paper, we investigate the W s , p W^{s,p} -boundedness for stationary wave operators of the Schrödinger operator with inverse-square potential L a = − Δ + a | x | 2 , a ≥ − ( d − 2 ) 2 4 , \begin{equation*} \mathcal L_a=-\Delta +\tfrac {a}{|x|^2}, \quad a\geq -\tfrac {(d-2)^2}{4}, \end{equation*} in dimension d ≥ 2 d\geq 2 . We construct the stationary wave operators in terms of integrals of Bessel functions and spherical harmonics, and prove that they are W s , p W^{s,p} -bounded for certain p p and s s which depend on a a . As corollaries, we solve some open problems associated with the operator L a \mathcal L_a , which include the dispersive estimates and the local smoothing estimates in dimension d ≥ 2 d\geq 2 . We also generalize some known results such as the uniform Sobolev inequalities, the equivalence of Sobolev norms and the Mikhlin multiplier theorem, to a larger range of indices. These results are important in the description of linear and nonlinear dynamics for dispersive equations with inverse-square potential.

BV Capacity for the Schrödinger Operator with an Inverse-Square Potential

BV Capacity for the Schrödinger Operator with an Inverse-Square Potential