Dimensional Potential Research Articles

Decay estimates for Beam equations with potential in dimension three

This paper is devoted to studying time decay estimates of the solution for Beam equation (higher order type wave equation) with a potentialutt+(Δ2+V)u=0,u(0,x)=f(x),ut(0,x)=g(x) in dimension three, where V is a real-valued and decaying potential. Assume that zero is a regular point of H=Δ2+V, we first prove the following optimal time decay estimates of the solution operators‖cos⁡(tH)Pac(H)‖L1→L∞≲|t|−32and‖sin⁡(tH)HPac(H)‖L1→L∞≲|t|−12. Moreover, if zero is a resonance of H, then time decay of the solution operators also is considered. It is noted that a first-kind resonance does not affect the decay rates of the propagator operators cos⁡(tH) and sin⁡(tH)H, but their decay will be significantly changed for the second and third-kind resonances.

ZK‐type limit for the Euler–Poisson system with a realistic electrostatic potential in dimension three

We explore the Zakharov–Kuznetsov‐type limit for the three‐dimensional Euler–Poisson systems with external magnetic fields when the plasma has a realistic electrostatic potential. It reveals that both homogeneous and inhomogeneous Zakharov–Kuznetsov‐type equations can be obtained from the Euler–Poisson system through the application of the Gardner–Morikawa transformation. By analyzing the remaining system, we employ the energy method within Sobolev spaces to provide estimates, leading to the emergence of Zakharov–Kuznetsov‐type dynamics over a time interval of the order .

Solutions of Gross–Pitaevskii equation with periodic potential in dimension three

Quasiperiodic solutions of the Gross–Pitaevskii equation with a periodic potential in dimension three are studied. It is proved that there is an extensive “nonresonant” set G ⊂ R 3 \mathcal {G}\subset \mathbb {R}^3 such that for every \vv k\in \mathcal {G} there is a solution asymptotically close to a plane wave Ae^{i\langle \vv {k},\vv {x}\rangle } as |\vv k|\to \infty , given A A is sufficiently small.

Resolvent bounds for Lipschitz potentials in dimension two and higher with singularities at the origin

We consider, for h,E>0 , the semiclassical Schrödinger operator -h^{2}\Delta + V - E in dimension two and higher. The potential V and its radial derivative \partial_{r}V are bounded away from the origin, have long-range decay and V is bounded by r^{-\delta} near the origin while \partial_{r}V is bounded by r^{-1-\delta} , where 0\leq\delta < 4(\sqrt{2}-1) . In this setting, we show that the resolvent bound is exponential in h^{-1} , while the exterior resolvent bound is linear in h^{-1} .

Quasilinear Schrödinger equations with unbounded or decaying potentials in dimension 2

AbstractWe establish the existence of nontrivial solutions for the following class of quasilinear Schrödinger equations: where κ is a positive parameter, and are continuous functions that can be singular at the origin, unbounded or vanishing at infinity, and the nonlinearity has critical exponential growth motivated by the Trudinger–Moser inequality. To prove our main result, we apply variational methods together with careful ‐estimates.

Radial power-like potentials: From the Bohr–Sommerfeld S-state energies to the exact ones

Following our previous study of the Bohr–Sommerfeld (BS) quantization condition for one-dimensional case [J. C. del Valle and A. V. Turbiner, Int. J. Mod. Phys. A 36, 2150221 (2021)], we extend it to [Formula: see text]-dimensional power-like radial potentials. The BS quantization condition for [Formula: see text]-states of the [Formula: see text]-dimensional radial Schrödinger equation is proposed. Based on numerical results obtained for the spectra of power-like potentials, [Formula: see text] with [Formula: see text], the correctness of the proposed BS quantization condition is established for various dimensions [Formula: see text]. It is demonstrated that by introducing the WKB correction [Formula: see text] into the right-hand side of the BS quantization condition leads to the so-called exact WKB quantization condition, which reproduces the exact energies, while [Formula: see text] remains always very small. For [Formula: see text] (any integer [Formula: see text]) and for [Formula: see text] (at [Formula: see text]) the WKB correction [Formula: see text]: for [Formula: see text] states the BS spectra coincides with the exact ones. Concrete calculations for physically important cases of linear, cubic, quartic and sextic oscillators, as well as Coulomb and logarithmic potentials in dimensions [Formula: see text] are presented. Radial quartic anharmonic oscillator is considered briefly.

Scattering of the focusing energy-critical NLS with inverse square potential

We consider the Cauchy problem for the focusing energy-critical nonlinear Schrödinger equation with an inverse square potential in dimension $ d = 4, 5, 6 $. We show that if the supremum of the kinetic energy of a solution over its maximal lifespan is less than the kinetic energy of the ground state, then the solution must exist globally in time and scatter in both time directions. We develop the 'long-term kinetic energy decoupling' associated with the appearance of inverse square potential. No radial assumption is made on the initial data. This extends the result in [22] by the first author to the non-radial case.

Global-in-Time $L^p−L^q$ Estimates for Solutions of the Kramers-Fokker-Planck Equation

In this work, we prove an optimal global-in-time L p --L q estimate for solutions to the Kramers-Fokker-Planck equation with short range potential in dimension three. Our result shows that the decay rate as t $\rightarrow$ +$\infty$ is the same as the heat equation in x-variables and the divergence rate as t $\rightarrow$ 0 + is related to the sub-ellipticity with loss of 1/3 derivatives of the Kramers-Fokker-Planck operator.

A remark on Strichartz estimates for Schrödinger equations with slowly decaying potentials

In this short note, we prove Strichartz estimates for Schrödinger operators with slowly decaying singular potentials in dimension two. This is a generalization of the recent results by Mizutani, which are stated for dimension greater than two. The main ingredient of the proof is a variant of Kato’s smoothing estimate with a singular weight.

Dynamics of Threshold Solutions for Energy Critical NLS with Inverse Square Potential

We consider the focusing energy critical NLS with inverse square potential in dimension $d= 3, 4, 5$ with the details given in $d=3$ and remarks on results in other dimensions. Solutions on the energy surface of the ground state are characterized. We prove that solutions with kinetic energy less than that of the ground state must scatter to zero or belong to the stable/unstable manifolds of the ground state. In the latter case they converge to the ground state exponentially in the energy space as $t\to \infty$ or $t\to -\infty$. (In 3-dim without radial assumption, this holds under the compactness assumption of non-scattering solutions on the energy surface.) When the kinetic energy is greater than that of the ground state, we show that all radial $H^1$ solutions blow up in finite time, with the only two exceptions in the case of 5-dim which belong to the stable/unstable manifold of the ground state. The proof relies on the detailed spectral analysis, local invariant manifold theory, and a global Virial analysis.

An Explicit Formula of the Dirichlet-to-Neumann Map for a Radial Potential in Dimension 3

In this paper, we provide an explicit expression for the full Dirichlet-to-Neumann map corresponding to a radial potential for the Schrödinger equation in 3-dimensional. We numerically implement the coefficients of the explicit formulas. In this work, Lipschitz type stability is established near the edge of the domain with giving estimation constant. That is necessary for the reconstruction of the potential from Dirichlet-to-Neuman map.

Numerical approximation of the scattering amplitude in elasticity

We propose a numerical method to approximate the scattering amplitudes for the elasticity system with a non-constant matrix potential in dimensions d=2 and 3. This requires to approximate first the scattering field, for some incident waves, which can be written as the solution of a suitable Lippmann-Schwinger equation. In this work we adapt the method introduced by Vainikko (Res Rep A 387:3–18, 1997) to solve such equations when considering the Lamé operator. Convergence is proved for sufficiently smooth potentials. Implementation details and numerical examples are also given.

Sharp bounds for eigenvalues of biharmonic operators with complex potentials in low dimensions

AbstractWe derive sharp quantitative bounds for eigenvalues of biharmonic operators perturbed by complex‐valued potentials in dimensions one, two and three.

Large time behavior of solutions to Schrödinger equation with complex-valued potential

We study the large-time behavior of the solutions to the Schrödinger equation associated with a quickly decaying potential in dimension three. We establish the resolvent expansions at threshold zero and near positive resonances. The large-time expansions of solutions are obtained under different conditions, including the existence of positive resonances and zero resonance or/and zero eigenvalue.

Global semiclassical limit from Hartree to Vlasov equation for concentrated initial data

We prove a quantitative and global in time semiclassical limit from the Hartree to the Vlasov equation in the case of a singular interaction potential in dimension d≥3, including the case of a Coulomb singularity in dimension d=3. This result holds for initial data concentrated enough in the sense that some space moments are initially sufficiently small. As an intermediate result, we also obtain quantitative bounds on the space and velocity moments of even order and the asymptotic behavior of the spatial density due to dispersion effects, uniform in the Planck constant ħ.

Refined mass-critical Strichartz estimates for Schrödinger operators

We develop refined Strichartz estimates at $L^2$ regularity for a class of time-dependent Schr\"{o}dinger operators. Such refinements begin to characterize the near-optimizers of the Strichartz estimate, and play a pivotal part in the global theory of mass-critical NLS. On one hand, the harmonic analysis is quite subtle in the $L^2$-critical setting due to an enormous group of symmetries, while on the other hand, the spacetime Fourier analysis employed by the existing approaches to the constant-coefficient equation are not adapted to nontranslation-invariant situations, especially with potentials as large as those considered in this article. Using phase space techniques, we reduce to proving certain analogues of (adjoint) bilinear Fourier restriction estimates. Then we extend Tao's bilinear restriction estimate for paraboloids to more general Schr\"{o}dinger operators. As a particular application, the resulting inverse Strichartz theorem and profile decompositions constitute a key harmonic analysis input for studying large data solutions to the $L^2$-critical NLS with a harmonic oscillator potential in dimensions $\ge 2$. This article builds on recent work of Killip, Visan, and the author in one space dimension.

Inequalities involving Aharonov–Bohm magnetic potentials in dimensions 2 and 3

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions 2 and 3.

Scattering of the energy-critical NLS with inverse square potential

We consider the initial value problem of the energy critical nonlinear Schrödinger equation with an inverse square potential in dimension d=4,5,6. In the focusing case, we achieve the global well-posedness and scattering when the kinetic energy and energy of the solution are both below the threshold of the ground state solution. In the defocusing case, we prove that any H˙1 bounded initial data leads to the global and scattering solution. No radial assumption is made on the initial data.

On the Kramers–Fokker–Planck equation with decreasing potentials in dimension one

For quickly decreasing potentials with one position variable, the first threshold zero is always a resonance of the Kramers–Fokker–Planck operator. In this article we study low-energy spectral properties of the operator and calculate large time asymptotics of solutions in terms of the Maxwellian.

Global center stable manifold for the defocusing energy critical wave equation with potential

In this paper we consider the defocusing energy critical wave equation with a trapping potential in dimension $3$. We prove that the set of initial data for which solutions scatter to an unstable excited state $(\phi, 0)$ forms a finite co-dimensional path connected $C^1$ manifold in the energy space. This manifold is a global and unique center-stable manifold associated with $(\phi,0)$. It is constructed in a first step locally around any solution scattering to $\phi$, which might be very far away from $\phi$ in the $\dot{H}^1\times L^2(\mathbb{R}^3)$ norm. In a second crucial step a no-return property is proved for any solution which starts near, but not on the local manifolds. This ensures that the local manifolds form a global one. Scattering to an unstable steady state is therefore a non-generic behavior, in a strong topological sense in the energy space. This extends our previous result [18] to the nonradial case. The new ingredients here are (i) application of the reversed Strichartz estimate from [3] to construct a local center stable manifold near any solution that scatters to $(\phi, 0)$. This is needed since the endpoint of the standard Strichartz estimates fails nonradially. (ii) The nonradial channel of energy estimate introduced by Duyckaerts-Kenig-Merle [14], which is used to show that solutions that start off but near the local manifolds associated with $\phi$ emit some amount of energy into the far field in excess of the amount of energy beyond that of the steady state $\phi$.