Nonnegative Potential Research Articles

Hardy type estimates for Riesz transforms associated with Schrödinger operators

Let $$\mathcal {L}=-\Delta +V$$ be a Schrodinger operator on $$\mathbb {R}^n (n\ge 3),$$ where the nonnegative potential V belongs to reverse Holder class $$RH_{q_1}$$ for $$q_1>\frac{n}{2}.$$ Let $$H^p_\mathcal {L}(\mathbb {R}^n)$$ be the Hardy space related to $$\mathcal {L}.$$ In this paper, we consider the Hardy type estimates for the Riesz transform $$T_\alpha =V^\alpha (-\Delta +V)^{-\alpha }$$ with $$0<\alpha <n/2.$$ We show that $$T_\alpha $$ is bounded from $$H^p_\mathcal {L}(\mathbb {R}^n)$$ into $$L^p(\mathbb {R}^n)$$ for $$\frac{n}{n+\delta '}<p\le 1,$$ where $$\delta '=\min \{1, 2-n/q_0\},$$ and $$q_0$$ is the reverse Holder index of V. Moreover, we prove that the commutator $$[b,T_\alpha ],$$ which associated with $$T_\alpha $$ and a new BMO function b, maps $$H^{1}_\mathcal {L}(\mathbb {R}^n)$$ continuously into weak $$L^1(\mathbb {R}^n)$$ .

Carleson measures, BMO spaces and balayages associated to Schrödinger operators

Let $\L$ be a Schr\"odinger operator of the form $\L=-\Delta+V$ acting on $L^2(\mathbb R^n)$, $n\geq3$, where the nonnegative potential $V$ belongs to the reverse H\"older class $B_q$ for some $q\geq n.$ Let ${\rm BMO}_{{\mathcal{L}}}(\RR)$ denote the BMO space associated to the Schr\"odinger operator $\L$ on $\RR$. In this article we show that for every $f\in {\rm BMO}_{\mathcal{L}}(\RR)$ with compact support, then there exist $g\in L^{\infty}(\RR)$ and a finite Carleson measure $\mu$ such that $$ f(x)=g(x) + S_{\mu, {\mathcal P}}(x) $$ with $\|g\|_{\infty} +\||\mu\||_{c}\leq C \|f\|_{{\rm BMO}_{\mathcal{L}}(\RR)},$ where $$ S_{\mu, {\mathcal P}}=\int_{{\mathbb R}^{n+1}_+} {\mathcal P}_t(x,y) d\mu(y, t), $$ and ${\mathcal P}_t(x,y)$ is the kernel of the Poisson semigroup $\{e^{-t\sqrt{\L}}\}_{t> 0} $ on $L^2(\mathbb R^n)$. Conversely, if $\mu$ is a Carleson measure, then $S_{\mu, {\mathcal P}}$ belongs to the space ${\rm BMO}_{{\mathcal{L}}}(\RR)$. This extends the result for the classical John--Nirenberg BMO space by Carleson \cite{C} (see also \cite{U,GJ,W}) to the BMO setting associated to Schr\"odinger operators.

Superradiant instabilities for short-range non-negative potentials on Kerr spacetimes and applications

The wave equation □gM,aψ=0 on subextremal Kerr spacetimes (MM,a,gM,a), 0<|a|<M, does not admit real mode solutions, as was established by Shlapentokh-Rothman. In this paper, we show that the absence of real modes does not persist under the addition of an arbitrary short-range non-negative potential V to the wave equation or under changes of the metric gM,a in the far away region of MM,a (retaining the causality of the stationary Killing field T there).In particular, we first establish, for any 0<|a|<M, the existence of real mode solutions ψ to equation □gM,aψ−Vψ=0, for a suitably chosen time-independent real potential V with compact support in space, satisfying V≥0. Exponentially growing modes are also obtained after perturbing the potential V. Then, as an application of the above result, we construct a family of spacetimes (MM,a,gM,a(def)) which are compact in space perturbations of (MM,a,gM,a), have the same symmetries as (MM,a,gM,a) and moreover admit real and exponentially growing modes. These spacetimes contain stably trapped null geodesics, but we also construct a more complicated family of spacetimes with normally hyperbolic trapped set, admitting real and exponentially growing modes, at the expense of having conic asymptotics. The aforementioned results are in contrast with the case of stationary asymptotically flat (or conic) spacetimes (M,g) with a globally timelike Killing field T, where real modes for equation □gψ−Vψ=0 are always absent, giving a useful zero-frequency continuity criterion for showing stability for a smooth family of equations □gψ−Vλψ=0, with λ∈[0,1] and V0=0. We show explicitly that this criterion fails on Kerr spacetime.

Schrödinger operators involving singular potentials and measure data

We study the existence of solutions of the Dirichlet problem for the Schrödinger operator with measure data{−Δu+Vu=μin Ω,u=0on ∂Ω. We characterize the finite measures μ for which this problem has a solution for every nonnegative potential V in the Lebesgue space Lp(Ω) with 1≤p≤N2. The full answer can be expressed in terms of the W2,p capacity for p>1, and the W1,2 (or Newtonian) capacity for p=1. We then prove the existence of a solution of the problem above when V belongs to the real Hardy space H1(Ω) and μ is diffuse with respect to the W2,1 capacity.

Solutions of the Dirichlet–Sch problem and asymptotic properties of solutions for the Schrödinger equation

Let a ( X , y ) be a nonnegative radical potential in a cylinder. In this paper, we study the solutions of the Dirichlet–Sch problem associated with a stationary Schrödinger operator. Existence of solutions for the Schrödinger equation and asymptotic properties as y → + ∞ and y → − ∞ are also considered under suitable conditions.

Perturbations of Lane–Emden and Hamilton–Jacobi equations: The full space case

In this article we are interested in positive classical solutions of −Δu+a(x)⋅∇u+V(x)u=up+γuq in RN, and −Δu+a(x)⋅∇u=up+γ|∇u|q in RN, in the case of N≥4, p>N+1N−3 and γ∈R. We assume that V is a smooth nonnegative potential and a(x) is a smooth vector field, both of which satisfy natural decay assumptions. Under suitable assumptions on q we prove the existence of positive classical solutions.

Atomic decompositions for Hardy spaces related to Schrödinger operators

Let $\mathbf {L}^{U}= -\boldsymbol \Delta +U$ be a Schrödinger operator on ${\mathbb {R}^d}$, where $U\in L^1_{\rm loc}({\mathbb {R}^d})$ is a&nbsp;non-negative potential and $d\geq 3$. The Hardy space $H^1(\mathbf {L}^{U})$ is defined in terms of the max

Layered solutions to the vector Allen-Cahn equation in $\mathbb{R}^2$. Minimizers and heteroclinic connections

Let $W:\mathbb{R}^m\to \mathbb{R}$ be a nonnegative potential with exactly two nondegenerate zeros $a_-≠ a_+∈\mathbb{R}^m$. We assume that there are $N≥q 1$ distinct heteroclinic orbits connecting $a_-$ to $a_+$ represented by maps $\bar{u}_1,...,\bar{u}_N$ that minimize the one-dimensional energy $J_\mathbb{R}(u)=∈t_\mathbb{R}(\frac{\vert u^\prime\vert^2}{2}+W(u)){d} s$.We first consider the problem of characterizing the minimizers $u:\mathbb{R}^n\to \mathbb{R}^m$ of the energy $\mathcal{J}_Ω(u)=∈t_Ω(\frac{\vert\nabla u\vert^2}{2}+W(u)){d} x$. Under a nondegeneracy condition on $\bar{u}_j$, $j=1,...,N$ and in two space dimensions, we prove that, provided it remains away from $a_-$ and $a_+$ in corresponding half spaces $S_-$ and $S_+$, a bounded minimizer $u:\mathbb{R}^2\to \mathbb{R}^m$ is necessarily an heteroclinic connection between suitable translates $\bar{u}_-(·-η_-)$ and $\bar{u}_+(·-η_+)$ of some $\bar{u}_±∈\{\bar{u}_1,...,\bar{u}_N\}$.Then we focus on the existence problem and assuming $N=2$ and denoting $\bar{u}_-,\bar{u}_+$ the representations of the two orbits connecting $a_-$ to $a_+$ we give a new proof of the existence (first proved in [32]) of a solution $u:\mathbb{R}^2\to \mathbb{R}^m$ of $Δ u=W_u(u),$ that connects certain translates of $\bar{u}_±$.

Formula omitted]-fractional Kirchhoff equations involving critical nonlinearities

The paper deals with Kirchhoff type equations on the whole space RN, driven by the p-fractional Laplace operator, involving critical Hardy–Sobolev nonlinearities and nonnegative potentials. We present different variational approaches to overcome the lack of compactness at critical levels, due to the presence of critical terms as well as the possibly degenerate nature of the Kirchhoff problem.

Poisson semigroup, area function, and the characterization of Hardy space associated to degenerate Schrödinger operators

Let Lf(x)=−1ω(x)∑i,j∂i(aij(⋅)∂jf)(x)+V(x)f(x) be the degenerate Schrodinger operator, where ω is a weight from the Muckenhoupt class A2 and V is a nonnegative potential that belongs to a certain reverse Holder class with respect to the measure ω(x)dx. Based on some smoothness estimates of the Poisson semigroup e−tL, we introduce the area function SPL associated with e−tL to characterize the Hardy space associated with L.

Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators satisfying Gaussian estimates

Let $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$ whose heat kernels have the Gaussian upper bound estimates. Assume that the growth function $\varphi:\mathbb{R}^n\times[0,\infty) \to[0,\infty)$ satisfies that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ (the class of uniformly Muckenhoupt weights). Let $H_{\varphi,L}(\mathbb{R}^n)$ be the Musielak-Orlicz-Hardy space introduced via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors obtain several maximal function characterizations of the space $H_{\varphi,L}(\mathbb{R}^n)$, which, especially, answer an open question of L. Song and L. Yan under an additional mild assumption satisfied by Schrodinger operators on $\mathbb{R}^n$ with non-negative potentials belonging to the reverse Holder class, and second-order divergence form elliptic operators on $\mathbb{R}^n$ with bounded measurable real coefficients.

Continuity results and estimates for the Lyapunov exponent of Brownian motion in stationary potential

We collect some applications of the variational formula established by Schroeder [J. Funct. Anal. 77 (1988) 60-87] and Ruess [ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 679-709] for the quenched Lyapunov exponent of Brownian motion in stationary and ergodic nonnegative potential. We show, for example, that the Lyapunov exponent for nondeterministic potential is strictly lower than the Lyapunov exponent for the averaged potential. The behaviour of the Lyapunov exponent under independent perturbations of the underlying potential is examined. And with the help of counterexamples, we are able to give a detailed picture of the continuity properties of the Lyapunov exponent.

Standing waves with a critical frequency for nonlinear Schrödinger equations involving critical growth

We consider the following singularly perturbed Schrödinger equation − ε 2 Δ u + V ( x ) u = f ( u ) , u ∈ H 1 ( R N ) , where N ≥ 3 , V is a nonnegative continuous potential and the nonlinear term f is of critical growth. In this paper, with the help of a truncation approach, we prove that if V has a positive local minimum, then for small ε the problem admits positive solutions which concentrate at an isolated component of positive local minimum points of V as ε → 0 . In particular, the potential V is allowed to be either compactly supported or decay faster than ∣ x ∣ − 2 at infinity. Moreover, a general nonlinearity f is involved, i.e., the monotonicity of f ( s ) / s and the Ambrosetti–Rabinowitz condition are not required.

The boundedness of commutators associated with Schrödinger operators on Herz spaces

Let \(L=-\Delta+V\) be a Schrodinger operators on \(\mathbb{R}^{n}\), \(n\geq3\), where the nonnegative potential V belongs to the reverse Holder class \(B_{s}\) for \(s>\frac{d}{2}\). Let \(T_{\beta}=(-\Delta +V)^{-\beta}V^{\beta}\), \(\beta>0\), and \(R_{L}=\nabla(-\Delta+V)^{-1/2}\) be the Riesz transform associated to L. We prove that the operator \(T_{\beta}\) and \(R_{L}\) are bounded on Herz spaces \(\dot {K}_{q}^{\alpha ,p}(\mathbb{R}^{n})\) and \({K}_{q}^{\alpha,p}(\mathbb{R}^{n})\), respectively. Suppose that \(b\in \operatorname{BMO}_{\sigma}(\rho)\), which is larger than \(\operatorname{BMO}(\mathbb{R}^{n})\). By a maximal estimate, we obtain the boundedness of commutators \([b, T_{\beta}]\) and \([b, R_{L}]\) on Herz spaces.

No-scale inflation

Supersymmetry is the most natural framework for physics above the TeV scale, and the corresponding framework for early-Universe cosmology, including inflation, is supergravity. No-scale supergravity emerges from generic string compactifications and yields a non-negative potential, and is therefore a plausible framework for constructing models of inflation. No-scale inflation yields naturally predictions similar to those of the Starobinsky model based on gravity, with a tilted spectrum of scalar perturbations: , and small values of the tensor-to-scalar perturbation ratio , as favoured by Planck and other data on the cosmic microwave background (CMB). Detailed measurements of the CMB may provide insights into the embedding of inflation within string theory as well as its links to collider physics.

A note for Riesz transforms associated with Schrödinger operators on the Heisenberg Group

Let \({\mathbb {H}^n}\) be the Heisenberg group and \(Q=2n+2\) be its homogeneous dimension. The Schrodinger operator is denoted by \( - {\Delta _{{\mathbb {H}^n}}} + V\), where \({\Delta _{{\mathbb {H}^n}}}\) is the sub-Laplacian and the nonnegative potential V belongs to the reverse Holder class \({B_{{q_1}}}\) for \({q_1} \ge \frac{Q}{2}\). Let \(H^p_L(\mathbb {H}^n)\) be the Hardy space associated with the Schrodinger operator for \(\frac{Q}{Q+\delta _0}<p\le 1\), where \(\delta _0=\min \{1,2-\frac{Q}{q_1}\}\). In this note we show that the operators \({T_1} = V{( - {\Delta _{{\mathbb {H}^n}}} + V)^{ - 1}}\) and \({T_2} = {V^{1/2}}{( - {\Delta _{{\mathbb {H}^n}}} + V)^{ - 1/2}}\) are bounded from \(H_L^p({\mathbb {H}^n})\) into \({L^p}({\mathbb {H}^n})\). Our results are also valid on the stratified Lie group.

Perturbations of Lane–Emden and Hamilton–Jacobi equations II: Exterior domains

In this article we are interested in the existence of positive classical solutions of(1){−Δu+a(x)⋅∇u+V(x)u=up+γuq in Ωu=0 on ∂Ω, and(2){−Δu+a(x)⋅∇u+V(x)u=up+γ|∇u|q in Ωu=0 on ∂Ω, where Ω is a smooth exterior domain in RN in the case of N≥4, p>N+1N−3 and γ∈R. We assume that V is a smooth nonnegative potential and a(x) is a smooth vector field, both of which satisfy natural decay assumptions. Under suitable assumptions on q we prove the existence of an infinite number of positive classical solutions.We also consider the case of N+2N−2<p<N+1N−3 under further symmetry assumptions on Ω, a and V.

A Calderón–Zygmund operator of higher order Schrödinger type

We consider higher order Schrödinger type operators with nonnegative potentials. We assume that the potential belongs to the reverse Hölder class which includes nonnegative polynomials. We show that an operator of higher order Schrödinger type is a Calderón–Zygmund operator. We also show that there exist potentials which satisfy our assumptions but are not nonnegative polynomials.

Concentrations for the simple random walk in un- bounded nonnegative potentials

We consider the simple random walk in i.i.d.nonnegative potentials on the multidimensional cubic lattice. Our goal is to investigate the cost paid by the simple random walk for traveling from the origin to a remote location in a landscape of potentials. In particular, we obtain concentration inequalities for the travel cost in unbounded nonnegative potentials.

Endpoint estimates for commutators of singular integrals related to Schrödinger operators

Let L= -\Delta+ V be a Schrödinger operator on \mathbb R^d , d\geq 3 , where V is a nonnegative potential, V\ne 0 , and belongs to the reverse Hölder class RH_{d/2} . In this paper, we study the commutators [b,T] for T in a class \mathcal K_L of sublinear operators containing the fundamental operators in harmonic analysis related to L . More precisely, when T\in \mathcal K_L , we prove that there exists a bounded subbilinear operator \mathfrak R= \mathfrak R_T\colon H^1_L(\mathbb R^d)\times {\rm BMO}(\mathbb R^d)\to L^1(\mathbb R^d) such that (\star)\qquad |T(\mathfrak S(f,b))|- \mathfrak R(f,b)\leq |[b,T](f)|\leq \mathfrak R(f,b) + |T(\mathfrak S(f,b))|, where \mathfrak S is a bounded bilinear operator from H^1_L(\mathbb R^d)\times {\rm BMO}(\mathbb R^d) into L^1(\mathbb R^d) which does not depend on T . The subbilinear decomposition (\star) allows us to explain why commutators with the fundamental operators are of weak type (H^1_L,L^1) , and when a commutator [b,T] is of strong type (H^1_L,L^1) . Also, we discuss the H^1_L -estimates for commutators of the Riesz transforms associated with the Schrödinger operator L .