Nonnegative Potential Research Articles

On the spectral gap of one-dimensional Schrödinger operators on large intervals

We study the effect of non-negative potentials on the spectral gap of one-dimensional Schrödinger operators in the limit of large intervals. We derive upper bounds on the gap for different classes of potentials and show, as a main result, that the spectral gap of a Schrödinger operator with a non-zero and sufficiently fast decaying potential closes strictly faster than the gap of the free Laplacian. We show optimality of this result in some sense and establish a conjecture towards the actual decay rate of the spectral gap.

Minimization of the lowest positive Neumann-Dirichlet eigenvalue for general indefinite Sturm-Liouville problems

The aim of this paper is to obtain the sharp estimate for the lowest positive eigenvalue λ0ND+ for the general Sturm–Liouville problemy″=q(t)y+λm(t)y, with the Neumann-Dirichlet boundary conditions, where q is a nonnegative potential and another potential m admits to change sign. First, we will study the optimal lower bound for the smallest positive eigenvalue in the measure differential equations to make our results more applicable. Second, based on the relationship between the minimization problem of the smallest positive eigenvalue for the ODE and the one for the MDE, we find the explicit optimal lower bound of the smallest positive eigenvalue for the general Sturm–Liouville equation.

Semilinear elliptic equations on manifolds with nonnegative Ricci curvature

In this paper, we prove classification results for solutions to subcritical and critical semilinear elliptic equations with a nonnegative potential on noncompact manifolds with nonnegative Ricci curvature. We show in the subcritical case that all nonnegative solutions vanish identically. Moreover, under some natural assumptions, in the critical case we prove a strong rigidity result, namely we classify all nontrivial solutions showing that they exist only if the potential is constant and the manifold is isometric to the Euclidean space.

On the Dirichlet problem for the elliptic equations with boundary values in variable Morrey-Lorentz spaces

Under a metric measure space setting, we show that a solution to the elliptic equation with a non-negative potential, defined on the upper half-space, is in the essentially-bounded-Morrey-Lorentz space with variable exponent if and only if it can be represented as the Poisson integral of a variable Morrey-Lorentz function, where the doubling property, the pointwise upper bound on the heat kernel, the mean value property and the Liouville property are assumed. This extends the known result of Stein-Weiss in 1971 from the Euclidean space to the metric measure space, and from the Lebesgue function to the Morrey-Lorentz function with variable exponent.

Heteroclinic traveling waves of two-dimensional parabolic Allen–Cahn systems

In this paper we show the existence of traveling waves w\colon [0,+\infty) \times \mathbb{R}^{2} \to \mathbb{R}^{k} ( k \geq 2 ) for the parabolic Allen–Cahn system \partial_{t} w - \Delta w = -\nabla_{u} V(w) in [0,+\infty) \times \mathbb{R}^{2} , satisfying some heteroclinic conditions at infinity. The potential V is a nonnegative and smooth multi-well potential, which means that its null set is finite and contains at least two elements. The traveling wave w propagates along the horizontal axis according to a speed c^{\star}>0 and a profile \mathfrak{U} . The profile \mathfrak{U} joins as x_{1} \to \pm \infty (in a suitable sense) two locally minimizing one-dimensional heteroclinics which have different energies, and the speed c^{\star} satisfies certain uniqueness properties. The proof is variational and, in particular, it requires the assumption of an upper bound, depending on V , on the difference between the energies of the one-dimensional heteroclinics.

Perturbation theory for the Gross–Pitaevskii equation modeling stationary Bose–Einstein condensates

In this article a perturbative solution of the Gross–Pitaevskii(GP) equation in the D-dimensional space RD with a general continuous non-negative external potential is studied. The solution describes the condensate wave-function of a gas containing N Bose particles. A criteria for the validity of the perturbative solution is developed. Furthermore expressions for the particle density, the chemical potential, the internal energy and the mean-square radius of the condensate are derived corrected to first order in the coupling constant. The scheme is then applied to obtain the solution of the GP equation in D=1,2,3 for external harmonic potentials, and the results are compared with the numerical solutions. It is shown, in each case, that if N exceeds a certain value the perturbative solution breaks down.

Counting eigenvalues of Schrödinger operators using the landscape function

We prove an upper and a lower bound on the rank of the spectral projections of the Schrödinger operator -\Delta + V in terms of the volume of the sublevel sets of an effective potential \frac{1}{u} . Here, u is the ‘landscape function’ of G. David, M. Filoche, and S. Mayboroda [Adv. Math. 390 (2021), article no. 107946], namely a solution of (-\Delta + V)u = 1 in \mathbb R^{d} . We prove the result for non-negative potentials satisfying a Kato-type and a doubling condition, in all spatial dimensions, in infinite volume, and show that no coarse-graining is required. Our result yields in particular a necessary and sufficient condition for discreteness of the spectrum. In the case of nonnegative polynomial potentials, we prove that the spectrum is discrete if and only if no directional derivative vanishes identically.

Minimizations of positive periodic and Dirichlet eigenvalues for general indefinite Sturm-Liouville problems

The aim of this paper is to develop an analytical approach to obtain the sharp estimates for the lowest positive periodic eigenvalue and all Dirichlet eigenvalues of a general Sturm-Liouville problemy″=q(t)y+λm(t)y, where q is a nonnegative potential and another potential m admits to change sign. A typical example of such problems is the well-known Camassa-Holm equations with indefinite potentials, which corresponds to the case q(t)≡14. It is shown that the solution of the minimization problems of the lowest positive periodic eigenvalues and Dirichlet eigenvalues will lead to more general distributions of potentials which have no densities with respect to the Lebesgue measure. As a result, it is very natural to choose the general setting of the measure differential equationsdy•=y(t)dμ(t)+λydν(t), to understand the eigenvalues and their minimization, where μ and ν are two suitable measures. The variational characterization of lowest positive eigenvalues, together with a strong continuous dependence of eigenvalues on the potentials, will play crucial roles in our analysis. Different from the periodic case, we are able to obtain the optimal bounds for higher Dirichlet eigenvalues.

Semiclassical states for coupled nonlinear Schrödinger equations with critical frequency

In this paper, we are concerned with the coupled nonlinear Schrödinger system − ε 2 Δ u + a ( x ) u = μ 1 u 3 + β v 2 u in R N , − ε 2 Δ v + b ( x ) v = μ 2 v 3 + β u 2 v in R N , where 1 ⩽ N ⩽ 3, μ 1 , μ 2 , β > 0, a ( x ) and b ( x ) are nonnegative continuous potentials, and ε > 0 is a small parameter. We show the existence of positive ground state solutions for the system above and also establish the concentration behaviour as ε → 0, when a ( x ) and b ( x ) achieve 0 with a homogeneous behaviour or vanish in some nonempty open set with smooth boundary.

Ground‐state solutions of Schrödinger‐type equation with magnetic field

In this paper, the nonlinear Schrödinger‐type equation is considered in the presence of magnetic field, where , , denotes the Riesz potential, is a positive potential for some , is a nonnegative potential for some , and is assumed to be asymptotically linear at infinity. Under suitable assumptions regarding , , , and , variational methods are used to establish the existence of ground‐state solutions of the above equation for sufficiently small values of the parameter .

Boundedness for Littlewood-Paley functions on weighted Morrey-type spaces related to certain nonnegative potentials

Boundedness for Littlewood-Paley functions on weighted Morrey-type spaces related to certain nonnegative potentials

On the uniqueness of Gibbs distributions with a non-negative and subcritical pair potential

Nous prouvons que la loi d’un processus de Gibbs avec potentiel d’interaction positif est déterminée de façon unique dès qu’un modèle de connexion aléatoire (RCM) associé, dirigé par un processus de Poisson, ne percole pas. Notre preuve combine le couplage de désaccords dans le continuum avec le couplage d’un processus de Gibbs et d’un RCM. L’amélioration par rapport aux résultats d’unicité antérieurs est illustrée à la fois de manière théorique et avec des simulations.

Axially symmetric solutions for planar Schrödinger–Poisson systems with critical exponential growth and non-negative potential

This paper is concerned with the following planar Schrödinger–Poisson system −Δu+V(x)u+ϕu=f(u),x∈R2,Δϕ=u2,x∈R2,where V∈C(R2,[0,∞)) is axially symmetric and f∈C(R,R) has critical exponential growth in the sense of Trudinger–Moser. This system can be converted into the integro-differential equation with logarithmic convolution potential. We prove the existence of axially symmetric solutions by some new useful estimates on logarithmic convolution potential. Our result not only improves the ones of Chen–Tang (2020), but covers the zero mass case that V(x)≡0.

Matrix-valued Schrödinger operators over finite adeles

Let [Formula: see text] be an algebraic number field. With [Formula: see text] we associate the ring of finite adeles [Formula: see text] In this paper we give a path integral formula for the propagator of a quantum mechanical system over the abelian group [Formula: see text] Specifically, we consider matrix-valued Hamiltonian operators [Formula: see text] where [Formula: see text] is the Vladimirov operator and [Formula: see text] is a non-negative definite potential. The free part of the Hamiltonian gives rise to a measure on the Skorokhod space of paths which allows us to prove the Feynman–Kac formula for the Schrödinger semigroup generated by [Formula: see text] This formula is given in terms of the ordered time exponentials.

Normalized solution for fractional Choquard equation with potential and general nonlinearity

In this paper, we consider the following fractional Choquard equation: 0, \\end{cases} \\] ]]> { ( − Δ ) s u + V ( x ) u + λu = ( I α ∗ F ( u ) ) f ( u ) in R N , ∫ R N u 2 d x = ρ 2 , ρ > 0 , where s ∈ ( 0 , 1 ) , α ∈ ( 0 , N ) , N>2s and V : R N → R is a continuous nonnegative potential and vanishes at infinity. Under some mild assumptions imposed on V and f, we establish the existence of L 2 -normalized solution ( u , λ ) ∈ H s ( R N ) × R + .

Weak Type (1,1) Bounds for Schrödinger Groups

Let L be a nonnegative self-adjoint operator acting on L2(X), where X is a space of homogeneous type of dimension n. Suppose that the heat kernel of L satisfies a Gaussian upper bound. It is known that the operator (I+L)−seitL is bounded on Lp(X) for s>n|1/2−1/p| and p∈(1,∞) (see, e.g., [7, 22, 33]). The index s=n|1/2−1/p| was only obtained recently in [9, 10], and this range of s is sharp since it is precisely the range known in the case where L is the Laplace operator Δ on X=Rn [30]. In this paper, we establish that for p=1, the operator (1+L)−n/2eitL is of weak type (1,1), that is, there is a constant C, independent of t and f, such that μ({x:|(I+L)−n/2eitLf(x)|>λ})≤Cλ−1(1+|t|)n/2‖f‖ L1(X),t∈R (for λ>0 when μ(X)=∞ and λ>μ(X)−1‖f‖L1(X) when μ(X)<∞). Moreover, we also show that the index n/2 is sharp when L is the Laplacian on Rn by providing an example. Our results are applicable to Schrödinger groups for large classes of operators including elliptic operators on compact manifolds, Schrödinger operators with nonnegative potentials and Laplace operators acting on Lie groups of polynomial growth or irregular nondoubling domains of Euclidean spaces.

Orlicz estimates for parabolic Schrödinger operators with non-negative potentials on nilpotent Lie groups

&lt;abstract&gt;&lt;p&gt;In this paper, we study the Orlicz estimates for the parabolic Schrödinger operator&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ L = {\partial _t} - {\Delta _X} + V, $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;where the nonnegative potential $ V $ belongs to a reverse Hölder class on nilpotent Lie groups $ {\Bbb G} $ and $ {\Delta _X} $ is the sub-Laplace operator on $ {\Bbb G} $. Under appropriate growth conditions of the Young function, we obtain the regularity estimates of the operator $ L $ in the Orlicz space by using the domain decomposition method. Our results generalize some existing ones of the $ L^{p} $ estimates.&lt;/p&gt;&lt;/abstract&gt;

Cluster Expansions: Necessary and Sufficient Convergence Conditions

We prove a new convergence condition for the activity expansion of correlation functions in equilibrium statistical mechanics with possibly negative pair potentials. For non-negative pair potentials, the criterion is an if and only if condition. The condition is formulated with a sign-flipped Kirkwood–Salsburg operator and known conditions such as Kotecký–Preiss and Fernández–Procacci are easily recovered. In addition, we deduce new sufficient convergence conditions for hard-core systems in mathbb {R}^d and mathbb {Z}^d as well as for abstract polymer systems. The latter improves on the Fernández–Procacci criterion.

On the structure of minimizers of the Allen-Cahn energy for $$Z_N$$ symmetric nonnegative potentials with $$N+1$$ zeros

On the structure of minimizers of the Allen-Cahn energy for $$Z_N$$ symmetric nonnegative potentials with $$N+1$$ zeros

Ground states solutions for a modified generalized Choquard fractional Schrödinger equation

ABSTRACT In this paper, we consider the modified fractional Schrödinger system of Choquard type: where is the fractional Laplacian with , are nonnegative potentials, are positive constant and , . Under some assumptions on and G, by using the variational method, we obtain ground state solutions to the system.