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In this paper, we prove that the dimension of the space of positive (bounded, respectively) -harmonic functions on a complete Riemannian manifold with -regular ends is equal to the number of ends (-nonparabolic ends, respectively). This result is a solution of an open problem of Grigor'yan related to the Liouville property
for the Schr?dinger operator . We also prove that if a given complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying
the volume doubling condition, the Poincaré inequality and the finite covering condition on each end, then the dimension of
the space of positive (bounded, respectively,) solutions for the Schr?dinger operator with a potential satisfying a certain
decay rate on the manifold is equal to the number of ends (-nonparabolic ends, respectively). This is a partial answer of the question, suggested by Li, related to the regularity of
ends of a complete Riemannian manifold. Especially, our results directly generalize various earlier results of Yau, of Li
and Tam, of Grigor'yan, and of present authors, but with different techniques that the peculiarity of the Schr?dinger operator
demands.
Received: 4 April 2000; in final form: 19 September 2000 / Published online: 25 June 2001 相似文献
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Jardel M. Pereira 《Acta Appl Math》2014,134(1):173-183
A generalized discrete nonlinear Schrödinger equation $$i\dot{u}_n(t)+\sum_{m=-\infty}^{+\infty} J(n-m)u_m(t)+g\bigl(u_n(t)\bigr)+i\gamma u_n(t)=f_n,\quad n\in\mathbb{Z}, $$ with long-range interactions in weighted spaces \(\ell_{\mathbf{{q}}}^{2}\) is considered. Under suitable assumptions on the coupling constants J(m), the damping γ and the weight \(\mathbf{{q}}=(q_{n})_{n\in \mathbb{Z}}\) , the existence of a global attractor is proved. 相似文献
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Arne Jensen 《Arkiv f?r Matematik》1998,36(2):363-377
Results are obtained on the scattering theory for the Schrödinger equation $i\partial _t u(t,x) = - \Delta _x u(t,x) + V(t,x)u(t,x) + F(u(t,x))$ in spacesL r (R;L q (R d )) for a certain range ofr, q, the so-called space-time scattering. In the linear case (i.e.F≡)) the relation with usual configuration space scattering is established. 相似文献
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Laurent Michel 《偏微分方程通讯》2013,38(7):1198-1215
We study the nonlinear Schrödinger equation with time-depending magnetic field without smallness assumption at infinity. We obtain some results on the Cauchy problem, WKB asymptotics and instability. 相似文献
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Solitary Waves for the Generalized
Nonautonomous Dual-power Nonlinear
Schrödinger Equations with Variable Coeffcients
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In this paper, we study the solitary waves for the generalized
nonautonomous dual-power nonlinear Schrödinger equations (DPNLS) with
variable coefficients, which could be used to describe the saturation of the
nonlinear refractive index and the solitons in photovoltaic-photorefractive ma-
terials such as LiNbO3, as well as many nonlinear optics problems. We gen-
eralize an explicit similarity transformation, which maps generalized nonau-
tonomous DPNLS equations into ordinary autonomous DPNLS. Moreover,
solitary waves of two concrete equations with space-quadratic potential and
optical super-lattice potential are investigated. 相似文献
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Philipp Bader Arieh Iserles Karolina Kropielnicka Pranav Singh 《Foundations of Computational Mathematics》2014,14(4):689-720
The computation of the semiclassical Schrödinger equation presents major challenges because of the presence of a small parameter. Assuming periodic boundary conditions, the standard approach consists of semi-discretisation with a spectral method, followed by an exponential splitting. In this paper we sketch an alternative strategy. Our analysis commences with the investigation of the free Lie algebra generated by differentiation and by multiplication with the interaction potential: it turns out that this algebra possesses a structure which renders it amenable to a very effective form of asymptotic splitting: exponential splitting where consecutive terms are scaled by increasing powers of the small parameter. This leads to methods which attain high spatial and temporal accuracy and whose cost scales as \({\mathcal {O}}\!\left( M\log M\right) \) , where \(M\) is the number of degrees of freedom in the discretisation. 相似文献
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In this paper we present analogues of the maximum principle and of some parabolic inequalities for the regularized time-dependent Schrödinger operator on open manifolds using Günter derivatives. Moreover, we study the uniqueness of bounded solutions for the regularized Schrödinger–Günter problem and obtain the corresponding fundamental solution. Furthermore, we present a regularized Schrödinger kernel and prove some convergence results. Finally, we present an explicit construction for the fundamental solution to the Schrödinger–Günter problem on a class of conformally flat cylinders and tori. 相似文献
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Wei Mingjun 《高校应用数学学报(英文版)》2003,18(1):30-34
Based on the methods introduced by Klainerman and Ponce, and Cohn, a lower hounded estimate of the existence time for a kind of semilinear Schrödinger equation is ohtained in this paper. The implementation of this method depends on the L p ? L q estimate and the energy estimate. 相似文献
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We prove a Poisson type formula for the Schrödinger group. Such a formula had been derived in a previous article by the authors, as a consequence of the study of the asymptotic behavior of nonlinear wave operators for small data. In this note, we propose a direct proof, and extend the range allowed for the power of the nonlinearity to the set of all short range nonlinearities. Moreover, H 1-critical nonlinearities are allowed. 相似文献
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VIJAY KUMAR SOHANI 《Proceedings Mathematical Sciences》2013,123(4):525-537
We obtain homogeneous Strichartz estimate for the Schrödinger propagator e $^{-itL_{\alpha}}$ for the Laguerre operator L α on ${\mathbb R}_+^n$ . We follow regularization technique as introduced in J. Funct. Anal. 224(2) (2005) 371–385. We also establish inhomogeneous Strichartz estimates for different admissible pairs. 相似文献
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A system of nonlinear Schrödinger equations u
k
} / t=ia
k u
k+f
k
(u,u
*), t>0, k=1,... ,m; u
k
(0,x)=u
k0
(x), where f
k
are homogeneous functions of order 1+4/n, is considered. Sufficient conditions for the globality of the solution are obtained. The existence of the explicit blow-up solution is proved. 相似文献
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In this paper, we study the blow-up theory for the L 2-critical coupled nonlinear Schrödinger system in the Euclidean plane ${\mathbb{R}^{2}}In this paper, we study the blow-up theory for the L
2-critical coupled nonlinear Schr?dinger system in the Euclidean plane
\mathbbR2{\mathbb{R}^{2}} .We show that at the finite time blow-up, similar results of Weinstein, Merle-Tsutsumi for scalar Schrodinger equations are
true for the system. 相似文献
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We consider a derivative nonlinear Schrödinger equation with a general nonlinearity. This equation has a two-parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the nonlinearity and, in some instances, on the velocity. We illustrate these results with numerical simulations. 相似文献
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In this paper, we consider global solutions for the following nonlinear Schrödinger equation
in
with
and
We show that no nontrivial solution can decay faster than the solutions of the free Schrödinger equation, provided that u(0) lies in the weighted Sobolev space
in the energy space, namely
or in
according to the different cases. 相似文献
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