共查询到20条相似文献，搜索用时 15 毫秒
1.
本文主要研究如下带有三次方项的拟线性薛定谔方程begin{equation*}label{eq1}left{begin{aligned}&-Delta u+V(|x|)u-frac{1}{2}Delta(|u|^2)u=lambda|u|^2u,quadmbox{in }mathbb{R}^N, &uto 0, qquad as |x|toinfty, end{aligned}right.end{equation*}其中$Ngeq 3,lambda>0$, $V(|x|)$ 表示正的径向对称位势函数. 利用能量比较方法和变分方法,并结合极限方法,可得上述方程存在任意给定$k$次径向节点解 $U_{k,4}^lambda$,其中正整数$kgeq1$. 进一步, $U_{k,4}^lambda$的能量关于$k$单调递增.同时任意给定序列${lambda_n}$,在子列意义下, 当$lambda_nto +infty$时,$lambda_n^{frac{1}{2}}U_{k,4}^{lambda_n}$收敛于$bar{U}_{k,4}^0$,其中$bar{U}_{k,4}^0$是如下经典薛定谔方程 $k$次径向节点解begin{equation*}left{begin{aligned}&-Delta u+V(|x|)u=|u|^2uquadmbox{in }mathbb{R}^N, &uto 0 qquad as |x|toinfty.end{aligned}right.end{equation*}本文研究结果将非线性项超三次方的情形推广到三次方的情形,拓展了已有文献中的结果. 相似文献
2.
Mikló s Horvá th Má rton Kiss 《Proceedings of the American Mathematical Society》2006,134(5):1425-1434
For Schrödinger operators with nonnegative single-well potentials ratios of eigenvalues are extremal only in the case of zero potential. To prove this, we investigate some monotonicity properties of Prüfer-type variables.
3.
Giorgio Metafune Diego Pallara 《Proceedings of the American Mathematical Society》2002,130(6):1779-1786
We give estimates on the bottom of the essential spectrum of Schrödinger operators in .
4.
Atanas Stefanov 《Proceedings of the American Mathematical Society》2001,129(5):1395-1401
We prove an endpoint Strichartz estimate for radial solutions of the two-dimensional Schrödinger equation:
5.
Alexander Pankov 《Proceedings of the American Mathematical Society》2008,136(7):2565-2570
We present general results on exponential decay of finite energy solutions to stationary nonlinear Schrödinger equations. Under certain natural assumptions we show that any such solution is continuous and vanishes at infinity. This allows us to interpret the solution as a finite multiplicity eigenfunction of a certain linear Schrödinger operator and, hence, apply well-known results on the decay of eigenfunctions.
6.
7.
Alexander Kiselev 《Journal of the American Mathematical Society》2005,18(3):571-603
We construct examples of potentials satisfying where the function is growing arbitrarily slowly, such that the corresponding Schrödinger operator has an imbedded singular continuous spectrum. This solves one of the fifteen ``twenty-first century\" problems for Schrödinger operators posed by Barry Simon. The construction also provides the first example of a Schrödinger operator for which Möller wave operators exist but are not asymptotically complete due to the presence of a singular continuous spectrum. We also prove that if the singular continuous spectrum is empty. Therefore our result is sharp.
8.
Ill-posedness is established for the initial value problem (IVP) associated to the derivative nonlinear Schrödinger equation for data in , . This result implies that best result concerning local well-posedness for the IVP is in . It is also shown that the (IVP) associated to the generalized Benjamin-Ono equation for data below the scaling is in fact ill-posed.
9.
The authors study, by applying and extending the methods developed by Cazenave(2003), Dias and Figueira(2014), Dias et al.(2014), Glassey(1994–1997), Kato(1987), Ohta and Todorova(2009) and Tsutsumi(1984), the Cauchy problem for a damped coupled system of nonlinear Schrdinger equations and they obtain new results on the local and global existence of H~1-strong solutions and on their possible blowup in the supercritical case and in a special situation, in the critical or supercritical cases. 相似文献
10.
Hon-kit Wai 《Transactions of the American Mathematical Society》1999,351(6):2141-2182
Given an -invariant Morse function and an -invariant Riemannian metric , a family of finite dimensional subcomplexes , , of the Witten deformation of the -equivariant de Rham complex is constructed, by studying the asymptotic behavior of the spectrum of the corresponding Laplacian as . In fact the spectrum of can be separated into the small eigenvalues, finite eigenvalues and the large eigenvalues. Then one obtains as the complex of eigenforms corresponding to the small eigenvalues of . This permits us to verify the -equivariant Morse inequalities. Moreover suppose is self-indexing and satisfies the Morse-Smale condition, then it is shown that this family of subcomplexes converges as to a geometric complex which is induced by and calculates the -equivariant cohomology of .
11.
We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of mathematical physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved Combes-Thomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than any polynomial.
12.
David Damanik Serguei Tcheremchantsev 《Journal of the American Mathematical Society》2007,20(3):799-827
We develop a general method to bound the spreading of an entire wavepacket under Schrödinger dynamics from above. This method derives upper bounds on time-averaged moments of the position operator from lower bounds on norms of transfer matrices at complex energies.
This general result is applied to the Fibonacci operator. We find that at sufficiently large coupling, all transport exponents take values strictly between zero and one. This is the first rigorous result on anomalous transport.
For quasi-periodic potentials associated with trigonometric polynomials, we prove that all lower transport exponents and, under a weak assumption on the frequency, all upper transport exponents vanish for all phases if the Lyapunov exponent is uniformly bounded away from zero. By a well-known result of Herman, this assumption always holds at sufficiently large coupling. For the particular case of the almost Mathieu operator, our result applies for coupling greater than two.
13.
David Damanik Dirk Hundertmark 《Proceedings of the American Mathematical Society》2004,132(7):1957-1962
We prove a criterion for absence of decaying solutions for one-dimensional Schrödinger operators. As necessary input, we require infinitely many centers of local reflection symmetry and upper and lower bounds for the traces of the associated transfer matrices.
14.
Christian Remling 《Transactions of the American Mathematical Society》1999,351(6):2479-2497
We investigate one-dimensional Schrödinger operators with sparse potentials (i.e. the potential consists of a sequence of bumps with rapidly growing barrier separations). These examples illuminate various phenomena related to embedded singular continuous spectrum.
15.
Daniella Bekiranov Takayoshi Ogawa Gustavo Ponce 《Proceedings of the American Mathematical Society》1997,125(10):2907-2919
An interaction equation of the capillary-gravity wave is considered. We show that the Cauchy problem of the coupled Schrödinger-KdV equation,
is locally well-posed for weak initial data . We apply the analogous method for estimating the nonlinear coupling terms developed by Bourgain and refined by Kenig, Ponce, and Vega.
16.
Yong Hah Lee 《Proceedings of the American Mathematical Society》2005,133(11):3411-3420
We pose and solve the asymptotic Dirichlet problem for the Schrödinger operator via rough isometries on a certain class of Riemannian manifolds. With suitable potentials, we give the solvability of the problem for a naturally defined class of data functions.
17.
We consider discrete one-dimensional Schrödinger operators whose potentials decay asymptotically like an inverse square. In the super-critical case, where there are infinitely many discrete eigenvalues, we compute precise asymptotics of the number of eigenvalues below a given energy as this energy tends to the bottom of the essential spectrum.
18.
Kasso A. Okoudjou Robert S. Strichartz 《Proceedings of the American Mathematical Society》2007,135(8):2453-2459
In this note we investigate the asymptotic behavior of spectra of Schrödinger operators with continuous potential on the Sierpinski gasket . In particular, using the existence of localized eigenfunctions for the Laplacian on we show that the eigenvalues of the Schrödinger operator break into clusters around certain eigenvalues of the Laplacian. Moreover, we prove that the characteristic measure of these clusters converges to a measure. Results similar to ours were first observed by A. Weinstein and V. Guillemin for Schrödinger operators on compact Riemannian manifolds.
19.
Christian Remling 《Proceedings of the American Mathematical Society》2007,135(10):3329-3340
We point out finite propagation speed phenomena for discrete and continuous Schrödinger operators and discuss various types of kernel estimates from this point of view.
20.
We consider the semilinear Schrödinger equation , , where , are periodic in for , 0$\">, is of subcritical growth and 0 is in a gap of the spectrum of . We show that under suitable hypotheses this equation has a solution . In particular, such a solution exists if and .