共查询到20条相似文献,搜索用时 15 毫秒
1.
Let λ i (Ω,V) be the i th eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain $\Omega \subset \mathbb{R}^nLet λ
i
(Ω,V) be the i
th
eigenvalue of the Schr?dinger operator with Dirichlet boundary conditions on a bounded domain
and with the positive potential V. Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential V
*, we prove that λ2(Ω,V) ≤ λ2(S
1,V
*). Here S
1 denotes the ball, centered at the origin, that satisfies the condition λ1(Ω,V)=λ1(S
1,V
*).Further we prove under the same convexity assumptions on a spherically symmetric potential V, that λ2(B
R
, V) / λ1(B
R
, V) decreases when the radius R of the ball B
R
increases.We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density.R.B. was supported by FONDECYT project # 102-0844.H.L. gratefully acknowledges financial support from DIPUC of the Pontifí cia Universidad Católica de Chile and from CONICYT. 相似文献
2.
We consider a discrete Schrödinger operator H=–+V acting in l
2(
d
), with periodic potential V supported by the subspace surface {0}×
d
2. We prove that the spectrum of H is purely absolutely continuous, and that surface waves oscillate in the longitudinal directions to the surface. We also find an explicit formula for the generalized spectral shift function introduced by the author in Helv. Phys. Acta.
72 (1999), 93–122. 相似文献
3.
We discuss properties of eigenvalues of non-self-adjoint Schrödinger operators with complex-valued potential V. Among our results are estimates of the sum of powers of imaginary parts of eigenvalues by the L p -norm of \({{\Im{V}}}\). 相似文献
4.
5.
6.
We study the one-dimensional random dimer model, with Hamiltonian H ω =Δ+V ω , where for all x∈ $\mathbb{Z}$ , V ω(2x)=V ω(2x+1) and where the V ω(2x) are i.i.d. Bernoulli random variables taking the values ±V, V>0. We show that, for all values of Vand with probability one in ω, the spectrum of His pure point. If V≤1 and V≠1/ $\sqrt 2$ , the Lyapunov exponent vanishes only at the two critical energies given by E=±V. For the particular value V=1/ $\sqrt 2$ , respectively, V= $\sqrt 2$ , we show the existence of new additional critical energies at E=±3/ $\sqrt 2$ , respectively, E=0. On any compact interval Inot containing the critical energies, the eigenfunctions are then shown to be semi-uniformly exponentially localized, and this implies dynamical localization: for all q>0 and for all ψ∈ $\ell$ 2( $\mathbb{Z}$ ) with sufficiently rapid decrease $${\mathop {\sup }\limits_t} r_{\psi ,I}^{\left( q \right)} {\kern 1pt} \left( t \right): = {\mathop {\sup }\limits_t} \left\langle {P_I \left( {H\omega } \right)\psi _t ,\left| X \right|^q P_I \left( {H\omega } \right)\psi _t } \right\rangle < \infty $$ Here $\psi _t = e^{- iH_{\omega ^t}} \psi$ , and P I(H ω) is the spectral projector of H ωonto the interval I. In particular, if V>1 and V≠ $\sqrt 2$ , these results hold on the entire spectrum [so that one can take I=σ(H ω)]. 相似文献
7.
J. Bourgain 《Journal of statistical physics》2013,153(1):1-9
We consider the random Schrödinger operator on a strip of width W, assuming the site distribution of bounded density. It is shown that the positive Lyapounov exponents satisfy a lower bound roughly exponential in ?W for W→∞. The argument proceeds directly by establishing Green’s function decay, but does not appeal to Furstenberg’s random matrix theory on the strip. One ingredient involved is the construction of ‘barriers’ using the random Schrödinger operator theory on $\mathbb{Z}$ . 相似文献
8.
9.
Let H=−Δ+V(x) be a three dimensional Schrödinger operator. We study the time decay in Lp spaces of scattering solutions e−itHPcu, where Pc is the orthogonal projection onto the continuous spectral subspace of L2(R3) for H. Under suitable decay assumptions on V(x) it is shown that they satisfy the so-called Lp-Lq estimates ||e−itHPcu||p≤(4π|t|)−3(1/2−1/p)||u||q for all 1≤q≤2≤p≤∞ with 1/p+1/q=1 if H has no threshold resonance and eigenvalue; and for all 3/2<q≤2≤p<3 if otherwise. 相似文献
10.
We consider random Schrödinger operators of the form \({\Delta+\xi}\), where \({\Delta}\) is the lattice Laplacian on \({\mathbb{Z}^{d}}\) and \({\xi}\) is an i.i.d. random field, and study the extreme order statistics of the Dirichlet eigenvalues for this operator restricted to large but finite subsets of \({\mathbb{Z}^{d}}\). We show that, for \({\xi}\) with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class, and the corresponding eigenfunctions are exponentially localized in regions where \({\xi}\) takes large, and properly arranged, values. The picture we prove is thus closely connected with the phenomenon of Anderson localization at the spectral edge. Notwithstanding, our approach is largely independent of existing methods for proofs of Anderson localization and it is based on studying individual eigenvalue/eigenfunction pairs and characterizing the regions where the leading eigenfunctions put most of their mass. 相似文献
11.
Given a one dimensional perturbed Schrödinger operator H = ? d 2/dx 2 + V(x), we consider the associated wave operators W ± , defined as the strong L 2 limits $\lim_{s\to\pm\infty}e^{isH}e^{-isH_{0}}Given a one dimensional perturbed Schr?dinger operator H = − d
2/dx
2 + V(x), we consider the associated wave operators W
± , defined as the strong L
2 limits
. We prove that W
± are bounded operators on L
p
for all 1 < p < ∞, provided
, or else
and 0 is not a resonance. For p = ∞ we obtain an estimate in terms of the Hilbert transform. Some applications to dispersive estimates for equations with
variable rough coefficients are given. 相似文献
12.
The transformations of all the Schrödinger operators with point interactions in dimension one under space reflection P, time reversal T and (Weyl) scaling W are presented. In particular, those operators which are invariant (possibly up to a scale) are selected. Some recent papers on related topics are commented upon. 相似文献
13.
We study the final problem for the nonlinear Schrödinger equation where\(\lambda \in{\bf R},n=1,2,3\). If the final data\(u_{+}\in {\bf H}^{0,\alpha }=\left\{ \phi \in {\bf L}^{2}:\left( 1+\left\vert x\right\vert \right) ^{\alpha }\phi \in {\bf L}^{2}\right\} \) with\(\frac{ n}{2} < \alpha < \min \left( n,2,1+\frac{2}{n}\right) \) and the norm\(\Vert \widehat{u_{+}}\Vert _{{\bf L}^{\infty }}\) is sufficiently small, then we prove the existence of the wave operator in L 2. We also construct the modified scattering operator from H 0,α to H 0,δ with\(\frac{n}{2} < \delta < \alpha\).
相似文献
$i{\partial }_{t}u+\frac{1}{2}\Delta u=\lambda|u|^{\frac{2}{n}}u,\quad (t,x)\in {\mathbf{R}}\times \mathbf{R}^{n},$
14.
Shu Nakamura 《Communications in Mathematical Physics》2000,214(3):565-572
We study the behavior of the density states at the lower edge of the spectrum for Schr?dinger operators with random magnetic
fields. We use a new estimate on magnetic Schr?dinger operators, which is similar to the Avron–Herbst–Simon estimate but the
bound is always nonnegative.
Received: 3 January 2000 / Accepted: 18 April 2000 相似文献
15.
We show that the Schrödinger equation for a free particle on the surface of a spindle torus is quasi-exactly solvable. Our result complements former ones in an interesting way: it is known that the Schrödinger equation for a free particle on a ring torus is non-solvable, whereas it is exactly solvable for a particle on a horn torus. 相似文献
16.
We study semiclassical eigenvalues and eigenfunctions of the Schrödinger operator on a geometric graph. We show that nontrivial boundary conditions at vertices lead to the existence of eigenfunctions, concentrated near a single vertex. We also construct semiclassical eigenfunctions, localized near edges and discuss general construction of spectral series which correspond to a general subgraph. 相似文献
17.
For a two-dimensional Schrödinger operator H α V = ?Δ ?αV with the radial potential V(x) = F(|x|), F(r) ≥ 0, we study the behavior of the number N ?(H α V ) of its negative eigenvalues, as the coupling parameter α tends to infinity. We obtain the necessary and sufficient conditions for the semi-classical growth N ?(H α V ) = O(α) and for the validity of the Weyl asymptotic law. 相似文献
18.
We study Dirichlet Laplacian in a screw-shaped region, i.e. a straight twisted tube of a non-circular cross section. It is
shown that a local perturbation which consists of “slowing down” the twisting in the mean gives rise to a non-empty discrete
spectrum
Mathematical Subject Classifications: 35P05, 81Q10. 相似文献
19.
We show that the knowledge of the set of the Cauchy data on the boundary of a bounded open set in ${\mathbb{R}^n}$ , ${n \geq 3}$ , for the magnetic Schrödinger operator with L ∞ magnetic and electric potentials, determines the magnetic field and electric potential inside the set uniquely. The proof is based on a Carleman estimate for the magnetic Schrödinger operator with a gain of two derivatives. 相似文献
20.
V. P. Ruban 《JETP Letters》2012,95(9):486-491
A nonlinear Schrödinger equation with variable coefficients for surface waves on a large-scale steady nonuniform current has been derived without the assumption of a relative smallness of the velocity of the current. This equation can describe with good accuracy the loss of modulation stability of a wave coming to a counter current, leading to the formation of so-called rogue waves. Some theoretical estimates are compared to the numerical simulation with the exact equations for a two-dimensional potential motion of an ideal fluid with a free boundary over a nonuniform bottom at a nonzero average horizontal velocity. 相似文献