共查询到20条相似文献,搜索用时 15 毫秒
1.
Matania Ben-Artzi 《Israel Journal of Mathematics》1981,40(3-4):259-274
LetH=?Δ+V(r) be a Schrödinger operator with a spherically symmetric exploding potential, namely,V(r)=V S(r)+V L(r), whereV S(r) is short-range and the exploding partV L(r) satisfies the following assumptions: (a) Λ=lim sup r→∞ V L(r)<∞ (but Λ=?∞ is possible). Denote Λ+= max(Λ,0). (b)V L(r)∈C 2k (r 0, ∞) and, with someδ>0 such that 2kδ>1: (d/dr) j V L(r) · (Λ+?V L(r))?1=O(r jδ) asr → ∞,j=1, ..., 2k. (c) ∫ r0 ∞ dr|V L(r|1/2 dr|V L(r)|1/2=∞. (d) (d/dr)V L(r)≦0. Under these assumptions a limiting absorption principle forR(z)=(H?z)?1 is established. More specifically, ifK ?C +={zImz≧0} is compact andK ∩ (?∞, Λ]=Ø thenR (z) can be extended as a continuous map ofK intoB (Y, Y*) (with the uniform operator topology), whereY ?L 2(R n) is a weighted-L 2 space. To ensure uniqueness of solutions of (H?z)u=f, z ∈K, a suitable radiation condition is introduced. 相似文献
2.
For one-dimensional Schrödinger operators with complex-valued potentials, we construct pseudomodes corresponding to large pseudoeigenvalues. We develop a first systematic non-semi-classical approach, which results in a substantial progress in achieving optimal conditions and conclusions as well as in covering a wide class of previously inaccessible potentials, including discontinuous ones. Applications of the present results to higher-dimensional Schrödinger operators are also discussed. 相似文献
3.
In this paper we consider the Schrödinger operator ?Δ + V on \({\mathbb R^d}\), where the nonnegative potential V belongs to the reverse Hölder class \({B_{q_{_1}}}\) for some \({q_{_1}\geq \frac{d}{2}}\) with d ≥ 3. Let \({H^1_L(\mathbb R^d)}\) denote the Hardy space related to the Schrödinger operator L = ?Δ + V and \({BMO_L(\mathbb R^d)}\) be the dual space of \({H^1_L(\mathbb R^d)}\). We show that the Schrödinger type operator \({\nabla(-\Delta +V)^{-\beta}}\) is bounded from \({H^1_L(\mathbb R^d)}\) into \({L^p(\mathbb R^d)}\) for \({p=\frac{d}{d-(2\beta-1)}}\) with \({ \frac{1}{2}<\beta<\frac{3}{2} }\) and that it is also bounded from \({L^p(\mathbb R^d)}\) into \({BMO_L(\mathbb R^d)}\) for \({p=\frac{d}{2\beta-1}}\) with \({ \frac{1}{2}<\beta< 2}\). 相似文献
4.
Teruo Ikebe 《Journal of Functional Analysis》1975,20(2):158-177
A spectral representation for the self-adjoint Schrödinger operator H = ?Δ + V(x), x? R3, is obtained, where V(x) is a long-range potential: , grad , being the Laplace-Beltrami operator on the unit sphere Ω. Namely, we shall construct a unitary operator from PL2(R3) onto being the orthogonal projection onto the absolutely continuous subspace for H, such that for any Borel function α(λ), . 相似文献
5.
《Journal of Computational and Applied Mathematics》2002,148(1):1-28
Several recent papers have obtained bounds on the distribution of eigenvalues of non-self-adjoint Schrödinger operators and resonances of self-adjoint operators. In this paper we describe two new methods of obtaining such bounds when the potential decays more slowly than previously permitted. 相似文献
6.
Karl-Theodor Sturm 《manuscripta mathematica》1992,76(1):367-395
We investigate the Feynman-Kac semigroupP t V and its densityp V(t,.,.),t>0, associated with the Schrödinger operator ?1/2Δ+V on ?d\{0}.V will be a highly singular, oscillating potential like $V\left( x \right) = k \cdot \left\| x \right\|^{ - 1} \cdot \sin \left( {\left\| x \right\|^{ - m} } \right)$ with arbitraryk, l, m>0. We derive conditions (onk,l,m) which are sufficientand necessary for the existence of constants α, β, γ, ∈ ? such that for allt, x, y p V(t, x, y)≤γ·p(βt, x, y)·eat. On the other hand, also conditions are derived which imply thatp V (t, x, y)≡∞ for allt, x, y. The aim is to see to which extent quick oscillations can lead to annihilations of the singularities ofV. For this purpose, we analyse the above example in great detail. Note that forl≥2 the potential is so singular that none of the usual perturbation techniques applies. 相似文献
7.
A. A. Oblomkov 《Theoretical and Mathematical Physics》1999,121(3):1574-1584
We consider one-dimensional monodromy-free Schrödinger operators with quadratically increasing rational potentials. It is shown that all these operators can be obtained from the operator -?2 + x2 by finitely many rational Darboux transformations. An explicit expression is found for the corresponding potentials in terms of Hermite polynomials. 相似文献
8.
9.
A construction of “sparse potentials,” suggested by the authors for the lattice
\mathbbZd {\mathbb{Z}^d} , d > 2, is extended to a large class of combinatorial and metric graphs whose global dimension is a number D > 2. For the Schr?dinger operator − Δ − αV on such graphs, with a sparse potential V, we study the behavior (as α → ∞) of the number N_(−Δ − αV) of negative eigenvalues of − Δ − αV. We show that by means of sparse potentials one can realize any prescribed asymptotic behavior of N_(−Δ − αV) under very mild regularity assumptions. A similar construction works also for the lattice
\mathbbZ2 {\mathbb{Z}^2} , where D = 2. Bibliography: 13 titles. 相似文献
10.
Ian Knowles 《Journal of Mathematical Analysis and Applications》1978,66(3):574-585
A sufficient condition is given for the operator T0: C∞0(Rm) → L2(Rm) given by to be essentially self-adjoint. This condition is sufficiently general to admit certain potentials q having unbounded oscillations in a neighborhood of ∞. 相似文献
11.
Consider the Schrödinger operator
with a complex-valued
potential v of period
Let
and
be the eigenvalues of L that are close to
respectively, with periodic (for n even),
antiperiodic (for n odd), and Dirichelet
boundary conditions on [0,1], and let
be the diameter of the spectral
triangle with vertices
We prove the following statement: If
then v(x) is a Gevrey function, and moreover
相似文献
12.
13.
We prove that a deformation of a hypersurface in an (n + 1)-dimensional real space form \({{\mathbb S}^{n+1}_{p,1}}\) induces a Hamiltonian variation of the normal congruence in the space \({{\mathbb L}({\mathbb S}^{n+1}_{p,1})}\) of oriented geodesics. As an application, we show that every Hamiltonian minimal submanifold in \({{\mathbb L}({\mathbb S}^{n+1})}\) (resp. \({{\mathbb L}({\mathbb H}^{n+1})}\)) with respect to the (para-)Kähler Einstein structure is locally the normal congruence of a hypersurface \({\Sigma}\) in \({{\mathbb S}^{n+1}}\) (resp. \({{\mathbb H}^{n+1}}\)) that is a critical point of the functional \({{\mathcal W}(\Sigma) = \int_\Sigma\left(\Pi_{i=1}^n|\epsilon+k_i^2|\right)^{1/2}}\), where ki denote the principal curvatures of \({\Sigma}\) and \({\epsilon \in \{-1, 1\}}\). In addition, for \({n = 2}\), we prove that every Hamiltonian minimal surface in \({{\mathbb L}({\mathbb S}^{3})}\) (resp. \({{\mathbb L}({\mathbb H}^{3})}\)), with respect to the (para-)Kähler conformally flat structure, is the normal congruence of a surface in \({{\mathbb S}^{3}}\) (resp. \({{\mathbb H}^{3}}\)) that is a critical point of the functional \({{\mathcal W}\prime(\Sigma) = \int_\Sigma\sqrt{H^2-K+1}}\) (resp. \({{\mathcal W}\prime(\Sigma) = \int_\Sigma\sqrt{H^2-K-1}}\)), where H and K denote, respectively, the mean and Gaussian curvature of \({\Sigma}\). 相似文献
14.
Wei Wei 《Czechoslovak Mathematical Journal》2014,64(2):539-566
We give a new representation of solutions to a class of time-dependent Schrödinger type equations via the short-time Fourier transform and the method of characteristics. Moreover, we also establish some novel estimates for oscillatory integrals which are associated with the fractional power of negative Laplacian \({( - \Delta )^{\kappa /2}}\) with 1 ? κ ? 2. Consequently the classical Hamiltonian corresponding to the previous Schrödinger type equations is studied. As applications, a series of new boundedness results for the corresponding propagator are obtained in the framework of modulation spaces. The main results of the present article include the case of wave equations. 相似文献
15.
The aim of this note is to prove the following theorem. Let
where P(ix) is a nonnegative homogeneous elliptic polynomial on R
d
and V is a nonnegative polynomial potential. Then for every 1 < p < ∞ and every α > 0 there exist constants C
1, C
2 > 0 such that
and
for f in the Schwartz class . We take advantage of the Christ inversion theorem for singular integral operators with a small amount of smoothness on
nilpotent Lie groups, the maximal subelliptic L
2-estimates for the generators of stable semi-groups of measures, and the principle of transference of Coifman–Weiss.
In memory of Tadek Pytlik, our teacher and friend.
Research supported by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis,
Nonlinear Analysis and Probability” MTKD-CT-2004-013389 and by Polish funds for science in years 2005–2008 (research project
1P03A03029). 相似文献
16.
Lorenzo Brasco Giovanni Franzina Berardo Ruffini 《Journal of Functional Analysis》2018,274(6):1825-1863
We consider the Schrödinger operator for negative potentials V, on open sets with positive first eigenvalue of the Dirichlet–Laplacian. We show that the spectrum of is positive, provided that V is greater than a negative multiple of the logarithmic gradient of the solution to the Lane–Emden equation (for some ). In this case, the ground state energy of is greater than the first eigenvalue of the Dirichlet–Laplacian, up to an explicit multiplicative factor. This is achieved by means of suitable Hardy-type inequalities, that we prove in this paper. 相似文献
17.
18.
We study Schrödinger operatorsT+Q, whereT=?Δ is the Laplace operator andQ is the multiplication operator by a generalized function (distribution). We also consider generalizations for the case of the polyharmonic operatorT = (-δ) n 相似文献
19.
Emil Kieri 《BIT Numerical Mathematics》2014,54(3):729-748
When numerically solving the time-dependent Schrödinger equation for the electrons in an atom or molecule, the Coulomb singularity poses a challenge. The solution will have limited regularity, and high-order spatial discretisations, which are much favoured in the chemical physics community, are not performing to their full potential. By exploiting knowledge about the jumps in the derivatives of the solution we construct a correction, and show how this improves the convergence rate of Fourier collocation from second to fourth order. This allows for a substantial reduction in the number of grid points. The new method is applied to the higher harmonic generation from atomic hydrogen. 相似文献