共查询到20条相似文献,搜索用时 15 毫秒
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We study the Schrödinger operator H = ?Δ + V(x) in dimension two, V(x) being a limit-periodic potential. We prove that the spectrum of H contains a semiaxis, and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves exp( $i\left\langle {\overrightarrow k ,\overrightarrow x } \right\rangle $ ) at the high energy region. Second, the isoenergetic curves in the space of momenta $\overrightarrow k $ corresponding to these eigenfunctions have the form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous. 相似文献
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We provide a complete spectral characterization of a new method of constructing isospectral (in fact, unitary) deformations of general Schrödinger operatorsH=?d 2/dx 2+V in $H = - d^2 /dx^2 + V in \mathcal{L}^2 (\mathbb{R})$ . Our technique is connected to Dirichlet data, that is, the spectrum of the operatorH D onL 2((?∞,x 0)) ⊕L 2((x 0, ∞)) with a Dirichlet boundary condition atx 0. The transformation moves a single eigenvalue ofH D and perhaps flips which side ofx 0 the eigenvalue lives. On the remainder of the spectrum, the transformation is realized by a unitary operator. For cases such asV(x)→∞ as |x|→∞, whereV is uniquely determined by the spectrum ofH and the Dirichlet data, our result implies that the specific Dirichlet data allowed are determined only by the asymptotics asE→∞. 相似文献
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We study the discreteness of the spectrum of Schrödinger operators which are defined on a class of radial N-dimensional rooted trees of a finite or infinite volume, and are subject to a certain mixed boundary condition. We present a method to estimate their eigenvalues using operators on a one-dimensional tree. These operators are called width-weighted operators, since their coefficients depend on the section width or area of the N-dimensional tree. We show that the spectrum of the width-weighted operator tends to the spectrum of a one-dimensional limit operator as the sections width tends to zero. Moreover, the projections to the one-dimensional tree of eigenfunctions of the N-dimensional Laplace operator converge to the corresponding eigenfunctions of the one-dimensional limit operator. 相似文献
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Doklady Mathematics - 相似文献
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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
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Shu Nakamura 《偏微分方程通讯》2013,38(10):1385-1419
The existence and the location of shape resonances are stud-ied for Schrödinger operators in the semiclassical limit. The potential is assumed to be a sum of an analytic function and an exponentially decaying smooth function. Analytic distortion which is local in momen-tum space is employed and h-pseuddodifferential operator methods are used. 相似文献
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We consider Schrödinger operators with periodic magnetic field having zero flux through a fundamental cell of the period lattice. We show that, for a generic small magnetic field and a generic small Fermi energy, the corresponding Fermi surface is convex and not invariant under inversion in any point. 相似文献
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《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1998,326(9):1145-1150
We use the method of the conjugate operator to prove a limiting absorption principle and the absence of the singular continuous spectrum for discrete Schrödinger operators. We also obtain local decay estimates. Our results apply to a large class of perturbating potentials V decaying arbitrarily slowly to zero at infinity. 相似文献
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Helge Krüger 《Journal d'Analyse Mathématique》2011,115(1):343-387
A variant of multiscale analysis for ergodic Schrödinger operators is developed. This enables us to prove positivity of Lyapunov exponents, given initial scale estimates and an initial Wegner estimate. This postivivity is then applied to high-dimensional skew-shifts at small coupling, where initial conditions are checked using the Pastur-Figotin formalism. 相似文献
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Andreas M. Hinz 《Proceedings Mathematical Sciences》2002,112(1):85-98
Since it became clear that the band structure of the spectrum of periodic Sturm-Liouville operatorst = - (d2/dr2) +q(r) does not survive a spherically symmetric extension to Schrödinger operatorsT =- Δ+ V with V(x) =q(¦x¦) for x ∈ ?d,d ∈ ? 1, a wealth of detailed information about the spectrum of such operators has been acquired. The observation of eigenvalues embedded in the essential spectrum [μ0, ∞[ ofT with exponentially decaying eigenfunctions provided evidence for the existence of intervals of dense point spectrum, eventually proved by spherical separation into perturbed Sturm-Liouville operatorst c = t +(c/r 2). Subsequently, a numerical approach was employed to investigate the distribution of eigenvalues ofT more closely. An eigenvalue was discovered below the essential spectrum in the cased = 2, and it turned out that there are in fact infinitely many, accumulating at μ0. Moreover, a method based on oscillation theory made it possible to count eigenvalues oft c contributing to an interval of dense point spectrum ofT. We gained evidence that an asymptotic formula, valid forc → ∞, does in fact produce correct numbers even for small values of the coupling constant, such that a rather precise picture of the spectrum of radially periodic Schrödinger operators has now been obtained. 相似文献
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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
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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