共查询到20条相似文献,搜索用时 16 毫秒
1.
The problem of absence of eigenvalues imbedded into the continuous spectrum is considered for a Schrödinger operator with a periodic potential perturbed by a sufficiently fast decaying "impurity" potential. Results of this type have previously been known for the one-dimensional case only. Absence of embedded eigenvalues is shown in dimensions two and three if the corresponding Fermi surface is irreducible modulo natural symmetries. It is conjectured that all periodic potentials satisfy this condition. Separable periodic potentials satisfy it, and hence in dimensions two and three Schrödinger operator with a separable periodic potential perturbed by a sufficiently fast decaying "impurity" potential has no embedded eigenvalues 相似文献
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《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
4.
We study one-dimensional Schrödinger operators with complex measures as potentials and present an improved criterion for absence of eigenvalues which involves a weak local periodicity condition. As an application we consider quasiperiodic measures as potentials. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
5.
Alberto Parmeggiani 《Annali di Matematica Pura ed Applicata》1988,152(1):237-258
Summary Let E(t, h) be the Fourier Integral Operator representing the parametrix of the propagator for the Schrödinger equation Q=iht, Q=– 1/2h2x + V(x), where V is a potential of quadratic growth inR
n. It is proved that E(t, h) coincides exactly with the Schwarte kernel of the propagator if and only if V is a positive quadratic form, i.e. if the corresponding physical system is a n-dimensional harmonic oscillator. 相似文献
6.
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
7.
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
8.
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
9.
A. F. Vakulenko 《Journal of Mathematical Sciences》1986,32(5):447-448
The operator –(d/dx)2+ where is a complex function, |J(x)(1+|x|1+|>, has no positive eigenvalues. This follows from the existence of a triangular Green function for the operator –(d/dx)z–,gl>0, and from Hardy type inequalities. We give the multidimensional analogues of these inequalities and we prove the absence of positive eigenvalues for the operator –+.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 33–34, 1984. 相似文献
10.
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
11.
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
12.
Suppose thatM n is a complete, noncompact, Riemannian manifold. If Δ denotes the Laplace operator ofM, one has associated Schrödinger operators ? Δ +V. Conditions onV are formulated, which ensures the essential self-adjointness of ? Δ +V. In particular, ifV ∈ Qα,loc (M n), the local Stummel class, andV ≥ ? c outside of a compact set, then ? Δ +V is essentially self-adjoint on C 0 ∞ (M n). In addition, essential self-adjointness is proved for potentials which are strongly singular at a point. The absence of eigenvalues of ?Δ +V is also studied. This relies upon Rellich-type identities. The results on strongly singular potentials make use of a generalization of the classical uncertainty principle, inR n, to Riemannian manifolds with a pole. 相似文献
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14.
《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
15.
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
16.
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
17.
The eigenvalues of the self-adjoint Schrödinger operators on a bounded domain with Dirichlet and Neumann boundary conditions are characterized by the singularities of an associated Dirichlet-to-Neumann map and its inverse, respectively. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
18.
M. Burak Erdoğan Michael Goldberg William R. Green 《Journal of Functional Analysis》2018,274(7):2139-2161
Let be a Schrödinger operator on with real-valued potential V, and let . If V has sufficient pointwise decay, the wave operators are known to be bounded on for all if zero is not an eigenvalue or resonance. We show that if there is an s-wave resonance or an eigenvalue only at zero, then the wave operators are bounded on for . This result stands in contrast to results in higher dimensions, where the presence of zero energy obstructions is known to shrink the range of valid exponents p. 相似文献
19.
Andreas Enblom 《Journal of Mathematical Sciences》2009,156(4):699-723
We prove that [0, ∞) is an essential support for the absolutely continuous part of the spectral measure associated with the
magnetic Schr?dinger operator (i∇ + A)2 in , given certain conditions on the decay of A. Bibliography: 8 titles. Illustrations: 1 figure.
Translated from Problemy Matematicheskogo Analiza, No. 38, 2008, pp. 121–143. 相似文献
20.
Ihyeok Seo 《Monatshefte für Mathematik》2016,180(4):893-912
In this paper we find a new condition on a real periodic potential for which the self-adjoint Schrödinger operator may be defined by a quadratic form and the spectrum of the operator is purely absolutely continuous. This is based on resolvent estimates and spectral projection estimates in weighted \(L^2\) spaces on the torus, and an oscillatory integral theorem. 相似文献