共查询到20条相似文献,搜索用时 15 毫秒
1.
Let m
, 0 m+ in Kato's class. We investigate the spectral function s( + m) where s( + m) denotes the upper bound of the spectrum of the Schrödinger operator + m. In particular, we determine its derivative at 0. If m- is sufficiently large, we show that there exists a unique 1 > 0 such that s( + 1m) = 0. Under suitable conditions on m+ it follows that 0 is an eigenvalue of + 1m with positive eigenfunction. 相似文献
2.
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. The criterion leads to sharp quantitative bounds on the eigenvalues. We apply our result to quasiperiodic measures as potentials. 相似文献
3.
Yanheng Ding Andrzej Szulkin 《Calculus of Variations and Partial Differential Equations》2007,29(3):397-419
We study the existence and the number of decaying solutions for the semilinear Schrödinger equations \({-\varepsilon^{2}\Delta u + V(x)u = g(x,u)}\), \({\varepsilon > 0}\) small, and \({-\Delta u + \lambda V(x)u = g(x,u)}\), \({\lambda > 0}\) large. The potential V may change sign and g is either asymptotically linear or superlinear (but subcritical) in u as \({|u| \to \infty}\) . 相似文献
4.
Maximal dissipative Schrödinger operators are studied in L
2((–,);E) (dimE=n<) that the extensions of a minimal symmetric operator with defect index (n,n) (in limit-circle case at – and limit point-case at ). We construct a selfadjoint dilation of a dissipative operator, carry out spectral analysis of a dilation, use the Lax–Phillips scattering theory, and find the scattering matrix of a dilation. We construct a functional model of the dissipative operator, determine its characteristic function in terms of the Titchmarsh–Weyl function of selfadjoint operator and investigate its analytic properties. Finally, we prove a theorem on completeness of the eigenvectors and associated vectors of a dissipative Schrödinger operators. 相似文献
5.
Itaru Sasaki 《Integral Equations and Operator Theory》2007,58(4):563-571
Schr?dinger operators
with rapidly oscillating potentials V such as
are considered. Such potentials are not relatively compact with respect to the free Schr?dinger operator −Δ. We show that
the oscillating potential V do not change the essential spectrum of −Δ. Moreover we derive upper bounds for negative eigenvalue sums of Ĥ. 相似文献
6.
Let be a domain in , . Let be a divergence form uniformly elliptic operator with Dirichlet boundary conditions on and Neumann boundary conditions on , where is a closed subset of . We prove intrinsic ultracontractivity for the semigroup associated to the Schrödinger operator , where is a potential in the Kato class, provided that is locally Lipschitz and is given by the boundary of either a Hölder domain of order or a uniformly Hölder domain of order , . Our results extend to the mixed boundary case the results of Bañuelos, Bass and Burdzy, Bass and Hsu, and Davies and Simon. 相似文献
7.
Let \(x \in \mathbb {R}^{d}\), d ≥ 3, and \(f: \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ i, j ≤ d, let \(a_{ij} : \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated towhere \(J: \mathbb {R}^{d} \times \mathbb {R}^{d} \rightarrow \mathbb {R}\) is a symmetric measurable function. Let \(q: \mathbb {R}^{d} \rightarrow \mathbb {R}.\) We specify assumptions on a, q, and J so that non-negative bounded solutions to satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to \(\mathcal {L}f = 0.\)
相似文献
$$\mathcal{L}f(x) = \frac12 \sum\limits_{i=1}^{d} \sum\limits_{j=1}^{d} \frac{\partial}{\partial x_{i}} \left( a_{ij}(\cdot) \frac{\partial f}{\partial x_{j}}\right)(x) + {\int}_{\mathbb{R}^{d}\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy $$
$$\mathcal{L}f + qf = 0 $$
8.
We prove local smoothing estimates for the Schrödinger initial value problem with data in the energy space L 2(? d ), d ≥ 2 and a general class of potentials. In the repulsive setting we have to assume just a power like decay (1 + |x|)?γ for some γ > 0. Also attractive perturbations are considered. The estimates hold for all time and as a consequence a weak dispersion of the solution is obtained. The proofs are based on similar estimates for the corresponding stationary Helmholtz equation and Kato H-smooth theory. 相似文献
9.
Hendrik Vogt 《Annales Henri Poincare》2009,10(2):395-414
We study Schr?dinger operators on formally given by , where μ is a positive, compactly supported measure from the Kato class. Under the assumption that a certain condition on
the μ-volume of balls is satisfied and that Hμ has at least two eigenvalues below the essential spectrum , we derive a lower bound on the first spectral gap of Hμ. The assumption on the μ-volume of balls is in particular satisfied if μ is of the form , where M is a compact (n-1)-dimensional Lipschitz submanifold of , σM the surface measure on M, and .
Submitted: May 8, 2008.; Accepted: February 20, 2009. 相似文献
10.
Thomas Bartsch Zhi-Qiang Wang Juncheng Wei 《Journal of Fixed Point Theory and Applications》2007,2(2):353-367
We consider the existence of bound states for the coupled elliptic system
where n ≤ 3. Using the fixed point index in cones we prove the existence of a five-dimensional continuum of solutions (λ1, λ2, μ
1, μ
2, β, u
1, u
2) bifurcating from the set of semipositive solutions (where u
1 = 0 or u
2 = 0) and investigate the parameter range covered by .
Dedicated to Albrecht Dold and Edward Fadell 相似文献
11.
Fatma Ghribi 《Annales Henri Poincare》2008,9(3):595-624
This paper is devoted to the study of Lifshits tails for random Schr?dinger operator acting on of the form , where H
0 is a -periodic Schr?dinger operator, λ is a positive coupling constant, are i.i.d and bounded random variables and V is the single site potential with changing sign. We prove that, in the weak disorder regime, at an open band edge, a true
Lifshits tail for the random Schr?dinger operator occurs under a certain set of conditions on H
0 and on V.
Submitted: April 17, 2007. Accepted: December 13, 2007. 相似文献
12.
We study the spatial decay of eigenfunctions of non-local Schrödinger operators whose kinetic terms are generators of symmetric jump-paring Lévy processes with Kato-class potentials decaying at infinity. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps. We find that the decay rates of eigenfunctions depend on the process via specific preference rates in particular jump scenarios, and depend on the potential through the distance of the corresponding eigenvalue from the edge of the continuous spectrum. We prove that the conditions of the jump-paring class imply that for all eigenvalues the corresponding positive eigenfunctions decay at most as rapidly as the Lévy intensity. This condition is sharp in the sense that if the jump-paring property fails to hold, then eigenfunction decay becomes slower than the decay of the Lévy intensity. We furthermore prove that under reasonable conditions the Lévy intensity also governs the upper bounds of eigenfunctions, and ground states are comparable with it, i.e., two-sided bounds hold. As an interesting consequence, we identify a sharp regime change in the decay of eigenfunctions as the Lévy intensity is varied from sub-exponential to exponential order, and dependent on the location of the eigenvalue, in the sense that through the transition Lévy intensity-driven decay becomes slower than the rate of decay of the Lévy intensity. Our approach is based on path integration and probabilistic potential theory techniques, and all results are also illustrated by specific examples. 相似文献
13.
Let {K t } t>0 be the semigroup of linear operators generated by a Schrödinger operator ?L = Δ ? V (x) on ? d , d ≥ 3, where V (x) ≥ 0 satisfies Δ ?1 V ∈ L ∞. We say that an L 1-function f belongs to the Hardy space \({H^{1}_{L}}\) if the maximal function ? L f (x) = sup t>0 |K t f (x)| belongs to L 1 (? d ). We prove that the operator (?Δ)1/2 L ?1/2 is an isomorphism of the space \({H^{1}_{L}}\) with the classical Hardy space H 1(? d ) whose inverse is L 1/2(?Δ)?1/2. As a corollary we obtain that the space \({H^{1}_{L}}\) is characterized by the Riesz transforms \(R_{j}=\frac {\partial }{\partial x_{j}}L^{-1\slash 2}\) . 相似文献
14.
We prove existence of modified wave operators for one-dimensional Schrödinger equations with potential in
If in addition the potential is conditionally integrable, then the usual Möller wave operators exist. We also prove asymptotic completeness of these wave operators for some classes of random potentials, and for almost every boundary condition for any given potential. 相似文献
15.
M. Goldberg 《Geometric And Functional Analysis》2006,16(3):517-536
We prove a dispersive estimate for the time-independent Schr?dinger operator H = − Δ + V in three dimensions. The potential V(x) is assumed to lie in the intersection
p < 3/2 < q, and also to satisfy a generic zero-energy spectral condition. This class, which includes potentials that have pointwise
decay
is nearly critical with respect to the natural scaling of the Laplacian. No additional regularity, decay, or positivity of
V is assumed.
Received: December 2004 Accepted: February 2005 相似文献
16.
Tingjian Luo & Zhengping Wang 《偏微分方程(英文版)》2012,25(1):79-89
In this paper, we are concerned with the existence of nodal type bound state for the following stationary nonlinear Schrödinger equation $$-Δu(x)+V(x)u(x)=|u|^{p-1}u, x∈ R^N, N ≥ 3,$$ where 1 < p < (N+2)/(N-2) and the potential V(x) is a positive radial function and may decay to zero at infinity. Under appropriate assumptions on the decay rate of V(x), Souplet and Zhang [1] proved the above equation has a positive bound state. In this paper, we construct a nodal solution with precisely two nodal domains and prove that the above equation has a nodal type bound state under the same conditions on V(x) as in [1]. 相似文献
17.
Ovidiu Costin Roland Donninger Wilhelm Schlag Saleh Tanveer 《Annales Henri Poincare》2012,13(6):1371-1426
We consider semiclassical Schr?dinger operators on the real line of the form $$H(\hbar)=-\hbar^2 \frac{{\rm d}^2}{{\rm d}x^2}+V(\cdot;\hbar)$$ with ${\hbar >0 }$ small. The potential V is assumed to be smooth, positive and exponentially decaying towards infinity. We establish semiclassical global representations of Jost solutions ${f_\pm(\cdot,E;\hbar)}$ with error terms that are uniformly controlled for small E and ${\hbar}$ , and construct the scattering matrix as well as the semiclassical spectral measure associated with ${H(\hbar)}$ . This is crucial in order to obtain decay bounds for the corresponding wave and Schr?dinger flows. As an application we consider the wave equation on a Schwarzschild background for large angular momenta ? where the role of the small parameter ${\hbar}$ is played by ? ?1. It follows from the results in this paper and Donninger et al. (Commun Math Phys 2009, arXiv:0911.3179), that the decay bounds obtained in Donninger et al. (Adv Math 226(1):484–540, 2011) and Donninger and Wilhelm (Int Math Res Not IMRN 22:4276–4300, 2010) for individual angular momenta ? can be summed to yield the sharp t ?3 decay for data without symmetry assumptions. 相似文献
18.
We prove that singular Schrödinger equations with external magnetic field admit a representation with a positive Lagrangian density whenever their “nonmagnetic” counterpart is nonnegative. In this case the operator has a weighted spectral gap as long as the strength of the magnetic field is not identically zero. We provide estimates of the weight in the spectral gap, including the versions with L p -norm and with a magnetic gradient term, and applications to an increase of the best Hardy constant due to the presence of a magnetic field. The paper also shows existence of the ground state for the nonlinear magnetic Schrödinger equation with the periodic magnetic field. 相似文献
19.
Let L = L 0 + V be a Schrödinger type operator, where L 0 is a higher order elliptic operator with bounded complex coefficients in divergence form and V is a signed measurable function. Under the strongly subcritical assumption on V, we study the L q boundedness of Riesz transform ? m L ?1/2 for q ≤ 2 based on the off-diagonal estimates of semigroup e ?t L . Furthermore, the authors impose extra regularity assumptions on V to obtain the L q boundedness of Riesz transform ? m L ?1/2 for some q > 2. In particular, these results are applied to the more interesting Schrödinger operators L = P(D) + V, where P(D) is any homogeneous positive elliptic operator with constant coefficients. 相似文献
20.
We study the heat kernel p(x,y,t) associated to the real Schrödinger operator H=?Δ + V on \(L^{2}(\mathbb {R}^{n})\), n≥1. Our main result is a pointwise upper bound on p when the potential \(V \in A_{\infty }\). In the case that \(V\in RH_{\infty }\), we also prove a lower bound. Additionally, we compute p explicitly when V is a quadratic polynomial. 相似文献