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1.
Christian Remling 《Communications in Mathematical Physics》2007,271(1):275-287
I present an example of a discrete Schrödinger operator that shows that it is possible to have embedded singular spectrum and, at the same time, discrete eigenvalues that approach the edges of the essential spectrum (much) faster than exponentially. This settles a conjecture of Simon (in the negative). The potential is of von Neumann-Wigner type, with careful navigation around a previously identified borderline situation. 相似文献
2.
Christian Remling 《Proceedings of the American Mathematical Society》1996,124(7):2097-2100
In 1949, Hartman and Wintner showed that if the eigenvalue equations of a one-dimensional Schrödinger operator possess square integrable solutions, then the essential spectrum is nowhere dense. Furthermore, they conjectured that this statement could be improved and that under this condition the essential spectrum might always be void. This is shown to be false. It is proved that, on the contrary, every closed, nowhere dense set does occur as the essential spectrum of Schrödinger operators which satisfy the condition of existence of -solutions. The proof of this theorem is based on inverse spectral theory.
3.
We present a direct and rather elementary method for defining and analyzing one-dimensional Schrödinger operators H = −d2/dx2 + μ with measures as potentials. The basic idea is to let the (suitably interpreted) equation −f′′ + μ f = zf take center stage.We show that the basic results from direct and inverse spectral theory then carry over to Schrödinger operators with measures. 相似文献
4.
Christian Remling 《Proceedings of the American Mathematical Society》2007,135(10):3329-3340
We point out finite propagation speed phenomena for discrete and continuous Schrödinger operators and discuss various types of kernel estimates from this point of view.
5.
We study the pointwise behavior of the Fourier transform of the spectral measure for discrete one-dimensional Schr?dinger operators with sparse
potentials. We find a resonance structure which admits a physical interpretation in terms of a simple quasiclassical model.
We also present an improved version of known results on the spectrum of such operators.
Received: 17 May 2001 / Accepted: 28 June 2001 相似文献
6.
We investigate the location and nature of the spectrum of thefourth-order self-adjoint equation (p0 y')'+(p1 y')'+qy=zwy subject to certain asymptotic assumptions on the coefficients.The main tools are the theory of asymptotic integration andthe TitchmarshWeyl M-matrix. Asymptotic integration yieldsasymptotic formulae for the solutions of the differential equationwhich are then used to derive properties of the M-matrix. Thecharacterisation of spectral properties in terms of the boundarybehaviour of M leads to the desired results. 相似文献
7.
Christian Remling 《Communications in Mathematical Physics》1997,186(2):481-493
We construct (deterministic) potentials such that the Schr?dinger equation on has dense pure point spectrum in for almost all boundary conditions at . As a by-product, we also obtain power-decaying potentials for which the spectrum is purely singular continuous on for all boundary conditions.
Received: 8 November 1996 / Accepted: 8 January 1997 相似文献
8.
Christian Remling 《Mathematical Physics, Analysis and Geometry》2007,10(4):359-373
This paper deals with general structural properties of one-dimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and Denisov-Rakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper (Remling, The absolutely continuous spectrum of Jacobi matrices, http://arxiv.org/abs/0706.1101, 2007). The treatment of the continuous case in the present paper depends on the same basic ideas. 相似文献
9.
We study several related aspects of reflectionless Jacobi matrices. First, we discuss the singular part of the corresponding
spectral measures. We then show how to identify sets on which measures are reflectionless by looking at the logarithmic potentials
of these measures.
A. P.’s work is supported in part by NSF grant DMS 0800300. 相似文献
10.
Consider the Schr?dinger operator H=−d
2/dx
2+V(x) with power-decaying potential V(x)=O(x
−α). We prove that a previously obtained dimensional bound on exceptional sets of the WKB method is sharp in its whole range
of validity. The construction relies on pointwise bounds on finite gap potentials. These bounds are obtained by an analysis
of the Jacobi inversion problem on hyperelliptic Riemann surfaces.
Received: 14 March 2001 / Accepted: 27 June 2001 相似文献