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35
The absolutely continuous spectrum of onedimensional Schrödinger operators with decaying potentials
, 2008
"... This paper deals with general structural properties of onedimensional 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 spectr ..."
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Cited by 56 (7 self)
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This paper deals with general structural properties of onedimensional 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 DenisovRakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper [19]. The treatment of the continuous case in the present paper depends on the same basic ideas.
Spectra of selfadjoint extensions and applications to solvable Schrödinger operators
, 2007
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Boundary triples and Weyl functions for singular perturbations of selfadjoint operators. Methods Funct. Anal
 S. Kondej) Institute of Physics, University of Zielona
"... Abstract. Given the symmetric operator AN obtained by restricting the selfadjoint operator A to N, a linear dense set, closed with respect to the graph norm, we determine a convenient boundary triple for the adjoint A ∗ N and the corresponding Weyl function. These objects provide us with the selfa ..."
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Cited by 18 (2 self)
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Abstract. Given the symmetric operator AN obtained by restricting the selfadjoint operator A to N, a linear dense set, closed with respect to the graph norm, we determine a convenient boundary triple for the adjoint A ∗ N and the corresponding Weyl function. These objects provide us with the selfadjoint extensions of AN and their resolvents. 1.
Tater: Point interactions in a strip
, 1996
"... We study the behavior of a quantum particle confined to a hard–wall strip of a constant width in which there is a finite number N of point perturbations. Constructing the resolvent of the corresponding Hamiltonian by means of Krein’s formula, we analyze its spectral and scattering properties. The bo ..."
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Cited by 13 (5 self)
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We study the behavior of a quantum particle confined to a hard–wall strip of a constant width in which there is a finite number N of point perturbations. Constructing the resolvent of the corresponding Hamiltonian by means of Krein’s formula, we analyze its spectral and scattering properties. The bound state–problem is analogous to that of point interactions in the plane: since a two–dimensional point interaction is never repulsive, there are m discrete eigenvalues, 1 ≤ m ≤ N, the lowest of which is nondegenerate. On the other hand, due to the presence of the boundary the point interactions give rise to infinite series of resonances; if the coupling is weak they approach the thresholds of higher transverse modes. We derive also spectral and scattering properties for point perturbations in several related models: a cylindrical surface, both of a finite and infinite heigth, threaded by a magnetic flux, and a straight strip which supports a potential independent of the transverse coordinate. As for strips with an infinite number of point perturbations, we restrict ourselves to the situation when the latter are arranged periodically; we show that in distinction to the case of a point–perturbation array in the plane, the spectrum may exhibit any finite number of gaps. Finally, we study numerically conductance fluctuations in case of random point perturbations.
Spectral properties of Schrödinger operators with a strongly attractive δ interaction supported by a surface
 Proceedings of the NSF Summer Research Conference (Mt. Holyoke 2002); AMS “Contemporary Mathematics” Series
, 2003
"... Abstract. We investigate the operator − ∆ − αδ(x − Γ) in L 2 (R 3), where Γ is a smooth surface which is either compact or periodic and satisfies suitable regularity requirements. We find an asymptotic expansion for the lower part of the spectrum as α → ∞ which involves a “twodimensional ” compar ..."
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Cited by 12 (5 self)
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Abstract. We investigate the operator − ∆ − αδ(x − Γ) in L 2 (R 3), where Γ is a smooth surface which is either compact or periodic and satisfies suitable regularity requirements. We find an asymptotic expansion for the lower part of the spectrum as α → ∞ which involves a “twodimensional ” comparison operator determined by the geometry of the surface Γ. In the compact case the asymptotics concerns negative eigenvalues, in the periodic case Floquet eigenvalues. We also give a bandwidth estimate in the case when a periodic Γ decomposes into compact connected components. Finally, we comment on analogous systems of lower dimension and other aspects of the problem. 1.
DIRECT AND INVERSE SPECTRAL THEORY OF ONEDIMENSIONAL SCHRÖDINGER OPERATORS WITH MEASURES
, 2003
"... We present a direct and rather elementary method for defining and analyzing onedimensional Schrödinger operators H = −d 2 /dx 2 + µ with measures as potentials. The basic idea is to let the (suitably interpreted) equation −f ′ ′ +µf = zf take center stage. We show that the basic results from direct ..."
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Cited by 8 (0 self)
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We present a direct and rather elementary method for defining and analyzing onedimensional Schrödinger operators H = −d 2 /dx 2 + µ with measures as potentials. The basic idea is to let the (suitably interpreted) equation −f ′ ′ +µf = zf take center stage. We show that the basic results from direct and inverse spectral theory then carry over to Schrödinger operators with measures.
Eigenvalue asymptotics for the Schrödinger operator with a δinteraction on a punctured surface
, 2004
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The AllegrettoPiepenbrink Theorem for Strongly Local Dirichlet Forms
 DOCUMENTA MATH.
, 2009
"... The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. ..."
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Cited by 6 (5 self)
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The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
Generalized Qfunctions and DirichlettoNeumann maps for elliptic differential operators
"... Abstract. The classical concept of Qfunctions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the DirichlettoNeumann map in the theory of elliptic differential equations can be interpreted as a generalized Qfunction. For couplings ..."
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Abstract. The classical concept of Qfunctions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the DirichlettoNeumann map in the theory of elliptic differential equations can be interpreted as a generalized Qfunction. For couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit Krein type formulas for the difference of the resolvents and trace formulas in an H 2framework are obtained. 1.