Results 1  10
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28
Eigenfunctions, transfer matrices, and absolutely continuous spectrum of onedimensional Schrödinger operators
, 1999
"... ..."
Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum
 PHYS. ACTA
, 1997
"... We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential q of a onedimensional Schrödinger operator H = − d2 dx2 + q determine the potential completely. Included are theorems for finite intervals and for the whole l ..."
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Cited by 33 (6 self)
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We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential q of a onedimensional Schrödinger operator H = − d2 dx2 + q determine the potential completely. Included are theorems for finite intervals and for the whole line. In particular, we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of H on a finite interval and knowledge of q over a corresponding fraction of the interval. The methods employed rest on Weyl mfunction techniques and densities of zeros of a class of entire functions.
B: Pseudospectra of differential operators
 J. Oper. Theory
, 2000
"... We describe methods which have been used to analyze the spectrum of nonselfadjoint differential operators, emphasizing the differences from the selfadjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely ..."
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Cited by 27 (7 self)
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We describe methods which have been used to analyze the spectrum of nonselfadjoint differential operators, emphasizing the differences from the selfadjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the operator. AMS subject classifications:34L05, 35P05, 47A10, 47A12
Spectral pollution
 IMA J. Numer. Anal
, 2004
"... It is well known that computing the eigenvalues of a selfadjoint bounded or differential ..."
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Cited by 22 (1 self)
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It is well known that computing the eigenvalues of a selfadjoint bounded or differential
On isospectral sets of Jacobi operators
 Com. Math. Phys
, 1996
"... Abstract. We consider the inverse spectral problem for a class of reflectionless bounded Jacobi operators with empty singularly continuous spectra. Our spectral hypotheses admit countably many accumulation points in the set of eigenvalues as well as in the set of boundary points of intervals of abso ..."
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Cited by 21 (16 self)
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Abstract. We consider the inverse spectral problem for a class of reflectionless bounded Jacobi operators with empty singularly continuous spectra. Our spectral hypotheses admit countably many accumulation points in the set of eigenvalues as well as in the set of boundary points of intervals of absolutely continuous spectrum. The corresponding isospectral set of Jacobi operators is explicitly determined in terms of Dirichlettype data. 1.
OneDimensional Scattering Theory For Quantum Systems With Nontrivial Spatial Asymptotics
 ADV. DIFF. EQS
"... We provide a general framework of stationary scattering theory for onedimensional quantum systems with nontrivial spatial asymptotics. As a byproduct we characterize reflectionless potentials in terms of spectral multiplicities and properties of the diagonal Green's function of the underlying Schr ..."
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Cited by 15 (4 self)
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We provide a general framework of stationary scattering theory for onedimensional quantum systems with nontrivial spatial asymptotics. As a byproduct we characterize reflectionless potentials in terms of spectral multiplicities and properties of the diagonal Green's function of the underlying Schrodinger operator. Moreover, we prove that single (CrumDarboux) and double commutation methods to insert eigenvalues into spectral gaps of a given background Schrodinger operator produce reflectionless potentials (i.e., solitons) if and only if the background potential is reflectionless. Possible applications of our formalism include impurity (defect) scattering in (half)crystals and charge transport in mesoscopic quantuminterference devices.
Absolute Summability of the Trace Relation for Certain Schrödinger Operators
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1995
"... A recently established general trace formula for onedimensional Schrödinger operators is systematically studied in the context of shortrange potentials, potentials which approach different spatial asymptotes sufficiently fast, and appropriate impurity (defect) interactions in onedimensional solid ..."
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Cited by 15 (12 self)
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A recently established general trace formula for onedimensional Schrödinger operators is systematically studied in the context of shortrange potentials, potentials which approach different spatial asymptotes sufficiently fast, and appropriate impurity (defect) interactions in onedimensional solids. We prove the absolute summability of the trace formula and establish its connections with scattering quantities, such as reflection coefficients, in each case.
Transient and Recurrent Spectrum
, 1981
"... We deal primarily with spectral analysis of an abstract selfadjoint operator. H, on a Hilbert space, X”. We propose a further refinement of the absolutely continuous subspace,;F”a,, into the transient absolutely continuous subspace, &‘a,. which is the closure of those cp with (cp, ej/Ho) = O(t“) ..."
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Cited by 11 (2 self)
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We deal primarily with spectral analysis of an abstract selfadjoint operator. H, on a Hilbert space, X”. We propose a further refinement of the absolutely continuous subspace,;F”a,, into the transient absolutely continuous subspace, &‘a,. which is the closure of those cp with (cp, ej/Ho) = O(t“) for all N and the recurrent absolutely continuous subspace, 2 & = ‘qc nX&. We discuss general features of this breakup. In a subsequent paper, we construct analytic almost periodic functions, V, on (03, 03) so that H =d*/dx * + V(x) on L2(co, co) has only recurrent absolutely continuous spectrum in the sense that qa, =.;Y.
Inverse Scattering theory for onedimensional Schrödinger operators with steplike finitegap potentials
, 2008
"... We develop direct and inverse scattering theory for onedimensional Schrödinger operators with steplike potentials which are asymptotically close to different finitegap potentials on different halfaxes. We give a complete characterization of the scattering data, which allow unique solvability of t ..."
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Cited by 11 (11 self)
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We develop direct and inverse scattering theory for onedimensional Schrödinger operators with steplike potentials which are asymptotically close to different finitegap potentials on different halfaxes. We give a complete characterization of the scattering data, which allow unique solvability of the inverse scattering problem in the class of perturbations with finite second moment.
Sum rules and spectral measures of Schrödinger operatros with L 2 potentials
"... Abstract. Necessary and sufficient conditions are presented for a positive measure to be the spectral measure of a halfline Schrödinger operator with square integrable potential. 1. ..."
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Cited by 9 (2 self)
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Abstract. Necessary and sufficient conditions are presented for a positive measure to be the spectral measure of a halfline Schrödinger operator with square integrable potential. 1.