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Absolutely continuous spectrum of multidimensional Schrödinger operator
 Int. Math. Res. Not
"... Abstract. We prove that 3dimensional Schrödinger operator with slowly decaying potential has an a.c. spectrum that fills R +. Asymptotics of Green’s functions is obtained as well. Consider the Schrödinger operator H = − ∆ + V, x ∈ R d We are interested in finding the support of an a.c. spectrum of ..."
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Cited by 14 (8 self)
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Abstract. We prove that 3dimensional Schrödinger operator with slowly decaying potential has an a.c. spectrum that fills R +. Asymptotics of Green’s functions is obtained as well. Consider the Schrödinger operator H = − ∆ + V, x ∈ R d We are interested in finding the support of an a.c. spectrum of H for the slowly decaying potential V. The following conjecture is due to B. Simon [21] Conjecture. If V (x) is such that R d
On the absolutely continuous spectrum of Dirac operator
 Comm. Partial Diff. Eq
"... Abstract. We prove that the massless Dirac operator in R 3 with longrange potential has an a.c. spectrum which fills the whole real line. The Dirac operators with matrixvalued potentials are considered as well. 1. ..."
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Cited by 8 (5 self)
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Abstract. We prove that the massless Dirac operator in R 3 with longrange potential has an a.c. spectrum which fills the whole real line. The Dirac operators with matrixvalued potentials are considered as well. 1.
SCHRÖDINGER OPERATORS AND ASSOCIATED HYPERBOLIC PENCILS
, 908
"... Abstract. For a large class of Schrödinger operators, we introduce the hyperbolic quadratic pencils by making the coupling constant dependent on the energy in the very special way. For these pencils, many problems of scattering theory are significantly easier to study. Then, we give some application ..."
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Cited by 4 (2 self)
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Abstract. For a large class of Schrödinger operators, we introduce the hyperbolic quadratic pencils by making the coupling constant dependent on the energy in the very special way. For these pencils, many problems of scattering theory are significantly easier to study. Then, we give some applications to the original Schrödinger operators including onedimensional Schrödinger operators with L 2 – operatorvalued potentials, multidimensional Schrödinger operators with slowly decaying potentials. 1.
SPECTRAL PROPERTIES OF SCHRÖDINGER OPERATORS WITH DECAYING POTENTIALS
, 2005
"... Abstract. We review recent advances in the spectral theory of Schrödinger operators with decaying potentials. The area has seen spectacular progress in the past few years, stimulated by several conjectures stated by Barry Simon starting at the 1994 International Congress on Mathematical Physics in P ..."
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Cited by 4 (2 self)
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Abstract. We review recent advances in the spectral theory of Schrödinger operators with decaying potentials. The area has seen spectacular progress in the past few years, stimulated by several conjectures stated by Barry Simon starting at the 1994 International Congress on Mathematical Physics in Paris. The onedimensional picture is now fairly complete, and provides many striking spectral examples. The multidimensional picture is still far from clear and may require deep original ideas for further progress. It might hold the keys for better understanding of a wide range of spectral and dynamical phenomena for Schrödinger operators in higher dimensions. 1.
Wave propagation through sparse potential barriers
"... Abstract. We prove that 3dimensional Schrödinger operator with slowly decaying sparse potential has an a.c. spectrum that fills R +. A new kind of WKB asymptotics for Green’s function is obtained. The absence of positive eigenvalues is established as well. Consider the Schrödinger operator H = − ∆ ..."
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Cited by 3 (3 self)
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Abstract. We prove that 3dimensional Schrödinger operator with slowly decaying sparse potential has an a.c. spectrum that fills R +. A new kind of WKB asymptotics for Green’s function is obtained. The absence of positive eigenvalues is established as well. Consider the Schrödinger operator H = − ∆ + V, x ∈ R d We are interested in studying the scattering properties of H for the slowly decaying potential V. The following conjecture is due to Barry Simon [19] Conjecture. If V (x) is such that R d
Ito diffusions, modified capacity, and harmonic measure. Applications to Schrödinger operators
, 2013
"... We observe that some special Itô diffusions are related to scattering properties of a Schrödinger operator on R d, d ≥ 2. We introduce FeynmanKac type formulae for these stochastic processes which lead us to results on the preservation of the a.c. spectrum of the Schrödinger operator. To better un ..."
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Cited by 1 (1 self)
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We observe that some special Itô diffusions are related to scattering properties of a Schrödinger operator on R d, d ≥ 2. We introduce FeynmanKac type formulae for these stochastic processes which lead us to results on the preservation of the a.c. spectrum of the Schrödinger operator. To better understand the analytic properties of the processes, we construct and study a special version of the potential theory. The modified capacity and harmonic measure play an important role in these considerations. Various applications to Schrödinger operators are also given. For example, we relate the presence of the absolutely continuous spectrum to the geometric properties of the support of the potential.
On the preservation of absolutely . . .
, 2004
"... We present general principles for the preservation of a.c. spectrum under weak perturbations. The Schrödinger operators on the strip and on the Caley tree (Bethe lattice) are considered. ..."
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We present general principles for the preservation of a.c. spectrum under weak perturbations. The Schrödinger operators on the strip and on the Caley tree (Bethe lattice) are considered.
MULTIDIMENSIONAL L² CONJECTURE: A SURVEY
"... In this survey, we will give a short overview of the recent progress on the multidimensional L² conjecture. It can also serve as a quick introduction to the subject. ..."
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In this survey, we will give a short overview of the recent progress on the multidimensional L² conjecture. It can also serve as a quick introduction to the subject.
On the preservation of absolutely continuous spectrum . . .
, 2004
"... We present general principles for the preservation of a.c. spectrum under weak perturbations. The Schrödinger operators on the strip and on the Caley tree (Bethe lattice) are considered. ..."
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We present general principles for the preservation of a.c. spectrum under weak perturbations. The Schrödinger operators on the strip and on the Caley tree (Bethe lattice) are considered.