Scattering length and capacity

Kac, Marc; Luttinger, J. M.

doi:10.5802/aif.586

Cited by 16 publications

(8 citation statements)

References 1 publication

(1 reference statement)

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“…In , Kac and Luttinger studied a connection between the scattering length

Γ (V)

of a positive integrable potential V and Brownian motion

B_{t}

R^{3}

. They gave a probabilistic expression of

Γ (V)

\begin{matrix} true \\ right normalΓ (V) = \lim_{t \to \infty} \frac{1}{t} \int_{{double-struckR}^{3}} (() 1 - {boldE}_{x} [e^{- \int_{0}^{t} V (B_{s}) normald s}]) normald x, \end{matrix}

where

E_{x}

denotes the expectation of

B_{t}

started at

x \in {double-struckR}^{3}

.…”

Section: Introductionmentioning

confidence: 99%

On a scattering length for additive functionals and spectrum of fractional Laplacian with a non‐local perturbation

Kim

Matsuura

2019

Mathematische Nachrichten

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In this paper we study the scattering length for positive additive functionals of symmetric stable processes on ℝ . The additive functionals considered here are not necessarily continuous. We prove that the semi-classical limit of the scattering length equals the capacity of the support of a certain measure potential, thus extend previous results for the case of positive continuous additive functionals. We also give an equivalent criterion for the fractional Laplacian with a measure valued non-local operator as a perturbation to have purely discrete spectrum in terms of the scattering length, by considering the connection between scattering length and the bottom of the spectrum of Schrödinger operator in our settings. K E Y W O R D Sadditive functionals, Dirichlet forms, scattering lengths, Schrödinger operators M S C ( 2 0 1 0 ) Primary: 47D08, 60J45; Secondary: 31C15, 60G52, 60J55 www.mn-journal.org

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“…In , Kac and Luttinger studied a connection between the scattering length

Γ (V)

of a positive integrable potential V and Brownian motion

B_{t}

R^{3}

. They gave a probabilistic expression of

Γ (V)

\begin{matrix} true \\ right normalΓ (V) = \lim_{t \to \infty} \frac{1}{t} \int_{{double-struckR}^{3}} (() 1 - {boldE}_{x} [e^{- \int_{0}^{t} V (B_{s}) normald s}]) normald x, \end{matrix}

where

E_{x}

denotes the expectation of

B_{t}

started at

x \in {double-struckR}^{3}

.…”

Section: Introductionmentioning

confidence: 99%

On a scattering length for additive functionals and spectrum of fractional Laplacian with a non‐local perturbation

Kim

Matsuura

2019

Mathematische Nachrichten

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“…We shall start from the probabilistic expression of scattering length in [12], which makes sense for general Markov processes. Let A be a positive continuous additive functional of X as in §2 with Revuz measure μ. Define…”

Section: The Existence Of Scattering Lengthmentioning

confidence: 99%

“…In [11], [12], M. Kac and J. Luttinger gave a probabilistic expression in terms of Brownian motion B = (B t , P x ) on R 3 , Γ(V ) = lim…”

Section: Introductionmentioning

confidence: 99%

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A formula on scattering length of dual Markov processes

2011

Proc. Amer. Math. Soc.

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“…In [4] and [5], M. Kac and J. Luttinger studied the scattering length Γ(V ) of a positive L 1 -function V on the 3-dimensional Euclidean space R 3 . They gave a probabilistic expression of Γ(V ), Γ(V ) = lim t→∞ 1 t R 3 1 − E 3 , and proved in [7] that for a compact set K ⊂ R 3 with Kac regularity, Γ(α1 K ) converges to the capacity of K as α → ∞.…”

Section: Introductionmentioning

confidence: 99%

A formula on scattering length of positive smooth measures

Takeda

2009

Proc. Amer. Math. Soc.

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Abstract. M. Kac studied the scattering length probabilistically and conjectured that its semi-classical limit equals the capacity of the support of the potential. This conjecture has been proved independently by Taylor, Takahashi, and Tamura. In this paper we give another simple proof by the random time-change argument for Dirichlet forms and extend the previous results to positive measure potentials.

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Scattering length and capacity

Cited by 16 publications

References 1 publication

On a scattering length for additive functionals and spectrum of fractional Laplacian with a non‐local perturbation

On a scattering length for additive functionals and spectrum of fractional Laplacian with a non‐local perturbation

A formula on scattering length of dual Markov processes

A formula on scattering length of positive smooth measures

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