Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval

Kamil Kaleta. "Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval." Studia Mathematica 209.3 (2012): 267-287. <http://eudml.org/doc/286568>.

@article{KamilKaleta2012,
abstract = {We prove a uniform lower bound for the difference λ₂ - λ₁ between the first two eigenvalues of the fractional Schrödinger operator $(-Δ)^\{α/2\} + V$, α ∈ (1,2), with a symmetric single-well potential V in a bounded interval (a,b), which is related to the Feynman-Kac semigroup of the symmetric α-stable process killed upon leaving (a,b). “Uniform” means that the positive constant $C_\{α\}$ appearing in our estimate $λ₂ - λ₁ ≥ C_\{α\}(b-a)^\{-α\}$ is independent of the potential V. In the general case of α ∈ (0,2), we also find a uniform lower bound for the difference λ⁎ - λ₁, where λ⁎ denotes the smallest eigenvalue corresponding to an antisymmetric eigenfunction. One of our key arguments used in proving the spectral gap lower bound is a certain integral inequality which is known to be a consequence of the Garsia-Rodemich-Rumsey lemma. We also study some basic properties of the corresponding eigenfunctions.},
author = {Kamil Kaleta},
journal = {Studia Mathematica},
keywords = {spectral gap; fractional Schrödinger operator; symmetric single well; Feynman-Kac semigroup; symmetric stable process; interval; eigenfunctions},
language = {eng},
number = {3},
pages = {267-287},
title = {Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval},
url = {http://eudml.org/doc/286568},
volume = {209},
year = {2012},
}

TY - JOUR
AU - Kamil Kaleta
TI - Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval
JO - Studia Mathematica
PY - 2012
VL - 209
IS - 3
SP - 267
EP - 287
AB - We prove a uniform lower bound for the difference λ₂ - λ₁ between the first two eigenvalues of the fractional Schrödinger operator $(-Δ)^{α/2} + V$, α ∈ (1,2), with a symmetric single-well potential V in a bounded interval (a,b), which is related to the Feynman-Kac semigroup of the symmetric α-stable process killed upon leaving (a,b). “Uniform” means that the positive constant $C_{α}$ appearing in our estimate $λ₂ - λ₁ ≥ C_{α}(b-a)^{-α}$ is independent of the potential V. In the general case of α ∈ (0,2), we also find a uniform lower bound for the difference λ⁎ - λ₁, where λ⁎ denotes the smallest eigenvalue corresponding to an antisymmetric eigenfunction. One of our key arguments used in proving the spectral gap lower bound is a certain integral inequality which is known to be a consequence of the Garsia-Rodemich-Rumsey lemma. We also study some basic properties of the corresponding eigenfunctions.
LA - eng
KW - spectral gap; fractional Schrödinger operator; symmetric single well; Feynman-Kac semigroup; symmetric stable process; interval; eigenfunctions
UR - http://eudml.org/doc/286568
ER -