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

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Kaleta K.

Studia Mathematica

2012

tom 209

nr 3

267-287

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.

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1 Studia Mathematica

1 Kaleta K.

1 2012

Wyniki wyszukiwania

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