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

• Autorzy: Kamil Kaleta
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 47D08: Schr\"odinger and Feynman-Kac semigroups
• 60G52: Stable processes
• 47G30: Pseudodifferential operators
• 58C40: Spectral theory; eigenvalue problems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2012
• Tom: 209
• Numer: 3
• Strony: 267-287

Daty

wydano

2012

Twórcy

autor

Kamil Kaleta

• Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

Identyfikatory

DOI

10.4064/sm209-3-5