Abstract

We obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schrödinger operator with negative Hardy potential Δα/2 − λ|x|−αΔα/2−λ|x|−α\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ {\\Delta}^{\\alpha/2} -\\lambda |x|^{-\\alpha} $$\\end{document} on , where α ∈ (0, d ∧ 2) and λ > 0. The proof is purely analytical and elementary. In particular, for upper bounds of heat kernel we use the Chapman-Kolmogorov equation and adopt self-improving argument.

Highlights

  • Let d ∈ N+ := {1, 2, · · · } and α ∈ (0, d ∧ 2)

  • Theorem 1.1 For any δ ∈ (0, α), the Schrodinger operator L given by Eq 1.1 has the heat kernel p(t, x, y), which is jointly continuous on (0, ∞) × Rd × Rd, and satisfies two-sided estimates as follows p(t, x, y) ≈

  • According to [33, Theorem 3.4], when the potential belongs to the so-called Kato class, heat kernel estimates for Schrodinger perturbations of fractional

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Summary

Introduction

Let d ∈ N+ := {1, 2, · · · } and α ∈ (0, d ∧ 2). We consider the following Schrodinger operator. Theorem 1.1 For any δ ∈ (0, α), the Schrodinger operator L given by Eq 1.1 has the heat kernel p(t, x, y), which is jointly continuous on (0, ∞) × Rd × Rd , and satisfies two-sided estimates as follows p(t, x, y) ≈. According to [33, Theorem 3.4], when the potential belongs to the so-called Kato class, heat kernel estimates for Schrodinger perturbations of fractional. The study of heat kernel estimates for Schrodinger-type perturbations by the Hardy potential of fractional Laplacian is much more delicate. The heat kernel p(t, x, y) corresponding to the Schrodinger operator α/2 + κ|x|−α satisfies p(t, x, y) ≈ In this setting, Theorem 1.1 may be treated as both a fractional counterpart of the result obtained in [30] and the extension of Eq 1.4 to negative values of κ. As usual we write a ∧ b := min(a, b) and a ∨ b := max(a, b)

Fractional Laplacian and Rotationally Symmetric α-Stable Levy Process
Fractional Laplacian Schrodinger Operator and Feynman-Kac Formula
Integral Analysis for Fractional Laplacian Schrodinger Operator
Dirichlet Forms

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