Kernel Estimates for Schrödinger Type Operators with Unbounded Diffusion and Potential Terms

Anna Canale; Abdelaziz Rhandi; Cristian Tacelli

doi:10.4171/zaa/1593

JournalszaaVol. 36, No. 4pp. 377–392

Kernel Estimates for Schrödinger Type Operators with Unbounded Diffusion and Potential Terms

Anna Canale
Università degli Studi di Salerno, Fisciano, Italy
Abdelaziz Rhandi
Università degli Studi di Salerno, Fisciano, Italy
Cristian Tacelli
Università degli Studi di Salerno, Fisciano, Italy

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Abstract

We prove that the heat kernel associated to the Schrödinger type operator $A := (1 + ∣ x ∣^{α}) Δ - ∣ x ∣^{β}$ satisfies the estimate

k (t, x, y) \leq c_{1} e^{λ_{0} t} e^{c_{2} t^{- b}} \frac{( ∣ x ∣∣ y ∣ ) ^{- \frac{N - 1}{2} - \frac{β - α}{4}}}{1 + ∣ y ∣ ^{α}} e^{- \frac{2}{β - α + 2} ∣ x ∣^{\frac{β - α + 2}{2}}} e^{- \frac{2}{β - α + 2} ∣ y ∣^{\frac{β - α + 2}{2}}}

for $t > 0, ∣ x ∣, ∣ y ∣ \geq 1$ , where $c_{1}, c_{2}$ are positive constants and $b = \frac{β - α + 2}{β + α - 2}$ provided that $N > 2, α \geq 2$ and $β > α - 2$ . We also obtain an estimate of the eigenfunctions of $A$ .

Cite this article

Anna Canale, Abdelaziz Rhandi, Cristian Tacelli, Kernel Estimates for Schrödinger Type Operators with Unbounded Diffusion and Potential Terms. Z. Anal. Anwend. 36 (2017), no. 4, pp. 377–392

DOI 10.4171/ZAA/1593