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arXiv · 2408.02902

Fractional Laplace operator and related Schrödinger equations on locally finite graphs

Mengjie Zhang·Yong Lin·Yunyan Yang
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Abstract

In this paper, we first define a discrete version of the fractional Laplace operator $(-Δ)^{s}$ through the heat semigroup on a stochastically complete, connected, locally finite graph $G = (V, E, μ, w)$. Secondly, we define the fractional divergence and give another form of $(-Δ)^s$. The third point, and the foremost, is the introduction of the fractional Sobolev space $W^{s,2}(V)$, which is necessary when we study problems involving $(-Δ)^{s}$. Finally, using the mountain-pass theorem and the Nehari manifold, we obtain multiplicity solutions to a discrete fractional Schrödinger equation on $G$. We caution the readers that though these existence results are well known in the continuous case, the discrete case is quite different.

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Mengjie Zhang, Yong Lin, Yunyan Yang. 2025-06-09. Fractional Laplace operator and related Schrödinger equations on locally finite graphs. https://arxiv.org/abs/2408.02902

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