The equivalence of solutions to fractional and logarithmic Helmholtz equations
Abstract
We show the following equivalence of solutions to general fractional Helmholtz equations: u is a solution to Asu=λsu for \textit{some} fixed 0<s<2 and some λ>0 if and only if u is a solution to the same equation for \textit{all} s≠0. Furthermore, we show as well that u equivalently solves the logarithmic Helmholtz equation log(A)u=(logλ)u. Here, A is a nonnegative, linear operator on a Banach space X. In particular, our results hold whenever A=−L, where L is the infinitesimal generator of a C0-semigroup in a Banach space X satisfying mild assumptions that are typical in applications. As a particular case, we recover known results for the fractional Laplacian (−Δ)s in ℝn. More importantly, we provide a list of examples of other fractional power operators in settings where the Fourier transform is not available and for which our theorems apply. These include fractional powers and logarithms of second order elliptic operators in bounded domains, Laplace--Beltrami operators on Riemannian manifolds, space-time master equations, the discrete Laplacian, and fractional derivatives.
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BibTeX
@misc{biswas2026equivalence,
title = {{The equivalence of solutions to fractional and logarithmic Helmholtz equations}},
author = {A. Biswas and J. Groszkiewicz and P. R. Stinga},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07553},
eprint = {2609.07553},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07553},
}
RIS
TY - UNPB
TI - The equivalence of solutions to fractional and logarithmic Helmholtz equations
AU - A. Biswas
AU - J. Groszkiewicz
AU - P. R. Stinga
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07553
UR - https://arxiv.org/abs/2609.07553
AB - We show the following equivalence of solutions to general fractional Helmholtz equations: u is a solution to A^su=λ^su for \textit{some} fixed 0<s<2 and some λ>0 if and only if u is a solution to the same equation for \textit{all} s≠0. Furthermore, we show as well that u equivalently solves the logarithmic Helmholtz equation log(A)u=(logλ)u. Here, A is a nonnegative, linear operator on a Banach space X. In particular, our results hold whenever A=−L, where L is the infinitesimal generator of a C_0-semigroup in a Banach space X satisfying mild assumptions that are typical in applications. As a particular case, we recover known results for the fractional Laplacian (−Δ)^s in ℝ^n. More importantly, we provide a list of examples of other fractional power operators in settings where the Fourier transform is not available and for which our theorems apply. These include fractional powers and logarithms of second order elliptic operators in bounded domains, Laplace--Beltrami operators on Riemannian manifolds, space-time master equations, the discrete Laplacian, and fractional derivatives.
ER -