Extension theorems for logarithmic Schrödinger and discrete Laplacian operators

Abstract: In this paper we consider logarithmic operators in two different contexts: the adapted to (continuous) Schrödinger operators and the classical discrete setting. The Schrödinger operator $\mathcal L_V$ on $\mathbb R<sup>d$ is defined as $\mathcal L_V=-Δ+V$, where the potential $V$ is nonnegative and satisfies a reverse Hölder inequality and, as usual, $Δ$ denotes the Euclidean Laplacian, while the discrete Laplacian $Δ_d$ on $\mathbb Z$ is given by $(Δ_df)(n)=f(n+1)-2f(n)+f(n-1)$, $n\in \mathbb Z$. Both logarithmic operators $\log \mathcal L_V$ and $\log (-Δ_d)$ are nonlocal operators and we will define them through suitable extension problems. The extension problems for logarithmic operators are inspired by the one introduced by Caffarelli and Silvestre for the fractional Laplacian but, in this case, the logarithmic operators are obtained as the boundary values of the extension in a more involved way.

• The paper introduces novel extension formulations for logarithmic operators via degenerate elliptic PDE characterizations.
• It derives explicit pointwise integral representations using heat semigroup techniques to capture non-Markovian corrections.
• The work advances both continuous and discrete frameworks, providing a rigorous basis for further analysis of nonlocal logarithmic operators.

Analysis of Extension Theorems for Logarithmic Schrödinger and Discrete Laplacian Operators

Introduction and Context

The paper "Extension theorems for logarithmic Schrödinger and discrete Laplacian operators" (2604.03638) investigates extension problems related to logarithmic operators associated with both continuous Schrödinger operators on $\mathbb{R}^d$ and the discrete Laplacian on $\mathbb{Z}$. The work builds directly on the lineage of the Caffarelli–Silvestre extension theorem for fractional Laplacians, as well as recent advances in extension theorems for other nonlocal operators. This manuscript provides new extension results for logarithmic versions of such operators, thereby clarifying their functional-analytic and PDE-theoretic structure.

Logarithmic Operators and Pointwise Representations

The authors consider logarithmic operators $\log \mathcal{L}_V$, where $\mathcal{L}_V = -\Delta + V$ is a Schrödinger operator with potential $V \in RH_q$ for $q > d/2$, and also $\log(-\Delta_d)$ for the discrete Laplacian $\Delta_d$. These logarithmic operators are defined through the spectral calculus and have nonlocal nature; they do not admit a purely local differential characterization.

A crucial feature of the paper is the derivation and use of explicit pointwise integral representations for $\log \mathcal{L}_V$ and $\log(-\Delta_d)$, leveraging the underlying heat semigroup. For the Schrödinger context, the formula

$\mathbb{Z}$0

is central. The kernel $\mathbb{Z}$1 contains highly nontrivial corrections due to the non-Markovian nature of the Schrödinger semigroup (unlike the Laplacian case, $\mathbb{Z}$2).

For the discrete setting, the analogous representation involves the discrete heat kernel $\mathbb{Z}$3 and yields

$\mathbb{Z}$4

where $\mathbb{Z}$5 and $\mathbb{Z}$6 integrate $\mathbb{Z}$7 over finite and infinite time intervals, respectively.

Extension Problems: Main Results

Continuous (Schrödinger) Setting

The paper establishes that $\mathbb{Z}$8 admits a characterization via a degenerate elliptic extension problem: $\mathbb{Z}$9 where the trace and asymptotics of $\log \mathcal{L}_V$0 as $\log \mathcal{L}_V$1 encode the nonlocal operator.

Specifically, if $\log \mathcal{L}_V$2 is defined by

$\log \mathcal{L}_V$3

then for sufficiently regular $\log \mathcal{L}_V$4,

$\log \mathcal{L}_V$5

where $\log \mathcal{L}_V$6 contains additional kernel-dependent terms. Key boundary behaviors, such as

$\log \mathcal{L}_V$7

are established in $\log \mathcal{L}_V$8 or pointwise, depending on regularity. This thoroughly grounds the extension problem for $\log \mathcal{L}_V$9 in nonlocal analysis.

Discrete Laplacian Setting

For the discrete Laplacian on $\mathcal{L}_V = -\Delta + V$0, the extension problem is similarly formulated: $\mathcal{L}_V = -\Delta + V$1 with

$\mathcal{L}_V = -\Delta + V$2

where $\mathcal{L}_V = -\Delta + V$3. The logarithmic discrete operator then emerges as

$\mathcal{L}_V = -\Delta + V$4

with $\mathcal{L}_V = -\Delta + V$5 an explicit constant correction involving the Euler–Mascheroni constant and additional integrals.

Technical Highlights and Implications

The approach generalizes and amplifies the Caffarelli–Silvestre program by introducing an extension mechanism for the logarithmic case, in both continuous and discrete frameworks. Essential technical advances include:

• Uniform control on the heat kernels under reverse Hölder potential assumptions ($\mathcal{L}_V = -\Delta + V$6), extending significant regularity theorems to the logarithmic field.
• Precise analysis of non-Markovian corrections (i.e., nontrivial $\mathcal{L}_V = -\Delta + V$7 and $\mathcal{L}_V = -\Delta + V$8 terms) that arise in the Schrödinger context.
• Careful treatment of various function spaces (Lipschitz, Dini, algebraic growth classes) required for domain and continuity properties.
• Extension to the discrete field, with control on convergence and kernel bounds, thus providing a basis for potential further extensions to more general graphs.

These results have implications for both theoretical analysis and applications to PDEs with nonlocal logarithmic diffusion. The extension characterization can serve as a tool for obtaining regularity, uniqueness, and structural properties of solutions.

Future Directions

The methodology supports several future avenues:

• Extension to more general geometric and combinatorial contexts: Including weighted graphs, metric measure spaces, and Riemannian manifolds, as initial investigations (e.g., [ChX], [Ch]) suggest.
• Analysis of nonlinear equations with logarithmic generators: The extension framework is suitable for adaptation to nonlinear and variational logarithmic problems, particularly those arising in quantum mechanics and probability.
• Spectral and functional inequalities: The explicit constructions open possibilities for sharp spectral bounds, trace inequalities, and related nonlocal variational identities.

Conclusion

This work offers a systematic semigroup and extension theory for logarithmic Schrödinger and discrete Laplacian operators. The main achievement is the precise extension problem formulation paralleling the fractional Laplacian paradigm, but exhibiting richer structural corrections due to the lack of Markovianity. The rigorous pointwise and distributional representations for both continuous and discrete settings provide foundational tools for further analysis and applications of logarithmic nonlocal operators in analysis and PDE theory (2604.03638).