On $m$-order logarithmic Schrödinger operator

Technical Summary: On m-Order Logarithmic Schrödinger Operator

Problem Statement
This paper investigates the $m$-order logarithm of the Schrödinger operator $L_V = -\Delta + V$ in $\mathbb{R}^d$, where $V$ is a nonnegative measurable potential. While the logarithmic Laplacian $\log(-\Delta)$ and its higher-order counterparts have been studied extensively in recent years (e.g., [5, 9]), the extension of these concepts to the Schrödinger setting with a potential $V$ presents specific analytical challenges. The primary objective is to define the operators $\log^m L_V$ for $m \in \mathbb{N}$ and utilize them to derive asymptotic expansions (Taylor series) for the fractional powers $L_V^s$ and $L_V^{-s}$ as the order $s \to 0^+$. The study focuses on establishing these expansions in $L^p(\mathbb{R}^d)$ spaces and pointwise for specific classes of functions, addressing the fact that the associated semigroup is not Markovian when $V > 0$.

Methodology
The authors employ a spectral theoretic approach combined with heat kernel estimates.

1. Spectral Definition: The operator $L_V$ is treated as a self-adjoint operator on $L^2(\mathbb{R}^d)$ associated with a closed sesquilinear form. Using the spectral measure $E_V$ supported on $[0, \infty)$, the fractional powers $L_V^s$ and logarithmic operators $\log^m L_V$ are defined via the spectral integral:
   $$\Phi(L_V)f = \int_0^\infty \Phi(\lambda) dE_V(\lambda)f.$$
2. Semigroup Extension: The definition of the logarithmic operators is extended from $L^2(\mathbb{R}^d)$ to a broader class of Lipschitz functions using the heat semigroup $\{T_t^V\}_{t>0}$ generated by $L_V$. The authors utilize the integral representation:
   $$L_V^s f = \frac{1}{\Gamma(-s)} \int_0^\infty \frac{T_t^V f - f}{t^{s+1}} dt,$$
   valid for $s \in (0, 1)$ and suitable $f$.
3. Potential Conditions: The analysis assumes $d \ge 3$ and that the potential $V$ satisfies a reverse Hölder inequality ($V \in RH_q$ with $q > d/2$). This condition allows for the use of the critical radius function $\rho(x)$ and improved heat kernel estimates (specifically, bounds involving $\rho(x)$ and the Euclidean heat kernel).
4. Asymptotic Analysis: The core technical work involves analyzing the behavior of the difference quotients $\frac{L_V^s f - f}{s}$ and their higher-order derivatives with respect to $s$ as $s \to 0^+$. The authors decompose integral representations into localized and non-localized parts to handle the singularity at $t=0$ and the behavior at infinity, utilizing the properties of the Gamma function and the critical radius function.

Key Contributions and Results

• Definition of Higher-Order Logarithmic Operators: The paper rigorously defines the $m$-order logarithmic Schrödinger operator $\log^m L_V$ via the spectral measure. It further extends the definition of the operator $\text{Log}^k L_V$ (acting as the pointwise limit of derivatives of fractional powers) to functions in local Lipschitz spaces intersected with weighted $L^1$ spaces.
• Asymptotic Expansions in $L^2$: Theorem 1.1 establishes Taylor expansions for $L_V^s f$ and $L_V^{-s} f$ in $L^2(\mathbb{R}^d)$ for functions in the domain of the logarithmic operators. Specifically, for $f \in D(\log^{m+1} L_V) \cap D(L_V^{s_0})$:
  $$L_V^s f - \sum_{k=0}^m \frac{s^k}{k!} (\log^k L_V)f = o(s^m) \quad \text{as } s \to 0^+.$$
  A similar expansion is provided for negative powers with alternating signs.
• Convergence in $L^p$ and $L^\infty$: Theorem 1.2 and Theorem 1.3 extend these results to $L^p(\mathbb{R}^d)$ ($1 < p < \infty$) and $L^\infty(K)$ for compact sets $K$. These results hold for functions in $C_c(\mathbb{R}^d) \cap \text{Lip}_{\text{loc,uni}}^\theta(\mathbb{R}^d)$ and $L^1 \cap L^\infty \cap \text{Lip}_{\text{loc,uni}}^\theta(\mathbb{R}^d)$. The expansions are given by:
  $$\lim_{s \to 0^+} \frac{L_V^s f - f - \sum_{k=1}^{n-1} \frac{s^k}{k!} (\text{Log}^k L_V)f}{s^n/n!} = (\text{Log}^n L_V)f.$$
• Pointwise Representation: The paper provides explicit pointwise integral representations for the operators $\text{Log}^k L_V$, generalizing the known representation for $\log L_V$ found in [1]. These representations involve integrals of the heat kernel $T_t^V(x, y)$ and the critical radius function $\rho(x)$, highlighting that the correction terms are not constant (unlike the pure Laplacian case) due to the non-Markovian nature of the semigroup.
• Application to Fractional Equations: Corollary 1.4 demonstrates that these expansions can be used to describe the solution $u_{f,s} = L_V^{-s} f$ of the equation $L_V^s u = f$ with vanishing boundary conditions at infinity, providing an asymptotic series in terms of the logarithmic operators as $s \to 0$.

Significance and Claims
The authors claim that this work generalizes previous results on the logarithmic Laplacian (specifically those by Chen [5] and Chen-Weth [9]) to the Schrödinger operator setting with non-trivial potentials. A significant aspect of the paper is the handling of the non-constant correction factors arising from the potential $V$, which necessitates the use of the critical radius function $\rho(x)$ and specific reverse Hölder conditions. The results provide a rigorous framework for understanding the behavior of fractional powers of Schrödinger operators near the identity (i.e., as the order approaches zero) through their logarithmic derivatives. The paper does not propose new experimental applications but rather establishes the theoretical foundation for asymptotic expansions in this specific operator-theoretic context.