Logarithmic Schrödinger operators

1. Problem Statement

The paper addresses the definition, characterization, and application of the logarithmic Schrödinger operator, denoted as $\log L_V$, where $L_V = -\Delta + V$ is the Schrödinger operator on $\mathbb{R}^d$ with a non-negative potential $V \not\equiv 0$.

While the fractional Schrödinger operator $L_V^s$ (for $s \in (0,1)$) and the logarithmic Laplacian $\log(-\Delta)$ (where $V=0$) are well-studied, the logarithmic operator for the general Schrödinger setting ($V \neq 0$) presents unique challenges. Specifically:

• The semigroup $\{T^V_t\}_{t>0}$ generated by $L_V$ is not Markovian (i.e., $T^V_t 1 \neq 1$), unlike the classical heat semigroup.
• This lack of the Markov property prevents the direct application of standard pointwise representation formulas used for the logarithmic Laplacian.
• The authors aim to define $\log L_V$ rigorously, provide a pointwise integral representation, and solve the associated initial value problem (Cauchy problem) using fractional integrals and adapted Lipschitz spaces.

2. Methodology

The authors employ a combination of spectral theory, semigroup theory, and harmonic analysis adapted to the Schrödinger setting.

• Spectral Definition: Using the spectral theorem, $L_V$ is defined via its spectral measure $E_V$. The operator $\log L_V$ is initially defined spectrally as:
  $$(\log L_V)f = \int_0^\infty \log(\lambda) \, dE_V(\lambda)f$$
  The domain $D(\log L_V)$ consists of functions where this integral converges in $L^2(\mathbb{R}^d)$.

• Semigroup Approach (Frullani Integral): The authors utilize the Frullani integral representation of the logarithm, $\log(\lambda) = \int_0^\infty \frac{e^{-t} - e^{-\lambda t}}{t} dt$, to define an alternative operator $\text{Log } L_V$ via the semigroup $\{T^V_t\}_{t>0}$:
  $$(\text{Log } L_V)f = \lim_{m \to \infty} \int_{1/m}^m \frac{e^{-t}I - T^V_t}{t} f \, dt$$
  They prove the equivalence between the spectral definition and this semigroup-based limit.

• Potential Assumptions: To derive pointwise representations, the potential $V$ is assumed to satisfy a Reverse Hölder inequality ($V \in RH_q$) with $q > d/2$. This condition allows for the introduction of the auxiliary function $\rho(x)$, which characterizes the local behavior of the potential and is crucial for obtaining precise heat kernel estimates.

• Heat Kernel Estimates: The proof relies heavily on sharp estimates for the heat kernel $T^V_t(x,y)$, including:

  • Upper bounds involving the Gaussian kernel and the function $\rho(x)$.
  • Regularity estimates (Hölder continuity in space).
  • Comparison estimates between $T^V_t(x,y)$ and the classical heat kernel $T_t(x-y)$.

3. Key Contributions and Results

A. Equivalence of Definitions (Theorem 1.1)

The authors establish that the spectral definition of $\log L_V$ coincides with the limit of fractional powers:
$$\lim_{s \to 0^+} \frac{L_V^s f - f}{s} = (\log L_V)f$$
They also prove that this limit is equivalent to the Frullani integral representation involving the semigroup $\{T^V_t\}$.

B. Pointwise Representation (Theorem 1.2)

This is a central result. For $f \in C_c^\infty(\mathbb{R}^d)$ and $V \in RH_q$ ($q > d/2$), the authors derive an explicit pointwise formula:
$$(\log L_V)f(x) = -\int_{B(x,1)} (f(y)-f(x)) \left( \int_0^\infty \frac{T^V_t(x,y)}{t} dt \right) dy - \int_{\mathbb{R}^d \setminus B(x,1)} f(y) \left( \int_0^\infty \frac{T^V_t(x,y)}{t} dt \right) dy - K(x)f(x)$$
Significance of the term $K(x)$:
Unlike the classical case where the constant term is a global constant, here $K(x)$ is a non-trivial function depending on $x$ and the potential $V$ (via $\rho(x)$).
$$K(x) = 2\log \rho(x) + \dots + \gamma$$
This reflects the non-Markovian nature of the Schrödinger semigroup. The formula extends previous results for the logarithmic Laplacian (where $V=1$ or $V=0$) to the general potential case.

C. Initial Value Problem (Theorem 1.3)

The paper solves the Cauchy problem:
$$\begin{cases} \frac{\partial u}{\partial t} = -(\log L_V)u, & \text{in } \mathbb{R}^d \times (0, \infty) \\ u(x,0) = f(x), & \text{in } \mathbb{R}^d \end{cases}$$
The solution is given by the fractional integral of the semigroup:
$$u(x,t) = L_V^{-t} f(x) = \frac{1}{\Gamma(t)} \int_0^\infty T^V_z(f)(x) z^{t-1} dz$$
The authors prove that for initial data $f$ in the intersection of $L^1$, $C^1$, and the Schrödinger-adapted Lipschitz space $\text{Lip}_V^\theta$, the function $u(x,t)$ is a classical solution (continuous in space and time, differentiable in time) satisfying the equation pointwise.

D. Lipschitz Spaces and Regularity

The paper utilizes the space $\text{Lip}_V^\theta$, defined by the condition $\|\rho^{-\theta} f\|_\infty < \infty$ and a Hölder-type condition. They show that the fractional integral operator $L_V^{-\alpha}$ maps $\text{Lip}_V^\theta$ to $\text{Lip}_V^{\theta+\alpha}$, which is essential for establishing the regularity of the solution to the Cauchy problem.

4. Significance and Impact

• Generalization: The work significantly extends the theory of logarithmic operators from the Euclidean setting ($-\Delta$) and the harmonic oscillator ($-\Delta + |x|^2$) to general Schrödinger operators with potentials satisfying Reverse Hölder conditions.
• Handling Non-Markovian Semigroups: The derivation of the pointwise formula with the variable coefficient $K(x)$ provides a new framework for analyzing nonlocal operators generated by semigroups that do not preserve constants. This is a critical technical advancement for harmonic analysis in the Schrödinger setting.
• Applications: The results provide the theoretical foundation for studying evolution equations involving logarithmic operators, which appear in various physical models (e.g., quantum mechanics with singular potentials, probability theory, and geometry).
• Methodological Rigor: The paper offers a comprehensive treatment of the domain of the operator, the convergence of fractional powers, and the regularity of solutions, filling a gap in the literature regarding the specific properties of $\log(-\Delta + V)$.

In summary, this paper successfully defines the logarithmic Schrödinger operator, overcomes the difficulties posed by non-Markovian semigroups to provide a concrete pointwise representation, and solves the associated time-evolution problem using adapted function spaces.