On $m$-order logarithmic Schrödinger operator

Abstract: In this paper we study the logarithm of order $m$ of the Schrödinger operator $\mathcal L_V$ in $\mathbb R<sup>d$, for certain nonnegative potentials $V$. First, the operator $\log^m\mathcal L_V$, $m\in \mathbb N$, is defined by using the spectral measure associated with the self-adjoint operator $\mathcal L_V$ on a suitable subspace of $L^2(\mathbb R<sup>d)$. Then, the semigroup of operators ${T_t<sup>V}_{t&gt;0}$ generated by $\mathcal L_V$ allows us to extend the definition of $\log^m\mathcal L_V$ to a wider class of Lipschitz functions. By using logarithmic operators $\log^m\mathcal L_V$, $m\in \mathbb N$, we prove Taylor expansions for the fractional powers $\mathcal L_V<sup>s$ and $\mathcal L_V<sup>{-s}$ with respect to the order $s\in (0,1)$, where the convergence is understood in $L^p(\mathbb R<sup>d)$, $1&lt;p&lt;\infty$.

• The paper establishes a spectral construction of m-order logarithmic Schrödinger operators using a refined functional calculus.
• It derives precise Taylor-type asymptotic expansions for fractional powers of the operator with convergence in Lp spaces.
• The study unifies nonlocal operator theory, offering new insights into potential-driven PDEs and spectral stability analysis.

Higher Order Logarithmic Schrödinger Operators: Spectral and Asymptotic Analysis

Introduction and Context

This work investigates the $m$-order logarithmic powers of the Schrödinger operator $\mathcal{L}_V = -\Delta + V$ on $\mathbb{R}^d$ with nonnegative, measurable potentials $V$, particularly focusing on the behavior of the operators $\log^m \mathcal{L}_V$ where $m \in \mathbb{N}$. The logarithmic Laplacian and, more generally, nonlocal pseudodifferential operators arising from functional calculi applied to Schrödinger-type operators, constitute a rapidly evolving subject in analysis, PDEs, and spectral theory. Previous work on the logarithmic Laplacian (e.g., [9], [5]) established its pointwise representations and its role in describing asymptotics of fractional Laplacians and related nonlocal potentials. The current paper generalizes these approaches to higher order logarithmic functionals of Schrödinger operators, employing a spectral measure-based construction and a systematic $L^p$-theoretic analysis.

Construction of the Operator Family and Functional Calculus Framework

The authors employ the self-adjointness of $\mathcal{L}_V$ (for $V \geq 0$) to define $\log^m \mathcal{L}_V$ for $\mathcal{L}_V = -\Delta + V$0 via the spectral representation: $\mathcal{L}_V = -\Delta + V$1 on an appropriate domain in $\mathcal{L}_V = -\Delta + V$2 determined by suitable integrability conditions on the spectral measure $\mathcal{L}_V = -\Delta + V$3. For fractional powers $\mathcal{L}_V = -\Delta + V$4 with $\mathcal{L}_V = -\Delta + V$5, the underlying analytic semigroup $\mathcal{L}_V = -\Delta + V$6 with heat kernel bounds (Feynman-Kac representation and off-diagonal decay adapted to $\mathcal{L}_V = -\Delta + V$7) allows for extension of the semigroups and spectral multipliers to $\mathcal{L}_V = -\Delta + V$8, $\mathcal{L}_V = -\Delta + V$9.

Asymptotic Expansions of Fractional Powers

The central technical contribution is the derivation of complete Taylor-type asymptotic expansions for the positive and negative fractional powers of $\mathbb{R}^d$0 for $\mathbb{R}^d$1 in terms of the higher order logarithmic operators: $\mathbb{R}^d$2 where all terms are interpreted in the $\mathbb{R}^d$3 sense for $\mathbb{R}^d$4 in suitable domains (see Theorems 1.1–1.3). This generalizes prior leading order results for $\mathbb{R}^d$5 using careful spectral and semigroup analysis.

Key properties include:

• Convergence in $\mathbb{R}^d$6: The expansions are shown to converge in $\mathbb{R}^d$7, $\mathbb{R}^d$8.
• Sharp quantitative rates: For $\mathbb{R}^d$9 in the intersection of the domains of fractional powers and iterated logarithmic operators, the remainder is $V$0.
• Compatibility with Riesz potentials and other nonlocal operators: The expansions parallel those previously established for the Laplacian and related spectral multipliers in [5].

Analysis on the Schrödinger Setting and Functional Spaces

To accommodate the nonuniformity induced by the potential $V$1, the analysis leverages the reverse Hölder class $V$2 for $V$3, the associated critical radius function $V$4 (see [28]), and heat kernel estimates adapted to the non-translation invariant setting. Notably, the differences in correction factors between Laplacian- and Schrödinger-based kernels (non-Markovian property of the Schrödinger semigroup) fundamentally affect the pointwise representations and require the development of new integral forms in the expansion.

Function spaces such as $V$5 and local weighted $V$6 spaces are used for existence, regularity, and pointwise convergence, reflecting the inevitable spatial inhomogeneity due to $V$7.

Justification for Higher Order Expansions and Representation Theory

A substantial technical advancement is the extension of the identification between the iterated limit processes, e.g., the analytic definition of $V$8 via the spectral measure and the pointwise limit definitions as $V$9 applied to $\log^m \mathcal{L}_V$0 (cf. Propositions 2.6–2.9).

Explicit formulas are obtained for distributional pointwise limits of the scaled differences $\log^m \mathcal{L}_V$1 where $\log^m \mathcal{L}_V$2 is the $\log^m \mathcal{L}_V$3-th order Taylor polynomial in $\log^m \mathcal{L}_V$4, ensuring the consistency of the spectral and heat semigroup approaches at every order.

Implications and Potential Directions

The results unify and generalize the spectral asymptotics for fractional powers of $\log^m \mathcal{L}_V$5 and link nonlocal Schrödinger-type semigroups to explicit algebraic identities involving higher order logarithmic spectral multipliers. On the practical side, these expansions may impact:

• Perturbative analysis in the limit $\log^m \mathcal{L}_V$6 for subelliptic and nonlocal PDEs with potentials,
• Sharp control of fractional and logarithmic regularity for evolution equations involving Schrödinger operators,
• Spectral theory of non-selfadjoint perturbations and stability problems, since the precise structure of the logarithmic expansions is crucial for understanding fine spectral properties (such as eigenvalue asymptotics, Green's function behavior, and functional inequalities).

The techniques introduced are extendable to more general functional calculi beyond powers and logarithms and may be adapted to geometric settings (e.g., Riemannian manifolds with $\log^m \mathcal{L}_V$7) and noncommutative analogs.

Conclusion

This paper provides a rigorous and comprehensive framework for higher order logarithmic operator calculi associated with Schrödinger operators. The established asymptotic expansions in $\log^m \mathcal{L}_V$8 and pointwise senses, supported by careful spectral and heat kernel estimates, greatly extend the analytic toolkit available for the nonlocal and spectral analysis of Schrödinger-type PDEs. The methodology and results set a foundation for further exploration of nonlocal phenomena, spectral multipliers, and the role of potential regularity in higher order nonlocal operator theory.