Complex-Time Schrödinger Operator

• Schrödinger Operator with Complex Time is defined via a propagator that combines oscillatory evolution with exponential damping, bridging unitary and dissipative dynamics.
• The analysis establishes precise Sobolev regularity thresholds and maximal operator bounds to ensure almost everywhere convergence for rough initial data.
• Advanced techniques like dyadic frequency decomposition and localization are used to construct sharp counterexamples and detail divergence set dimensions.

A Schrödinger operator with complex time refers to the unitary evolution generated by the Laplacian (or fractional Laplacian) where the time parameter is allowed to take nonzero imaginary values, specifically of the form $t + i t^{\gamma}$ with $\gamma > 0$. This complexification of time interpolates between the unitary (dispersive) Schrödinger group and the dissipative semigroup associated to the heat (or fractional heat) equation, and leads to rich regularity and convergence phenomena for solutions with rough initial data. This operator yields a damped evolution, and the balance between the dispersive and dissipative behaviors fundamentally alters pointwise convergence, maximal operator bounds, and the dimension of divergence sets, relative to the real-time Schrödinger flow.

1. Definition and Formalism

The complex-time Schrödinger propagator in $d$ dimensions is defined for $f \in \mathcal{S}(\mathbb{R}^d)$ by

$$P_{\gamma} f(x, t) = (2\pi)^{-d} \int_{\mathbb{R}^d} e^{i [x\cdot \xi + t|\xi|^2]} e^{- t^{\gamma} |\xi|^2} \widehat{f}(\xi) \, d\xi,$$

where $t>0$ and $\gamma>0$ (Wang et al., 26 Dec 2025). For $\gamma=0$, this reduces to the standard solution operator to the linear Schrödinger equation: $$e^{i t \Delta} f(x) = (2\pi)^{-d} \int_{\mathbb{R}^d} e^{i [x\cdot\xi + t|\xi|^2]} \widehat{f}(\xi) \, d\xi.$$ The general $P_\gamma$ can be interpreted as $e^{i (t + i t^{\gamma})\Delta} f(x)$, and thus the evolution is governed by a complex time variable $t + i t^{\gamma}$, where the dissipative term $e^{-t^\gamma |\xi|^2}$ dominates high frequencies as $t \to 0$ for $\gamma > 1$.

This construction admits direct generalization to fractional Laplacians and other dispersive operators, e.g., for $a > 0$,

$$P^{t}_{a,\gamma} f(x) = \frac{1}{2\pi}\int_{\mathbb{R}} e^{i x \xi} e^{i t |\xi|^a - t^{\gamma} |\xi|^a} \widehat{f}(\xi)\, d\xi$$

(Yuan et al., 2019, Bailey, 2011).

2. Main Results: Pointwise Convergence and Maximal Operators

The primary analytic object of study is the maximal operator

$$P^*_{a,\gamma} f(x) = \sup_{0 < t < 1} |P^t_{a,\gamma} f(x)|,$$

and the associated question: for which Sobolev exponents $s$ does $P^*_{a,\gamma}$ map $H^s$ to $L^2$? This determines the minimal regularity requirement for almost everywhere (a.e.) convergence of $P^t_{a,\gamma} f(x) \to f(x)$ as $t \to 0$. The regularity threshold depends acutely on the interplay between dispersion ($e^{i t |\xi|^a}$) and dissipation ($e^{- t^\gamma |\xi|^a}$):

• For the standard Schrödinger case ($a=2$) in $d\geq 2$, the sharp result (Wang et al., 26 Dec 2025):

$$s_0 = \min\left\{\,\frac{d}{2(d+1)},\; \frac{d}{d+1}\left(1 - \frac{1}{\gamma} \right)^+ \right\}$$

ensures a.e. convergence for all $f \in H^s$ with $s > s_0$.

• For $d=1$ and $0<\gamma\leq 1$, no positive Sobolev regularity is needed; $L^2$ data suffices (Bailey, 2011).
• For more general fractional operators ($a>0$), $s > \frac{a}{4}(1-\frac{1}{\gamma})^+$ characterizes the sharp $L^2$ bound for the global maximal operator (Bailey, 2011, Yuan et al., 2019).
• The local $L^2([−1,1])$ threshold is always $$s \geq \min\{\frac{a}{4}(1-\frac{1}{\gamma})^+, \frac{1}{4}\}$$.

When $\gamma > 1$, the dissipation $e^{- t^\gamma |\xi|^a}$ yields improved smoothing, and the regularity threshold can be strictly lower than in the pure Schrödinger flow. For $\gamma \leq 1$, the threshold recovers the Carleson–Bourgain value for the real-time operator.

A summary table of critical exponents for pointwise convergence:

Operator $a$	Dimension $d$	Threshold $s_0$	Reference
$a=2$	$d\geq2$	$$\min\{\frac{d}{2(d+1)},\,\frac{d}{d+1}(1-1/\gamma)^+\}$$	(Wang et al., 26 Dec 2025)
$a=2$	$d=1$	$0$ for $\gamma\leq1$	(Bailey, 2011)
arbitrary $a$	$d=1$	$\frac{a}{4}(1-1/\gamma)^+$	(Bailey, 2011)

Sharpness is demonstrated by explicit high-frequency counterexamples that saturate the maximal function and obstruct convergence below the threshold.

3. Proof Methods and Analytical Techniques

The analysis relies on a reduction to maximal operator bounds using Littlewood–Paley (dyadic frequency) decomposition and temporal localization. The approach consists in:

• Localizing $f$ to dyadic frequency shells $|\xi|\sim R$ and reducing to $L^2$ maximal estimates on balls of fixed radius.
• For "large" times $t > R^{-2/\gamma + \varepsilon}$, the exponential damping suppresses the contribution, leading to trivial $L^2$ bounds.
• For "small" times $t \leq R^{-2/\gamma+\varepsilon}$, periodic extension and Fourier expansion reduce the maximal problem for $P_\gamma$ to known estimates for the real-time propagator (Wang et al., 26 Dec 2025).
• Temporal localization (Lee–Vargas bound) gives:

$$\| \sup_{t\in J} |e^{it\Delta}g| \|_{L^2(B(0,1))} \leq C_\varepsilon R^{d/[2(d+1)]+\varepsilon} \|g\|_{L^2}$$

if $|J| > R^{-1}$, and refined bounds for shorter intervals. These are crucial to establish the correct exponent $s_0$.

• Bourgain-type examples show optimality by constructing high-frequency data with prescribed concentration and phase interaction, yielding divergence unless $s\geq s_0$.

For one-dimensional and fractional flows, the machinery includes van der Corput and stationary-phase estimates, counterexamples based on short, high-frequency Fourier supports, and sharp $L^1$-maximal bounds via Kolmogorov–Seliverstov–Plessner techniques (Yuan et al., 2019, Bailey, 2011).

4. Fractional Operators, Divergence Sets, and Dimension

For the fractional Schrödinger operator $(-\Delta)^{a/2}$, the same framework applies. The set of divergence points

$$E_{a,\gamma}(f) = \left\{ x \in \mathbb{R} : \limsup_{t\to 0^+} |P^t_{a,\gamma} f(x)| = \infty \right\}$$

admits sharp upper bounds for the Hausdorff dimension, reflecting the fine balance between dispersion and dissipation (Yuan et al., 2019). For $f \in H^s(\mathbb{R})$:

• If $a \neq 1$ and $\gamma \leq 1$ (or $\gamma>1$ with $s\in[\frac{1}{2},2]$): $\dim_{\mathcal{H}} E_{a,\gamma}(f) \leq 1-2s$.
• In more singular cases ($a < 1$, $\gamma>1$), subtler formulas involving $a, \gamma$, and $s$ appear, representing the dominance of dispersion or dissipation at small $t$. Frostman's lemma and $L^1$ maximal bounds against Borel measures yield the precise dimension estimates.

5. Propagators along Curves and Non-Tangential Convergence

The pointwise convergence problem has been further generalized to complex-time propagators evaluated along spatial curves: $$P_\gamma f(\Gamma(x,t), t) = \frac{1}{2\pi} \int_{\mathbb{R}} e^{i (\Gamma(x,t)\xi + t|\xi|^2)} e^{-t^\gamma |\xi|^2} \widehat{f}(\xi) d\xi,$$ for curves $\Gamma(x,t)$ bilipschitz in $x$ and Hölder in $t$ with exponent $\alpha$ (Wang et al., 5 Jul 2025). The regularity threshold for a.e. convergence depends explicitly on $\alpha$ and $\gamma$:

• For non-tangential curves ($\alpha \geq 1/2$), the threshold is $$s > \min\{ \frac{1}{2}(1-1/\gamma)^+, \frac{1}{4} \}$$.
• For more tangential approaches, the threshold increases, with detailed piecewise formulas capturing the interplay of curve geometry and dissipation.

The analogous results hold for fractional Laplacians, with optimal regularity described in terms of $m, \alpha,$ and $\gamma$.

6. Dispersion versus Dissipation: Heuristic and Significance

At high frequencies and small times, the fundamental symbol $e^{i t|\xi|^a - t^\gamma |\xi|^a}$ encodes a competition between oscillatory (dispersive) and exponential (dissipative) behavior. For $\gamma \leq 1$, dissipation is relatively weak, and the classical dispersive thresholds (Carleson-type results) are recovered. When $\gamma > 1$, dissipation dominates at small $t$ and high $|\xi|$, enabling improved regularity for convergence:

• For $a=2$, $d\geq2$, and $1<\gamma<2$, the critical exponent

$s_0 = \frac{d}{d+1}(1-1/\gamma)$

is strictly less than $\frac{d}{2(d+1)}$, reflecting an improved smoothing effect of the damping term (Wang et al., 26 Dec 2025).

• In one dimension, $s_0$ can drop to zero for $\gamma\leq1$, allowing $L^2$ data to propagate to a.e. convergence (Bailey, 2011).

This dynamic is mirrored in the Hausdorff dimension results: strong dissipation reduces the size of the divergence set, reflecting a higher degree of regularization in the evolution and convergence behavior (Yuan et al., 2019).

7. Endpoint Phenomena and Additional Consequences

All main results are sharp up to endpoints, with explicit counterexamples constructed whenever the regularity falls below the critical threshold. Notable consequences include:

• Rates of convergence and sequential convergence for $P_\gamma f(\cdot, t) \to f$ as $t\to0$, obtained via refined maximal-function arguments (Wang et al., 26 Dec 2025).
• Extension to nonstandard spatial domains, maximal operators along curves, and a full characterization for the entire scale $0<\gamma<\infty$.
• The endpoint sharpness principle: for every parameter regime, the minimal $s$ is both necessary and sufficient for the associated convergence or boundedness result.

These phenomena delineate the precise border between dispersive and parabolic behaviors in complex-time Schrödinger evolution, and offer a comprehensive blueprint for understanding maximal regularity, saturation, and singular set structure in both integer and fractional settings (Wang et al., 26 Dec 2025, Yuan et al., 2019, Bailey, 2011, Wang et al., 5 Jul 2025).