Maximal regularity for time-fractional Schrödinger equations and application to nonlinear equations

1. Problem Statement

The paper investigates the maximal regularity of abstract time-fractional Schrödinger equations of the form:
$$\partial_t^\alpha (u - u_0) - iAu = f, \quad t \in (0, T)$$
where:

• $\alpha \in (0, 1)$ is the order of the fractional derivative (Riemann-Liouville or Caputo type, defined via the inverse of the fractional integral).
• $A$ is a self-adjoint, negative-definite operator with a compact resolvent on a Hilbert space $H$.
• $f$ is a source term in $L^p(0, T; H)$.
• $u_0$ is the initial data.

The Core Challenge:
Maximal regularity implies that the solution $u$ possesses the same regularity as the source term $f$ regarding the time derivative $\partial_t^\alpha$ and the spatial operator $A$. Specifically, the authors aim to prove that if $f \in L^p(0, T; H)$, then $\partial_t^\alpha u \in L^p(0, T; H)$ and $Au \in L^p(0, T; H)$.

A significant theoretical hurdle is that the classical Schrödinger equation ($\alpha=1$) does not possess maximal regularity because the operator $iA$ does not generate an analytic semigroup (its spectrum lies on the imaginary axis). The authors investigate whether this property is recovered or altered in the fractional case ($\alpha < 1$).

2. Methodology

The authors employ a multi-step analytical approach combining spectral theory, special functions, and harmonic analysis:

A. Spectral Decomposition and Mittag-Leffler Functions

• The problem is analyzed using the spectral decomposition of the operator $A$. Since $A$ has a compact resolvent, there exists an orthonormal basis of eigenvectors $\{\phi_n\}$ with eigenvalues $-\lambda_n$ ($\lambda_n > 0$).
• The solution is represented via a series expansion involving the Mittag-Leffler function $E_{\alpha, \beta}(z)$. Specifically, the kernel of the solution involves $E_{\alpha, \alpha}(-i\lambda_n t^\alpha)$.
• Key Technical Innovation: Unlike previous works on fractional diffusion equations (where the argument is real and negative), the argument here is imaginary ($-i\lambda_n t^\alpha$). The authors explicitly avoid relying on the complete monotonicity of Mittag-Leffler functions, a property that is difficult to prove (or may not hold) for imaginary arguments. Instead, they utilize asymptotic expansions and decay estimates of $E_{\alpha, \beta}(z)$ for large $|z|$.

B. $L^2$-Regularity Proof

• For the case $p=2$, the authors prove maximal regularity using Young's convolution inequality.
• They establish that the integral of the kernel $t^{\alpha-1} |E_{\alpha, \alpha}(-i\lambda_n t^\alpha)|$ is uniformly bounded with respect to $\lambda_n$. This allows them to bound the $L^2$-norms of $\partial_t^\alpha u$ and $Au$ directly by the $L^2$-norm of $f$.

C. $L^p$-Regularity Proof ($1 < p < \infty$)

• To extend the result to general $p$, the authors utilize the operator-valued Mikhlin Multiplier Theorem.
• They define a Fourier multiplier operator $m(s)$ related to the resolvent of $A$ and the fractional derivative.
• By analyzing the symbol $m(s) = -i(is)^\alpha R(-i(is)^\alpha, A)$, they prove that $m(s)$ and $s m'(s)$ are bounded in the space of bounded linear operators $B(H)$.
• This relies on the fact that $A$ generates a bounded analytic semigroup, allowing the resolvent estimate $\|R(z, A)\| \leq M/|z|$ to hold in a sector containing the values $-i(is)^\alpha$.

D. Nonlinear Applications

• The maximal regularity results are applied to prove local well-posedness for quasilinear and semilinear equations using the Banach Contraction Mapping Theorem.
• They define a "maximal regularity space" $MR_{\alpha, p}(T)$ and establish continuous embeddings into spaces suitable for handling nonlinearities (e.g., $C([0, T]; X_{\alpha-\epsilon, p})$).

3. Key Contributions and Results

1. Maximal $L^2$-Regularity (Theorem 4.2)

The paper proves that for $f \in L^2(0, T; H)$, the unique weak solution satisfies:
$$\|u\|_{H^\alpha(0,T;H)} + \|Au\|_{L^2(0,T;H)} \leq C \|f\|_{L^2(0,T;H)}$$

• Significance: This result is achieved without the complete monotonicity of Mittag-Leffler functions, overcoming a limitation of previous methods used for subdiffusion equations.

2. Maximal $L^p$-Regularity (Theorem 4.5)

For any $p \in (1, \infty)$, if $f \in L^p(0, T; H)$, the solution satisfies:
$$\|u\|_{W^{\alpha,p}(0,T;H)} + \|Au\|_{L^p(0,T;H)} \leq C \|f\|_{L^p(0,T;H)}$$

• Significance: This generalizes the result to the full range of $L^p$ spaces using operator-valued harmonic analysis, a technique not previously applied to this specific fractional Schrödinger setting.

3. Homogeneous Problem Regularity (Theorem 4.6)

The authors provide sharp estimates for the homogeneous problem ($f=0$) with initial data $u_0$. They show that the regularity of the solution depends on the trace space $X_{\alpha, p} = (H, D(A))_{1-1/(\alpha p), p}$.

• For $p > 1/\alpha$, the solution enjoys maximal regularity in $L^p$.
• For $p \leq 1/\alpha$, the solution belongs to weak Lebesgue spaces $L^{p, \infty}$.

4. Nonlinear Well-Posedness (Theorems 5.2 and 5.4)

• Quasilinear Equations: Proven local well-posedness for equations where the operator $A$ depends on the solution $u$ (e.g., $A(u)u$), assuming Lipschitz continuity of the operator map.
• Semilinear Equations: Proven local well-posedness for equations with nonlinear source terms $F(u)$, utilizing compactness arguments and fixed-point theorems.

4. Significance and Novelty

• Distinction from Classical Theory: The paper highlights a fundamental difference between classical and fractional Schrödinger equations. While the classical equation ($\alpha=1$) fails to have maximal regularity (due to the lack of analytic semigroup generation by $iA$), the time-fractional version ($\alpha < 1$) does possess maximal regularity. This is attributed to the "smoothing" effect of the fractional derivative and the specific geometry of the spectrum in the complex plane relative to the fractional power.
• Methodological Breakthrough: By avoiding the reliance on complete monotonicity (which fails for imaginary arguments), the authors open the door to analyzing a broader class of fractional evolution equations where the operator is multiplied by a complex unit.
• Application to Nonlinear Models: The results provide the rigorous functional analytic framework necessary to study nonlinear quantum systems with memory effects, which are relevant in condensed matter physics, optics, and plasma physics.

5. Conclusion

The authors successfully establish a comprehensive theory of maximal regularity for time-fractional Schrödinger equations in Hilbert spaces. By combining spectral analysis of Mittag-Leffler functions with operator-valued multiplier theorems, they prove that these equations enjoy strong regularity properties ($L^p$-maximal regularity) that are absent in the classical integer-order case. These findings serve as a foundational tool for the analysis of nonlinear fractional quantum models.