AI 中文总结
该研究构造了1/2<α<1时幂核分数阶Sturm-Liouville算子的闭实现，推导其自伴延拓的奇异值渐近，给出逆算子形式，并应用于Caputo时间扩散方程的温和解。
AI 中文摘要
对于1/2<α<1，我们在L²(0,1)中构造算子ℒu=𝒟₁^α D₀^α u的闭实现，其定义域包含奇异模态x^(α-1)。该定义域通过分布Caputo复合定义，特征为I₀^(1-α)u∈H¹(0,1)且D₀^α u - c ∈ ran I₁^α；我们证明其元素的Volterra表示、图范数完备性，且ℒ是其零迹限制的伴随。四个有界端点迹构成普通边界三元组，因此所有自伴延拓对应ℂ⁴中的拉格朗日平面，包括耦合端点条件。对每个情况我们给出精确零特征值判据，当0∈ρ(ℒ_ω)时，给出形式为I₀^α I₁^α加显式有限秩修正的逆。精确奇异值渐近行为sₙ(ℒ_ω⁻¹)=(πn)^(-2α)(1+O(n⁻¹))，由此得ℒ_ω⁻¹∈𝔖ₚ当且仅当p>1/(2α)，含弱Schatten端点及Weyl渐近——适用于所有可逆边界平面，而非仅正参考问题。作为应用，我们得到C([0,T];L²(0,1))中Caputo时间扩散方程的温和解定理。
英文摘要
For $\tfrac12<α<1$ we construct a closed realization of $\mathcal L u=\mathcal D_1^αD_0^αu$ in $L^2(0,1)$ whose domain contains the singular mode $x^{α-1}$. The domain, defined via the distributional Caputo composition, is characterized by $I_0^{1-α}u\in H^1(0,1)$ and $D_0^αu-c\in\operatorname{ran}I_1^α$; we prove a Volterra representation for its elements, graph-norm completeness, and that $\mathcal L$ is the adjoint of its zero-trace restriction. Four bounded endpoint traces give an ordinary boundary triplet, so all self-adjoint extensions correspond to Lagrangian planes in $\mathbb C^4$, including coupled endpoint conditions. For each we give an exact zero-eigenvalue criterion and, when $0\inρ(\mathcal L_ω)$, an inverse of the form $I_0^αI_1^α$ plus an explicit finite-rank correction. Sharp singular-value asymptotics $s_n(\mathcal L_ω^{-1})=(πn)^{-2α}\bigl(1+O(n^{-1})\bigr)$ yield $\mathcal L_ω^{-1}\in\mathfrak S_p\iff p>1/(2α)$, with weak-Schatten endpoint and Weyl asymptotics --- now for \emph{every} invertible boundary plane, not only the positive reference problem. As an application we obtain a mild-solution theorem for a Caputo-time diffusion equation in $C([0,T];L^2(0,1))$.
Comments23 pages
