可精确求解的微分算子的半经典展开
Semiclassical expansion for exactly solvable differential operators
AI总结：
研究线性微分方程\(\MM (v(z,\eta))=\eta^M{v(z,\eta)}\)，证明其WKB解在斯托克斯区域的波莱尔可和性，对\(M = 3\)给出斯托克斯复形描述，还表明欧拉 - 柯西方程的WKB解在通常意义下收敛。
摘要：
我们研究线性微分方程\(\MM (v(z,\eta))=\eta^M{v(z,\eta)}\)，其中\(\eta>0\)是大谱参数，\(\MM=\sum_{k=1}^{M}\rho_{k}(z)\frac{d^k}{dz^k}\)，\(M\geq2\)是具有多项式系数的微分算子。证明了其WKB解在斯托克斯区域的波莱尔可和性。对于\(M = 3\)，在\(\rho_M\)有单零点假设下，给出了该方程斯托克斯复形的完整描述。最后表明欧拉 - 柯西方程的WKB解在通常意义下收敛。
英文摘要：
Below we study a linear differential equation $\MM (v(z,η))=η^M{v(z,η)}$, where $η>0$ is a large spectral parameter and $\MM=\sum_{k=1}^{M}ρ_{k}(z)\frac{d^k}{dz^k},\; M\ge 2$ is a differential operator with polynomial coefficients such that the leading coefficient $ρ_M(z)$ is a monic complex-valued polynomial with $\dgr{ρ_M }=M$ and other $ρ_k(z)$'s are complex-valued polynomials with $\dgr{ρ_k }\leq k$. We prove the Borel summability of its WKB-solutions in the Stokes regions. For $M=3$ under the assumption that $ρ_M$ has simple zeros, we give the full description of the Stokes complex (i.e. the union of all Stokes curves) of this equation. Finally, we show that for the Euler-Cauchy equations, their WKB-solutions converge in the usual sense.
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