论文标题
任意秩序的一维多谐操作员的决定因素
The determinant of one-dimensional polyharmonic operators of arbitrary order
论文作者
论文摘要
我们获得了多谐波操作员$ p_ {n} =（ - 1）^{n}（\ partial_x）^{2n} $ on $（0，t）$带有dirichlet边界条件和$ n $ a正面integer的$（0，t）$的明确表达式的明确表达式。 -n^2 \ log {n} + \ left [\ frac {7ζ（3）} {2π^2} + \ frac {3} {2} {2} + \ log \ left（\ frac {t} {t} {4} {4} {4} \ right）这是$ \ log {（\ det p_ {n}）} $对所有$ n $有效的敏锐上限和下限的结果，并且在订单$ n $的条款中重合。这些结果构成了分析具有非恒定系数的更多普通操作员的基础，并表明相应的决定因素具有相似的渐近行为。
We obtain an explicit expression for the regularised spectral determinant of the polyharmonic operator $P_{n}=(-1)^{n} (\partial_x)^{2n}$ on $(0,T)$ with Dirichlet boundary conditions and $n$ a positive integer, and show that it satisfies the asymptotics $\log{(\det P_{n})} = -n^2 \log{n} + \left[\frac{7ζ(3)}{2π^2}+ \frac{3}{2}+\log\left(\frac{T}{4}\right)\right] n^2 + {\rm O}(n)$ for large $n$. This is a consequence of sharp upper and lower bounds for $\log{(\det P_{n})}$ valid for all $n$ and which coincide in the terms up to order $n$. These results form the basis to analyse more general operators with nonconstant coefficients and show that the corresponding determinants have a similar asymptotic behaviour.
