The eigenvalue distribution of elliptic operators with Hölder continuous coefficients
Improvements are studied for the estimates of the remainder term in the asymptotic formula for the eigenvalue distribution of an elliptic operator \(A\) of order \(2m\) whose principal part has Hölder continuous coefficients with exponent \(\tau\), \(0<\tau\leq 1\). In some cases, mainly when \(n=1\), a better estimate may be obtained for the remainder term of the counting function \(N(T)\): \[ N(t)=\mu_ A(\Omega)t^{n/2m}+O(t^{(n-\tau)/2m})\quad \text{as} \quad t\to\infty. \] Conditions are determined under which this estimate is valid when the following asymptotic formula for the partition function (the trace of the heat kernel) holds: \[ U(t)=\Gamma({n \over 2m}+1)\mu_ A(\Omega)t^{-n/2m}+O(t^{(\tau-n)/2m}) \quad\text{as}\quad t\to 0, \] where \(\Gamma(x)\) is the Gamma function.
asymptotic formula for the eigenvalue distribution
scientific article; zbMATH DE number 91631