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Spectral properties of the Cauchy process on half-line and interval. (English) Zbl 1220.60029
The authors study the spectral properties of the generator of the scalar Cauchy process killed at the first exit of the positive half-line or the interval \((-1,1)\). It is given by the square-root of the one-dimensional Laplacian
\[ \mathcal{A} = \sqrt{-\frac{d^2}{dx^2}} \] with Dirichlet exterior conditions on a complement on the domain or a mixed Steklov problem in the half-plane respectively. This was established in a prior work [R. Bañuelos and T. Kulczycki, J. Funct. Anal. 211, No. 2, 355–423 (2004; Zbl 1055.60072)]. For the half-plane, the authors derive a closed formula for the solution of the Steklov problem and infer an explicit formula for the generalized eigenfunctions \(\psi_\lambda\in L^\infty\) of \(\mathcal{A}\). Via a fine analysis of a crucial complex function the authors are able to construct a spectral representation of \(\mathcal{A}\). In the sequel, an explicit formula for the transition density of the killed process (i.e., the heat kernel of the \(\mathcal{A}\) in \((0,\infty)\)), and for the distribution of the first exit time from the half-line, is given.
The results obtained, in particular the representation for \(\psi_\lambda\), are heavily exploited for the construction of approximations to eigenfunctions of \(\mathcal{A}\) in the interval. For the eigenvalues \(\lambda_n\) of \(\mathcal{A}\) in the interval, the asymptotic behavior
\[ \lambda_n = n \pi/2 - \pi/8 + O(1/n) \] is established, which improves the available asymptotics at present. All eigenvalues \(\lambda_n\) are proved to be simple. Finally, efficient numerical methods for the estimation of the eigenvalues \(\lambda_n\) are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to the ninth decimal place.

MSC:
60G52 Stable stochastic processes
35J25 Boundary value problems for second-order elliptic equations
35P15 Estimates of eigenvalues in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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