[Paper Review] Complex scaling for the Dirichlet Laplacian in a domain with asymptotically cylindrical end

Victor Kalvin|arXiv (Cornell University)|Jun 3, 2009

Spectral Theory in Mathematical Physics32 references3 citations

TL;DR

This paper develops the complex scaling method for the Dirichlet Laplacian in a domain with an asymptotically cylindrical end, establishing that resonances correspond to poles of meromorphically continued resolvent matrix elements. The key result is that the Dirichlet Laplacian has no singular continuous spectrum and its eigenvalues can accumulate only at threshold values, even under arbitrarily slow metric convergence at infinity, via analyticity assumptions on the diffeomorphism defining the end.

ABSTRACT

We develop the complex scaling method for the Dirichlet Laplacian in a domain with asymptotically cylindrical end. We define resonances as discrete eigenvalues of non-selfadjoint operators, obtained as deformations of the selfadjoint Dirichlet Laplacian $Δ$ by means of the complex scaling. The resonances are identified with the poles of the resolvent matrix elements $((Δ-μ)^{-1}F, G)$ meromorphic continuation in $μ$ across the essential spectrum of $Δ$, where $F$ and $G$ are elements of an explicitly given set of partial analytic vectors. It turns out that the Dirichlet Laplacian has no singular continuous spectrum, and its eigenvalues can accumulate only at threshold values of the spectral parameter.

Motivation & Objective

• To extend the complex scaling method to Dirichlet Laplacians in domains with asymptotically cylindrical ends, where the metric convergence at infinity is arbitrarily slow.
• To define resonances as poles of the meromorphically continued resolvent matrix elements $((\Delta - \mu)^{-1}F, G)$, with $F, G$ in a set of partial analytic vectors.
• To prove that the Dirichlet Laplacian has no singular continuous spectrum, even without strong decay assumptions on the metric's convergence at infinity.
• To show that eigenvalues can accumulate only at threshold values of the spectral parameter, under analyticity conditions on the diffeomorphism.

Proposed method

• The complex scaling method is applied by deforming the Dirichlet Laplacian via a complex change of variables, parameterized by a complex scaling parameter $\lambda$ in a sector $\mathcal{D}_\alpha$.
• The spectrum of the deformed operator $^\lambda\!\Delta$ is analyzed using the theory of m-sectorial operators and the Aguilar-Balslev-Combes framework.
• A dense set $\mathcal{A}$ of partial analytic vectors is constructed such that resolvent matrix elements $((\Delta - \mu)^{-1}F, G)$ for $F, G \in \mathcal{A}$ admit meromorphic continuation across the essential spectrum.
• The meromorphic continuation is established via analytic continuation of the resolvent identity, linking the physical-space resolvent to the scaled operator's resolvent.
• The method relies on the non-degeneracy of the $L^2$ inner product under the scaled metric and the density of $\vartheta_\lambda[\mathcal{A}]$ in $L^2(\mathcal{G})$.
• The identification of resonances as poles of the continuation is achieved by showing that poles of the scaled operator’s resolvent correspond to poles of the physical-space resolvent matrix elements.

Experimental results

Research questions

• RQ1 Can the complex scaling method be extended to Dirichlet Laplacians in domains with asymptotically cylindrical ends under arbitrarily slow metric convergence at infinity?
• RQ2 Are resonances of the Dirichlet Laplacian characterized by poles of the meromorphically continued resolvent matrix elements?
• RQ3 Does the Dirichlet Laplacian possess a singular continuous spectrum under the given geometric and analytic assumptions?
• RQ4 Can eigenvalues accumulate only at threshold values of the spectral parameter in such domains?
• RQ5 Is the spectrum of the Dirichlet Laplac independent of the choice of scaling function $v$ under analyticity assumptions?

Key findings

• The Dirichlet Laplacian has no singular continuous spectrum, as shown by the analytic continuation of resolvent matrix elements and the countability of the spectrum of the scaled operator.
• Resonances are identified as non-real poles of the meromorphically continued resolvent matrix elements, corresponding to the non-real eigenvalues of the complex-scaled operator.
• Eigenvalues of the Dirichlet Laplacian can accumulate only at threshold values of the spectral parameter, consistent with the Aguilar-Balslev-Combes theorem.
• The set of resonances and the essential spectrum are independent of the choice of the scaling function $v$, relying only on the analyticity of the diffeomorphism at infinity.
• The eigenvalues of the Dirichlet Laplacian are of finite multiplicity and can only accumulate from below at thresholds, as a consequence of the spectral separation by complex scaling.
• The resolvent matrix elements $((\Delta - \mu)^{-1}F, G)$ for $F, G \in \mathcal{A}$ admit a meromorphic continuation to a Riemann surface, with poles precisely at the resonances and embedded eigenvalues.