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Register a new userLocalization of eigenvalues for non-self-adjoint Dirac and Klein-Gordon operators
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Added by
Nico Michele Schiavone
Publication date
2021
fields
Physics
and research's language is
English
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This note aims to give prominence to some new results on the absence and localization of eigenvalues for the Dirac and Klein-Gordon operators, starting from known resolvent estimates already established in the literature combined with the renowned Birman-Schwinger principle.
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We prove the absence of eigenvaues of the three-dimensional Dirac operator with non-Hermitian potentials in unbounded regions of the complex plane under smallness conditions on the potentials in Lebesgue spaces. Our sufficient conditions are quantitative and easily checkable.
In this paper the spectral and scattering properties of a family of self-adjoint Dirac operators in $L^2(Omega; mathbb{C}^4)$, where $Omega subset mathbb{R}^3$ is either a bounded or an unbounded domain with a compact $C^2$-smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with Robin boundary conditions. Among the Dirac operators treated here is also the so-called MIT bag operator, which has been used by physicists and more recently was discussed in the mathematical literature. Our approach is based on abstract boundary triple techniques from extension theory of symmetric operators and a thorough study of certain classes of (boundary) integral operators, that appear in a Krein-type resolvent formula. The analysis of the perturbation term in this formula leads to a description of the spectrum and a Birman-Schwinger principle, a qualitative understanding of the scattering properties in the case that $Omega$ is unbounded, and corresponding trace formulas.
We present a survey of some recent results concerning the location and the Weyl formula for the complex eigenvalues of two non self-adjoint operators. We study the eigenvalues of the generator $G$ of the contraction semigroup $e^{tG}, : t geq 0,$ related to the wave equation in an unbounded domain $Omega$ with dissipative boundary conditions on $partial Omega$. Also one examines the interior transmission eigenvalues (ITE) in a bounded domain $K$ obtaining a Weyl formula with remainder for the counting function $N(r)$ of complex (ITE). The analysis is based on a semi-classical approach.
Let $Omega_-$ and $Omega_+$ be two bounded smooth domains in $mathbb{R}^n$, $nge 2$, separated by a hypersurface $Sigma$. For $mu>0$, consider the function $h_mu=1_{Omega_-}-mu 1_{Omega_+}$. We discuss self-adjoint realizations of the operator $L_{mu}=- ablacdot h_mu abla$ in $L^2(Omega_-cupOmega_+)$ with the Dirichlet condition at the exterior boundary. We show that $L_mu$ is always essentially self-adjoint on the natural domain (corresponding to transmission-type boundary conditions at the interface $Sigma$) and study some properties of its unique self-adjoint extension $mathcal{L}_mu:=overline{L_mu}$. If $mu e 1$, then $mathcal{L}_mu$ simply coincides with $L_mu$ and has compact resolvent. If $n=2$, then $mathcal{L}_1$ has a non-empty essential spectrum, $sigma_mathrm{ess}(mathcal{L}_{1})={0}$. If $nge 3$, the spectral properties of $mathcal{L}_1$ depend on the geometry of $Sigma$. In particular, it has compact resolvent if $Sigma$ is the union of disjoint strictly convex hypersurfaces, but can have a non-empty essential spectrum if a part of $Sigma$ is flat. Our construction features the method of boundary triplets, and the problem is reduced to finding the self-adjoint extensions of a pseudodifferential operator on $Sigma$. We discuss some links between the resulting self-adjoint operator $mathcal{L}_mu$ and some effects observed in negative-index materials.
We study the eigenvalues of the Laplacian with a strong attractive Robin boundary condition in curvilinear polygons. It was known from previous works that the asymptotics of several first eigenvalues is essentially determined by the corner openings, while only rough estimates were available for the next eigenvalues. Under some geometric assumptions, we go beyond the critical eigenvalue number and give a precise asymptotics of any individual eigenvalue by establishing a link with an effective Schrodinger-type operator on the boundary of the domain with boundary conditions at the corners.
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