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Register a new userOn Dirac operators with electrostatic delta-shell interactions of critical strength
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In this paper we prove that the Dirac operator $A_eta$ with an electrostatic $delta$-shell interaction of critical strength $eta = pm 2$ supported on a $C^2$-smooth compact surface $Sigma$ is self-adjoint in $L^2(mathbb{R}^3;mathbb{C}^4)$, we describe the domain explicitly in terms of traces and jump conditions in $H^{-1/2}(Sigma; mathbb{C}^4)$, and we investigate the spectral properties of $A_eta$. While the non-critical interaction strengths $eta ot= pm 2$ have received a lot of attention in the recent past, the critical case $eta = pm 2$ remained open. Our approach is based on abstract techniques in extension theory of symmetric operators, in particular, boundary triples and their Weyl functions.
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In this article Dirac operators $A_{eta, tau}$ coupled with combinations of electrostatic and Lorentz scalar $delta$-shell interactions of constant strength $eta$ and $tau$, respectively, supported on compact surfaces $Sigma subset mathbb{R}^3$ are studied. In the rigorous definition of these operators the $delta$-potentials are modelled by coupling conditions at $Sigma$. In the proof of the self-adjointness of $A_{eta, tau}$ a Krein-type resolvent formula and a Birman-Schwinger principle are obtained. With their help a detailed study of the qualitative spectral properties of $A_{eta, tau}$ is possible. In particular, the essential spectrum of $A_{eta, tau}$ is determined, it is shown that at most finitely many discrete eigenvalues can appear, and several symmetry relations in the point spectrum are obtained. Moreover, the nonrelativistic limit of $A_{eta, tau}$ is computed and it is discussed that for some special interaction strengths $A_{eta, tau}$ is decoupled to two operators acting in the domains with the common boundary $Sigma$.
In this note the two dimensional Dirac operator $A_eta$ with an electrostatic $delta$-shell interaction of strength $etainmathbb R$ supported on a straight line is studied. We observe a spectral transition in the sense that for the critical interaction strengths $eta=pm 2$ the continuous spectrum of $A_eta$ inside the spectral gap of the free Dirac operator $A_0$ collapses abruptly to a single point.
This paper deals with the massive three-dimensional Dirac operator coupled with a Lorentz scalar shell interaction supported on a compact smooth surface. The rigorous definition of the operator involves suitable transmission conditions along the surface. After showing the self-adjointness of the resulting operator we switch to the investigation of its spectral properties, in particular, to the existence and non-existence of eigenvalues. In the case of an attractive coupling, we study the eigenvalue asymptotics as the mass becomes large and show that the behavior of the individual eigenvalues and their total number are governed by an effective Schrodinger operator on the boundary with an external Yang-Mills potential and a curvature-induced potential.
The compression of the resolvent of a non-self-adjoint Schrodinger operator $-Delta+V$ onto a subdomain $Omegasubsetmathbb R^n$ is expressed in a Krein-Naimark type formula, where the Dirichlet realization on $Omega$, the Dirichlet-to-Neumann maps, and certain solution operators of closely related boundary value problems on $Omega$ and $mathbb R^nsetminusoverlineOmega$ are being used. In a more abstract operator theory framework this topic is closely connected and very much inspired by the so-called coupling method that has been developed for the self-adjoint case by Henk de Snoo and his coauthors.
Consider a regular $d$-dimensional metric tree $Gamma$ with root $o$. Define the Schroedinger operator $-Delta - V$, where $V$ is a non-negative, symmetric potential, on $Gamma$, with Neumann boundary conditions at $o$. Provided that $V$ decays like $x^{-gamma}$ at infinity, where $1 < gamma leq d leq 2, gamma eq 2$, we will determine the weak coupling behavior of the bottom of the spectrum of $-Delta - V$. In other words, we will describe the asymptotical behavior of $inf sigma(-Delta - alpha V)$ as $alpha to 0+$
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