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Register a new userOn Dirac operators in $mathbb{R}^3$ with electrostatic and Lorentz scalar $delta$-shell interactions
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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 $Sigma$.
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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.
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.
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.
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.
Let $Sigmasubsetmathbb{R}^d$ be a $C^infty$-smooth closed compact hypersurface, which splits the Euclidean space $mathbb{R}^d$ into two domains $Omega_pm$. In this note self-adjoint Schrodinger operators with $delta$ and $delta$-interactions supported on $Sigma$ are studied. For large enough $minmathbb{N}$ the difference of $m$th powers of resolvents of such a Schrodinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in $L^2(mathbb{R}^d)$ is written in terms of Neumann-to-Dirichlet maps on the boundary space $L^2(Sigma)$.
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