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Register a new userSpectral transition for Dirac operators with electrostatic $delta$-shell potentials supported on the straight line
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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.
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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 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.
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.
We consider the self-adjoint Schrodinger operator in $L^2(mathbb{R}^d)$, $dgeq 2$, with a $delta$-potential supported on a hyperplane $Sigmasubseteqmathbb{R}^d$ of strength $alpha=alpha_0+alpha_1$, where $alpha_0inmathbb{R}$ is a constant and $alpha_1in L^p(Sigma)$ is a nonnegative function. As the main result, we prove that the lowest spectral point of this operator is not smaller than that of the same operator with potential strength $alpha_0+alpha_1^*$, where $alpha_1^*$ is the symmetric decreasing rearrangement of $alpha_1$. The proof relies on the Birman-Schwinger principle and the reduction to an analogue of the P{o}lya-SzegH{o} inequality for the relativistic kinetic energy in $mathbb{R}^{d-1}$.
We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of $mathbb{R}^2$. Motivated by spectral geometric inequalities, we prove a non-linear variational formulation to characterize its principal eigenvalue. This characterization turns out to be very robust and allows for a simple proof of a Szego type inequality as well as a new reformulation of a Faber-Krahn type inequality for this operator. The paper is complemented with strong numerical evidences supporting the existence of a Faber-Krahn type inequality.
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