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Register a new userGeneral $delta$-shell interactions for the two-dimensional Dirac operator: self-adjointness and approximation
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In this work we consider the two-dimensional Dirac operator with general local singular interactions supported on a closed curve. A systematic study of the interaction is performed by decomposing it into a linear combination of four elementary interactions: electrostatic, Lorentz scalar, magnetic, and a fourth one which can be absorbed by using unitary transformations. We address the self-adjointness and the spectral description of the underlying Dirac operator, and moreover we describe its approximation by Dirac operators with regular potentials.
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We study the two-dimensional Dirac operator with an arbitrary combination of electrostatic and Lorentz scalar $delta$-interactions of constant strengths supported on a smooth closed curve. For any combination of the coupling constants a rigorous description of the self-adjoint realizations of the operators is given and the qualitative spectral properties are described. The analysis covers also all so-called critical combinations of coupling constants, for which there is a loss of regularity in the operator domain. In this case, if the mass is non-zero, the resulting operator has an additional point in the essential spectrum, and the position of this point inside the central gap can be made arbitrary by a suitable choice of the coupling constants. The analysis is based on a combination of the extension theory of symmetric operators with a detailed study of boundary integral operators viewed as periodic pseudodifferential operators.
We discuss connections between the essential self-adjointness of a symmetric operator and the constancy of functions which are in the kernel of the adjoint of the operator. We then illustrate this relationship in the case of Laplacians on both manifolds and graphs. Furthermore, we discuss the Greens function and when it gives a non-constant harmonic function which is square integrable.
We construct the propagator of the massless Dirac operator $W$ on a closed Riemannian 3-manifold as the sum of two invariantly defined oscillatory integrals, global in space and in time, with distinguished complex-valued phase functions. The two oscillatory integrals -- the positive and the negative propagators -- correspond to positive and negative eigenvalues of $W$, respectively. This enables us to provide a global invariant definition of the full symbols of the propagators (scalar matrix-functions on the cotangent bundle), a closed formula for the principal symbols and an algorithm for the explicit calculation of all their homogeneous components. Furthermore, we obtain small time expansions for principal and subprincipal symbols of the propagators in terms of geometric invariants. Lastly, we use our results to compute the third local Weyl coefficients in the asymptotic expansion of the eigenvalue counting functions of $W$.
Uniqueness and reconstruction in the three-dimensional Calderon inverse conductivity problem can be reduced to the study of the inverse boundary problem for Schrodinger operators $-Delta +q $. We study the Born approximation of $q$ in the ball, which amounts to studying the linearization of the inverse problem. We first analyze this approximation for real and radial potentials in any dimension. We show that this approximation is well-defined and obtain a closed formula that involves the spectrum of the Dirichlet-to-Neumann map associated to $-Delta + q$. We then turn to general real and essentially bounded potentials in three dimensions and introduce the notion of averaged Born approximation, which captures the invariance properties of the exact inverse problem. We obtain explicit formulas for the averaged Born approximation in terms of the matrix elements of the Dirichlet to Neumann map in the basis spherical harmonics. Motivated by these formulas we also study the high-energy behaviour of the matrix elements of the Dirichlet to Neumann map.
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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