Three lectures on global boundary conditions and the theory of self-adjoint extensions of the covariant Laplace-Beltrami and Dirac operators on Riemannian manifolds with boundary

Ibort, A.

doi:10.1063/1.4733360

Cited by 7 publications

(5 citation statements)

References 29 publications

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“…We will just quote the analysis of non-local extensions of elliptic operators by Grubb [Gr68] and the theory of singular perturbations of differential operators by Albeverio and Kurasov [Al99] because of their influence on this work (see also [Ib14] for a quadratic forms based analysis of the extensions of the Laplace-Beltrami operator and [Ib12] where the reader will find a reasonable list of references on the subject).…”

Section: Introductionmentioning

confidence: 99%

The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

Asorey

Ibort

Marmo

2015

Int. J. Geom. Methods Mod. Phys.

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The theory of self-adjoint extensions of first and second order elliptic differential operators on manifolds with boundary is studied via its most representative instances: Dirac and Laplace operators.The theory is developed by exploiting the geometrical structures attached to them and, by using an adapted Cayley transform on each case, the space M of such extensions is shown to have a canonical group composition law structure.The obtained results are compared with von Neumann's Theorem characterising the self-adjoint extensions of densely defined symmetric operators on Hilbert spaces. The 1D case is thoroughly investigated.The geometry of the submanifold of elliptic self-adjoint extensions M ellip is studied and it is shown that it is a Lagrangian submanifold of the universal Grassmannian Gr. The topology of M ellip is also explored and it is shown that there is a canonical cycle whose dual is the Maslov class of the manifold. Such cycle, called the Cayley surface, plays a relevant role in the study of the phenomena of topology change.Self-adjoint extensions of Laplace operators are discussed in the path integral formalism, identifying a class of them for which both treatments leads to the same results.A theory of dissipative quantum systems is proposed based on this theory and a unitarization theorem for such class of dissipative systems is proved.The theory of self-adjoint extensions with symmetry of Dirac operators is also discussed and a reduction theorem for the self-adjoint elliptic Grasmmannian is obtained.Finally, an interpretation of spontaneous symmetry breaking is offered from the

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Section: Introductionmentioning

confidence: 99%

The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

Asorey

Ibort

Marmo

2015

Int. J. Geom. Methods Mod. Phys.

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“…The generalisation of the definition of the delta function potential for an electromagnetic field can be achieved using the theory of self-adjoint extensions for the quadratic Yang-Mills operator around a point-like configuration (see Refs. [21][22][23]).…”

Section: A Self Adjoint Extensionmentioning

confidence: 99%

Dirac lattices, zero-range potentials, and self-adjoint extension

Bordag

Munõz-Castañeda²

2015

Phys. Rev. D

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We consider the electromagnetic field in the presence of polarizable point dipoles. In the corresponding effective Maxwell equation these dipoles are described by three dimensional delta function potentials. We review the approaches handling these: the selfadjoint extension, regularization/renormalisation and the zero range potential methods. Their close interrelations are discussed in detail and compared with the electrostatic approach which drops the contributions from the self fields.For a homogeneous two dimensional lattice of dipoles we write down the complete solutions, which allow, for example, for an easy numerical treatment of the scattering of the electromagnetic field on the lattice or for investigating plasmons. Using these formulas, we consider the limiting case of vanishing lattice spacing, i.e., the transition to a continuous sheet. For a scalar field and for the TE polarization of the electromagnetic field this transition is smooth and results in the results known from the continuous sheet. Especially for the TE polarization, we reproduce the results known from the hydrodynamic model describing a two dimensional electron gas. For the TM polarization, for polarizability parallel and perpendicular to the lattice, in both cases, the transition is singular. For the parallel polarizability this is surprising and different from the hydrodynamic model. For perpendicular polarizability this is what was known in literature. We also investigate the case when the transition is done with dipoles described by smeared delta function, i.e., keeping a regularization. Here, for TM polarization for parallel polarizability, when subsequently doing the limit of vanishing lattice spacing, we reproduce the result known from the hydrodynamic model. In case of perpendicular polarizability we need an additional renormalization to reproduce the result obtained previously by stepping back from the dipole approximation.

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“…In particular, those works have applied the theory of self-adjoint extensions to the computation of the so-called Casimir energy of scalar fields [8,9]. On the other hand, Ibort et al have developed the theory of local self-adjoint extensions for Dirac-type operators, and have studied applications in condensed matter [10][11][12]. More recent works have shown how the theory of self-adjoint extensions can be used to describe physical systems confined to cavities [13,14].…”

Section: Introductionmentioning

confidence: 99%

Field Fluctuations and Casimir Energy of 1D-Fermions

et al. 2019

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We investigate the self-adjoint extensions of the Dirac operator of a massive one-dimensional field of mass m confined in a finite filament of length L. We compute the spectrum of vacuum fluctuations of the Dirac field under the most general dispersionless boundary conditions. We identify its edge states in the mass gap within a set of values of the boundary parameters, and compute the Casimir energy of the discrete normal modes. Two limit cases are considered, namely, that of light fermions with m L ≪ 1 , and that of heavy fermions for which m L ≫ 1 . It is found that both positive and negative energies are obtained for different sets of values of the boundary parameters. As a consequence of our calculation we demonstrate that the sign of the quantum vacuum energy is not fixed for exchange-symmetric plates (parity-invariant configurations), unlike for electromagnetic and scalar fields.

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Three lectures on global boundary conditions and the theory of self-adjoint extensions of the covariant Laplace-Beltrami and Dirac operators on Riemannian manifolds with boundary

Cited by 7 publications

References 29 publications

The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

Dirac lattices, zero-range potentials, and self-adjoint extension

Field Fluctuations and Casimir Energy of 1D-Fermions

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