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Register a new userParticle Creation at a Point Source by Means of Interior-Boundary Conditions
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We consider a way of defining quantum Hamiltonians involving particle creation and annihilation based on an interior-boundary condition (IBC) on the wave function, where the wave function is the particle-position representation of a vector in Fock space, and the IBC relates (essentially) the values of the wave function at any two configurations that differ only by the creation of a particle. Here we prove, for a model of particle creation at one or more point sources using the Laplace operator as the free Hamiltonian, that a Hamiltonian can indeed be rigorously defined in this way without the need for any ultraviolet regularization, and that it is self-adjoint. We prove further that introducing an ultraviolet cut-off (thus smearing out particles over a positive radius) and applying a certain known renormalization procedure (taking the limit of removing the cut-off while subtracting a constant that tends to infinity) yields, up to addition of a finite constant, the Hamiltonian defined by the IBC.
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A recently proposed approach for avoiding the ultraviolet divergence of Hamiltonians with particle creation is based on interior-boundary conditions (IBCs). The approach works well in the non-relativistic case, that is, for the Laplacian operator. Here, we study how the approach can be applied to Dirac operators. While this has been done successfully already in 1 space dimension, and more generally for codimension-1 boundaries, the situation of point sources in 3 dimensions corresponds to a codimension-3 boundary. One would expect that, for such a boundary, Dirac operators do not allow for boundary conditions because they are known not to allow for point interactions in 3d, which also correspond to a boundary condition. And indeed, we confirm this expectation here by proving that there is no self-adjoint operator on (a truncated) Fock space that would correspond to a Dirac operator with an IBC at configurations with a particle at the origin. However, we also present a positive result showing that there are self-adjoint operators with IBC (on the boundary consisting of configurations with a particle at the origin) that are, away from those configurations, given by a Dirac operator plus a sufficiently strong Coulomb potential.
This paper deals with the study of the two-dimensional Dirac operatorwith infinite mass boundary condition in a sector. We investigate the question ofself-adjointness depending on the aperture of the sector: when the sector is convexit is self-adjoint on a usual Sobolev space whereas when the sector is non-convexit has a family of self-adjoint extensions parametrized by a complex number of theunit circle. As a byproduct of this analysis we are able to give self-adjointnessresults on polygones. We also discuss the question of distinguished self-adjointextensions and study basic spectral properties of the operator in the sector.
We conjecture that for a plasma in a spatial domain with a boundary, the specular reflection effect of the boundary can be approximated by a large magnetic confinement field in the near-boundary region. In this paper, we verify this conjecture for the 1.5D relativistic Vlasov-Maxwell system (RVM) on a bounded domain $Omega = (0, 1)$ with an external confining magnetic field.
We establish an integral variational principle for the spreading speed of the one dimensional reaction diffusion equation with Stefan boundary conditions, for arbitrary reaction terms. This principle allows to obtain in a simple way the dependence of the speed on the Stefan constant. As an application a generalized Zeldovich-Frank-Kamenetskii lower bound for the speed, valid for monostable and combustion reaction terms, is given.
We consider a massless scalar field in 1+1 dimensions satisfying a Robin boundary condition (BC) at a non-relativistic moving boundary. We derive a Bogoliubov transformation between input and output bosonic field operators, which allows us to calculate the spectral distribution of created particles. The cases of Dirichlet and Neumann BC may be obtained from our result as limiting cases. These two limits yield the same spectrum, which turns out to be an upper bound for the spectra derived for Robin BC. We show that the particle emission effect can be considerably reduced (with respect to the Dirichlet/Neumann case) by selecting a particular value for the oscillation frequency of the boundary position.
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