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Register a new userOn the two-dimensional Schrodinger operator with an attractive potential of the Bessel-Macdonald type
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Added by
Oswaldo Monteiro Del Cima
Publication date
2018
fields
Physics
and research's language is
English
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We analyze the Schrodinger operator in two-dimensions with an attractive potential given by a Bessel-Macdonald function. This operator is derived in the non-relativistic approximation of planar quantum electrodynamics (${rm QED}_3$) models as a framework for evaluation of two-quasiparticle scattering potentials. The analysis is motivated keeping in mind the fact that parity-preserving ${rm QED}_3$ models can provide a possible explanation for the behavior of superconductors. Initially, we study the self-adjointness and spectral properties of the Schrodinger operator modeling the non-relativistic approximation of these ${rm QED}_3$ models. Then, by using {em Set^o-type estimates}, an estimate is derived of the number of two-particle bound states which depends directly on the value of the effective coupling constant, $C$, for {em any} value of the angular momentum. In fact, this result in connection with the condition that guarantees the self-adjointness of the Schrodinger operator shows that there can always be a large number of two-quasiparticle bound states in planar quantum electrodynamics models. In particular, we show the existence of an isolated two-quasiparticle bound state if the effective coupling constant $C in (0,2)$ in case of zero angular momentum. To the best of our knowledge, this result has not yet been addressed in the literature. Additionally, we obtain an explicit estimate for the energy gap of two-quasiparticle bound states which might be applied to high-$T_c$ $s$-wave Cooper-type superconductors as well as to $s$-wave electron-polaron--electron-polaron bound states (bipolarons) in mass-gap graphene systems.
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The Brown-Ravenhall operator was initially proposed as an alternative to describe the fermion-fermion interaction via Coulomb potential and subject to relativity. This operator is defined in terms of the associated Dirac operator and the projection onto the positive spectral subspace of the free Dirac operator. In this paper, we propose to analyze a modified version of the Brown-Ravenhall operator in two-dimensions. More specifically, we consider the Brown-Ravenhall operator with an attractive potential given by a Bessel-Macdonald function (also known as $K_0$-potential) using the Foldy-Wouthuysen unitary transformation. The $K_0$-potential is derived of the parity-preserving ${rm QED}_3$ model as a framework for evaluation of the fermion-fermion interaction potential. We prove that the two-dimensional Brown-Ravenhall operator with $K_0$-potential is bounded from below when the coupling constant is below a specified critical value (a property also referred to as stability). As by product, it is shown that the operator is in fact positive. We also investigate the location and nature of the spectrum of the Brown-Ravenhall operator with $K_0$-potential.
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