The Brown–Ravenhall operator was initially proposed as an alternative to describe the fermion–fermion interaction via Coulomb potential and subject relativity. This is defined in terms of associated Dirac projection onto positive spectral subspace free operator. In this paper, we propose analyze a modified version two-dimensions. More specifically, consider with short-range attractive given by ...

In their ground-breaking work [42], D. Jerison and C. Kenig have studied the well-posedness of the Poisson problem for the Dirichlet Laplacian on Besov and Bessel potential spaces, ∆u = f ∈ B α (Ω), u ∈ B α+2(Ω), Tru = 0 on ∂Ω, (1.1) ∆u = f ∈ Lα(Ω), u ∈ Lpα+2(Ω), Tru = 0 on ∂Ω, (1.2) in a bounded Lipschitz domain Ω ⊂ R. Let GD be the Green operator associated with the Dirichlet Laplacian in Ω ⊂...

I describe a mechanism to understand the relation between chiral-symmetry breaking and eigenmodes of the Dirac operator in lattice QCD with Kogut-Susskind sea quarks. It can be shown that if chiral symmetry is spontaneously broken, the eigenvalues λi should behave as z(i)/V for large volume V , where z(i) is the i-th zero of the Bessel function. With neither chiral nor λ extrapolation, one can ...

Abstract. In the case of the heat equation ut = uxx + V u on the real line there are some remarkable potentials V for which the asymptotic expansion of the fundamental solution becomes a finite sum and gives an exact formula. We show that a similar phenomenon holds when one replaces the real line by the integers. In this case the second derivative is replaced by the second difference operator L...

The minimal representation π of the indefinite orthogonal group O(m+ 1, 2) is realized on the Hilbert space of square integrable functions on R with respect to the measure |x|dx1 · · ·dxm. This article gives an explicit integral formula for the holomorphic extension of π to a holomorphic semigroup of O(m + 3,C) by means of the Bessel function. Taking its ‘boundary value’, we also find the integ...

We prove the null controllability of a one-dimensional degenerate parabolic equation with drift and singular potential. study case when potential arises at left endpoint weighted Dirichlet boundary control is located this point. get spectral decomposition suitable operator, defined in Sobolev space, involving Bessel functions their zeros, then we use moment method by Fattorini Russell to obtain...