A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we study boundary value problems involving bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels in the upper-half space and the unit ball are constructed, which give us solutions to the Dirichlet problems with $L^p$ boundary data, $1 \leq p \leq \infty$. We also prove the uniqueness for solutions to the Dirichlet problems with continuous data for bosonic Laplacians and provide analogs of some properties of harmonic functions for null solutions of bosonic Laplacians, for instance, Cauchy's estimates, the mean-value property, Liouville's Theorem, etc.
A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita-Schwinger type operators and bosonic Laplacians, we solve Poisson's equation for bosonic Laplacians. A representation formula for bounde…
Elegant 'microlocal' methods have long since been extensively developed for the analysis of conventional Schroedinger eigenvalue problems. For technical reasons though these methods have not heretofore been applicable to quantum field theories. In this article however we initiate a 'Euclidean signature semi-classical' program to extend the scope of these analytical techniques to encompass the study of self-interacting scalar fields in 1 + 1, 2 + 1 and 3 + 1 dimensions. The basic microlocal approach entails, first of all, the solution of a single, nonlinear equation of Hamilton-Jacobi type foll…
We give a formula that express magnetic Berezin transforms associated with generalized Bargmann-Fock spaces as functions of the Euclidean Laplacian on Cn.
A simple formula is derived for the Ricci scalar curvature of any smooth level set ${ψ(x_0,x_1,...,x_n)=C}$ embedded in the Euclidean space $ \mathbb R^{n+1}$, in terms of the gradient $ \nablaψ$ and the Laplacian $ Δψ$. Some applications are given to the geometry of low-dimensional $p$-harmonic functions and high-dimensional harmonic functions.
For an arbitrary nonempty, open set $Ω\subset \mathbb{R}^n$, $n \in \mathbb{N}$, of finite (Euclidean) volume, we consider the minimally defined higher-order Laplacian $(- Δ)^m\big|_{C_0^{\infty}(Ω)}$, $m \in \mathbb{N}$, and its Krein--von Neumann extension $A_{K,Ω,m}$ in $L^2(Ω)$. With $N(λ,A_{K,Ω,m})$, $λ> 0$, denoting the eigenvalue counting function corresponding to the strictly positive eigenvalues of $A_{K,Ω,m}$, we derive the bound $$ N(λ,A_{K,Ω,m}) \leq (2 π)^{-n} v_n |Ω| \{1 + [2m/(2m+n)]\}^{n/(2m)} λ^{n/(2m)}, \quad λ> 0, $$ where $v_n := π^{n/2}/Γ((n+2)/2)$ denotes the (Euclidean) …
We develop a scattering theory for perturbations of powers of the Laplacian on asymptotically Euclidean manifolds. The (absolute) scattering matrix is shown to be a Fourier integral operator associated to the geodesic flow at time πon the boundary. Furthermore, it is shown that on \Real^n the asymptotics of certain short-range perturbations of Δ^k can be recovered from the scattering matrix at a finite number of energies.
This paper proves sharp lower bounds on a resonance counting function for obstacle scattering in even-dimensional Euclidean space without a need for trapping assumptions. Similar lower bounds are proved for some other compactly supported perturbations of the Laplacian on even-dimensional Euclidean space, for example, for the Laplacian for certain metric perturbations. The proof uses a Poisson formula for resonances, complementary to one proved by Zworski in even dimensions.
The Dirichlet Laplacian between two parallel hypersurfaces in Euclidean spaces of any dimension in the presence of a magnetic field is considered in the limit when the distance between the hypersurfaces tends to zero. We show that the Laplacian converges in a norm-resolvent sense to a Schroedinger operator on the limiting hypersurface whose electromagnetic potential is expressed in terms of principal curvatures and the projection of the ambient vector potential to the hypersurface. As an application, we obtain an effective approximation of bound-state energies and eigenfunctions in thin quantu…
While dealing with a class of generalized Bargmann spaces, we rederive their reproducing kernels from the knowledge of an orthonormal basis by using an addition formula for Laguerre polynomials involving the disk polynomials. We construct for each of these spaces a set of coherent states to apply a coherent states quantization method. This provides us with another way to recover the Berezin transforms attached to these spaces. Finally, two new formulae representing these transforms as functions of the Euclidean Laplacian are established and a possible physics direction for the application of s…
A metric more general than the Euclidean Schwarzschild-Tangherlini metric is considered. The cosmological constant is not necessarily Zero, and the hypersphere is replaced by an Einstein variety. A differential equation that derives from the equation of eigen-values of the Laplace operator in this metric is studied. Two cases are considered, (i) the cosmological constant is not Zero, and (ii) the cosmological constant is Zero, and the set of eigen-values is discrete. Infinity is a singular point of this differential equation. In these cases only, it is regular. Series of powers solve the diffe…
We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the hypersurfaces tends to zero. The asymptotics are uniform and local in the sense that the coefficients depend only on the extremal points where the ratio of the area of the Neumann boundary to the Dirichlet one is locally the biggest.
We prove stability bounds for local gauge-invariant scalar QCD quantum models, with multiflavored bosons replacing (anti)quarks. We take a compact, connected gauge Lie group G, and concentrate on G=U(N),SU(N). Let d(N)=N^2,(N^2-1) be their Lie algebra dimensions. We start on a finite hypercubic lattice Λ\subset aZ^d, d=2,3,4, a\in(0,1], with L sites on a side, Λ_s=L^d sites, and free boundary conditions. The action is a sum of a Bose-gauge part and a Wilson pure-gauge plaquette term. We employ a priori local, scaled scalar bosons with an a-dependent field-strength renormalization: a non-canoni…
This paper continues the work of our previous paper [8], where we generalize kth-powers of the Euclidean Dirac operator D_x to higher spin spaces in the case the target space is a degree one homogeneous polynomial space. In this paper, we reconsider the generalizations of D_x^3 and D_x^4 to higher spin spaces in the case the target space is a degree k homogeneous polynomial space. Constructions of 3rd and 4th order conformally invariant operators in higher spin spaces are given; these are the 3rd order fermionic and 4th order bosonic operators. Fundamental solutions and intertwining operators …
We study extremal properties of the determinant of Friederichs selfadjoint Laplacian on the Euclidean isosceles triangle envelopes of fixed area as a function of angles. Small-angle asymptotics show that the determinant grows without any bound as an angle of triangle envelope goes to zero. We prove that the equilateral triangle envelope (the most symmetrical geometry) always gives rise to a critical point of the determinant and find the critical value. Moreover, if the area of envelopes is not too large, then the determinant achieves its absolute minimum only on the equilateral triangle envelo…
