Abstract
"In this paper the authors consider the Heisenberg Laplacian in a domain Ω⊂ℝ3 with Dirichlet boundary conditions, formally given by
A(Ω)=−X21−X22,
where
X1=∂x1+x22∂x3,X2=∂x2−x12∂x3.
The main result of the paper is a uniform upper bound with remainder of the quantity
Tr(A(Ω)−λ)−,
that is, the sum of all eigenvalues of A(Ω) smaller than λ, counted according to their multiplicities.
Previous and optimal results on the leading term were known from A. M. Hansson and A. Laptev, in Groups and analysis, 100–115, London Math. Soc. Lecture Note Ser., 354, Cambridge Univ. Press, Cambridge, 2008; MR2528463, and improved estimates were obtained in H. Kovařík and T. Weidl, Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), no. 1, 145–160; MR3304579, where it was proved that for any bounded domain Ω⊂ℝ3 there exists a constant C(Ω)>0 such that
Tr(A(Ω)−λ)−≤max0,|Ω|96λ3−C(Ω)λ2.
In this paper, the authors improve the above estimate for cylindrical domains of the form Ω=ω×(a,b), where ω⊂ℝ2 is an open, simply connected, bounded set. Their main result (Theorem 2.3) is an estimate of the form
Tr(A(Ω)−λ)−≤max0,|Ω|96λ3−D(Ω)λ(2c+5)/(c+2),(1)
where c is the best Hardy constant for ω, and the constant D(Ω) depends explicitly on the cylindrical domain Ω. Notice that the correction term in (1) is of order larger than λ2.
For cylinders Ω=ω×(a,b) with convex cross-section ω, the above estimate reads:
Tr(A(Ω)−λ)−≤max0,|Ω|96λ3−λ2+1/427⋅35/2|Ω|R(ω)3/2,
where R(ω) is the Euclidean in-radius of ω.
The main techniques employed are the relation of A(Ω) with the magnetic Laplacian (with constant magnetic field) and Hardy inequalities."
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