Semibounded symmetric operators have a distinguished self-adjoint extension, the Friedrichs extension. The eigenvalues of the Friedrichs extension are given by a variational principle that involves only the domain of the symmetric operator. Although Dirac operators describing relativistic particles are not semibounded, the Dirac operator with Coulomb potential is known to have a distinguished extension. Similarly, for Dirac-type operators on manifolds with a boundary a distinguished self-adjoint extension is characterised by the Atiyah-Patodi-Singer boundary condition. In this paper we relate these extensions to a generalisation of the Friedrichs extension to the setting of operators satisfying a gap condition. In addition we prove, in the general setting, that the eigenvalues of this extension are also given by a variational principle that involves only the domain of the symmetric operator.for k ≥ 1 are the discrete spectrum of A F in the interval (−∞, sup k≥1 λ k ), counted with multiplicities d k := # {j ≥ 1 : λ j = λ k } 2010 Mathematics Subject Classification. 49R05, 49S05, 47B25, 81Q10.

Abstract. We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are H(ζ) = U + U −1 + V + ζV −1 and, where U and V are self-adjoint Weyl operators satisfying U V = q 2 V U with q = e iπb 2 , b > 0 and ζ > 0, m, n ∈ N. We prove that H(ζ) and Hm,n are self-adjoint operators with purely discrete spectrum on L 2 (R). Using the coherent state transform we find the asymptotical behaviour for the Riesz mean j≥1 (λ − λj)+ as λ → ∞ and prove the Weyl law for the eigenvalue counting function N (λ) for these operators, which imply that their inverses are of trace class.

We prove L p → L p ′ bounds for the resolvent of the Laplace-Beltrami operator on a compact Riemannian manifold of dimension n in the endpoint case p = 2(n+1)/(n+3). It has the same behavior with respect to the spectral parameter z as its Euclidean analogue, due to Kenig-Ruiz-Sogge, provided a parabolic neighborhood of the positive half-line is removed. This is region is optimal, for instance, in the case of a sphere.

No abstract

Abstract. In this paper we approximate a Schrödinger operator on L 2 (R) by Jacobi operators on ℓ 2 (Z) to provide new proofs of sharp Lieb-Thirring inequalities for the powers γ = 1 2 and γ = 3 2 . To this end we first investigate spectral inequalities for Jacobi operators. Using the commutation method we present a new, direct proof of a sharp inequality corresponding to a LiebThirring inequality for the power 3 2 on ℓ 2 (Z). We also introduce inequalities for higher powers of the eigenvalues as well as for matrix-valued potentials and compare our results to previously established bounds. Consider the Schrödinger operatorHere and below, a ± = (|a| ± a)/2 denotes the positive/negative part of a variable, a function or a self-adjoint operator. Subsequently (1) 2 ) and a Weyl-type asymptotic result for the left-hand-side of (1) was also shown to hold in the case γ ≥ 3 2 , d ≥ 1 by Laptev and Weidl [22]. The authors first established Buslaev-Faddeev-Zakharov trace formulae in the more general case of H being defined on L 2 (R, C m ), where the potential is a Hermitian matrix-valued function V : R → C m×m . This yielded sharp Lieb-Thirring inequalities in one dimension for matrix-valued potentials and γ ≥ 32 . An induction argument making use of the matrix nature of the results lifted the inequalities to higher dimensions.For γ = 1 2 , d = 1 Hundertmark, Lieb and Thomas [18] proved that the best constant is given by L 1/2,1 = 2L cl 1/2,1 = 1 2 . The inequality is then sharp for delta potentials. In [12] it was shown that for 1 ≤ γ < 3 2 the sharp constant can be bounded by L γ,1 ≤ π √ 3 L cl γ,1 , which was generalised to higher dimensions in [11].In our paper we provide a new proof for sharp Lieb-Thirring bounds in one dimension for γ = 1 2 and γ = 3 2 using an approximation of the Schrödinger operator by Jacobi operators on the lattice.