Semiclassical resolvent bounds for compactly supported radial potentials

We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schrödinger operator −h2Δ+V(|x|)−E in dimension n≥2, where h,E>0, and V:[0,∞)→ℝ is L∞ and compactly supported. We show that the weighted resolvent estimate grows no faster than exp(Ch−1), and prove an exterior weighted estimate which grows ∼h−1

Perfectly-matched-layer truncation is exponentially accurate at high frequency

We consider a wide variety of scattering problems including scattering by Dirichlet, Neumann, and penetrable obstacles. We consider a radial perfectly-matched layer (PML) and show that for any PML width and a steep-enough scaling angle, the PML solution is exponentially close, both in frequency and the tangent of the scaling angle, to the true scattering solution. Moreover, for a fixed scaling angle and large enough PML width, the PML solution is exponentially close to the true scattering solution in both frequency and the PML width. In fact, the exponential bound holds with rate of decay c (omicrontanθ − C)k where omicron is the PML width and θ is the scaling angle. More generally, the results of the paper hold in the framework of black-box scattering under the assumption of an exponential bound on the norm of the cutoff resolvent, thus including problems with strong trapping. These are the first results on the exponential accuracy of PML at high-frequency with non-trivial scatterers

Site-selective measurement of coupled spin pairs in an organic semiconductor.

From organic electronics to biological systems, understanding the role of intermolecular interactions between spin pairs is a key challenge. Here we show how such pairs can be selectively addressed with combined spin and optical sensitivity. We demonstrate this for bound pairs of spin-triplet excitations formed by singlet fission, with direct applicability across a wide range of synthetic and biological systems. We show that the site sensitivity of exchange coupling allows distinct triplet pairs to be resonantly addressed at different magnetic fields, tuning them between optically bright singlet ([Formula: see text]) and dark triplet quintet ([Formula: see text]) configurations: This induces narrow holes in a broad optical emission spectrum, uncovering exchange-specific luminescence. Using fields up to 60 T, we identify three distinct triplet-pair sites, with exchange couplings varying over an order of magnitude (0.3-5 meV), each with its own luminescence spectrum, coexisting in a single material. Our results reveal how site selectivity can be achieved for organic spin pairs in a broad range of systems.This work was supported by HFMLRU/ FOM and LNCMI-CNRS, members of the European Magnetic Field Laboratory (EMFL) and by EPSRC (UK) via its membership to the EMFL (grant no. EP/N01085X/1 and NS/A000060/1) and through grant no. EP/M005143/1. L.R.W. acknowledges support of the Gates-Cambridge and Winton Scholarships. We acknowledge support from Labex ANR-10-LABX-0039-PALM, ANR SPINEX, and DFG SPP-1601 (Bi-464/10-2)

The Helmholtz boundary element method does not suffer from the pollution effect

In d dimensions, approximating an arbitrary function oscillating with frequency ≲k requires ∼kd degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber k and in d dimensions) suffers from the pollution effect if, as k→∞, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold (i.e., faster than kd for domain-based formulations, such as finite element methods, and kd−1 for boundary-based formulations, such as boundary element methods). It is well known that the h-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth h and keeping the polynomial degree p fixed) suffers from the pollution effect, and research over the last ∼ 30 years has resulted in a near-complete rigorous understanding of how quickly the number of degrees of freedom must grow with k (and how this depends on both p and properties of the scatterer). In contrast to the h-FEM, at least empirically, the h-version of the boundary element method (BEM) does not suffer from the pollution effect (recall that in the boundary element method the scattering problem is reformulated as an integral equation on the boundary of the scatterer, with this integral equation then solved numerically using a finite-element-type approximation space). However, the current best results in the literature on how quickly the number of degrees of freedom for the h-BEM must grow with k fall short of proving this. In this paper, we prove that the h-version of the Galerkin method applied to the standard second-kind boundary integral equations for solving the Helmholtz exterior Dirichlet problem does not suffer from the pollution effect when the obstacle is nontrapping (i.e., does not trap geometric-optic rays)

Site-selective measurement of coupled spin pairs in an organic semiconductor

From organic electronics to biological systems, understanding the role of intermolecular interactions between spin pairs is a key challenge. Here we show how such pairs can be selectively addressed with combined spin and optical sensitivity. We demonstrate this for bound pairs of spin-triplet excitations formed by singlet fission, with direct applicability across a wide range of synthetic and biological systems. We show that the site sensitivity of exchange coupling allows distinct triplet pairs to be resonantly addressed at different magnetic fields, tuning them between optically bright singlet (S = 0) and dark triplet quintet (S = 1, 2) configurations: This induces narrow holes in a broad optical emission spectrum, uncovering exchange-specific luminescence. Using fields up to 60 T, we identify three distinct triplet-pair sites, with exchange couplings varying over an order of magnitude (0.3-5 meV), each with its own luminescence spectrum, coexisting in a single material. Our results reveal how site selectivity can be achieved for organic spin pairs in a broad range of systems