Three formulas for eigenfunctions of integrable Schrödinger operators

We give three formulas for meromorphic eigenfunctions (scatteringstates) of Sutherlandsintegrable N-body Schrödinger operators and their generalizations.The first is an explicit computation of the Etingof–Kirillov tracesof intertwining operators, the second an integral representationof hypergeometric type, and the third is a formula of Bethe ansatz type.The last two formulas are degenerations of elliptic formulasobtained previously in connection with theKnizhnik–Zamolodchikov–Bernardequation. The Bethe ansatz formulas in the elliptic case are reviewed and discussed in more detail here: Eigenfunctionsare parametrized by a Hermite–Bethe variety, a generalizationof the spectral variety of the Lamé operator.We also give the q-deformed version of ourfirst formula. In the scalar slN case, this gives common eigenfunctionsof the commuting Macdonald–Rujsenaars difference operators.     

Complete (1)H NMR spectral fingerprint of huperzine A

Complete analysis of the (1)H NMR spectrum of huperzine A, 1-amino-13-ethylidene-11-methyl-6-aza-tricyclo[7.3.1.0(2, 7)]trideca-2(7),3,10-trien-5-one, a Lycopodium alkaloid and anti-Alzheimer drug lead containing an ABCD(E)(MN)(OP)X(3)Y(3)-type system of 15 nonexchangeable proton spins, is reported for the first time, and earlier assignments are corrected. The complete (1)H parameter set of 11 chemical shifts clarifies the diastereotopism of both methylene groups, and provides a total of 38 observed H,H-couplings including 31 long-range ((4-6)J) connectivities. The NMR data is consistent with the comparatively rigid alicyclic backbone predicted by molecular mechanics calculations, and forms the basis for (1)H NMR fingerprint analysis for the purpose of dereplication, purity analysis, and elucidation of structural analogs.     

A scheme related to the Brauer loop model

We introduce the Brauer loop scheme, where • is a certain degeneration of the ordinary matrix product. Its components of top dimension, ⌊N²/2⌋, correspond to involutions π∈SN having one or no fixed points. In the case N even, this scheme contains the upper-upper scheme from [A. Knutson, Some schemes related to the commuting variety, J. Algebraic Geom., in press, math.AG/0306275] as a union of (N/2)! of its components. One of those is a degeneration of the commuting variety of pairs of commuting matrices.The Brauer loop model is an integrable stochastic process studied in [J. de Gier, B. Nienhuis, Brauer loops and the commuting variety, J. Stat. Mech. (2005) P01006, math.AG/0410392], based on earlier related work in [M.J. Martins, B. Nienhuis, R. Rietman, An intersecting loop model as a solvable super spin chain, Phys. Rev. Lett. 81 (1998) 504-507, cond-mat/9709051], and some of the entries of its Perron-Frobenius eigenvector were observed (conjecturally) to equal the degrees of the components of the upper-upper scheme.Our proof of this equality follows the program outlined in [P. Di Francesco, P. Zinn-Justin, Inhomogeneous model of crossing loops and multidegrees of some algebraic varieties, math-ph/0412031]. In that paper, the entries of the Perron-Frobenius eigenvector were generalized from numbers to polynomials, which allowed them to be calculated inductively using divided difference operators. We relate these polynomials to the multidegrees of the components of the Brauer loop scheme, defined using an evident torus action on E. As a consequence, we obtain a formula for the degree of the commuting variety, previously calculated up to 4×4 matrices.     

SESAME-HSQC for simultaneous measurement of NH and CH scalar and residual dipolar couplings

We present a novel pulse sequence, SESAME-HSQC, for the simultaneous measurement of several NH and CH scalar and residual dipolar couplings in double labeled proteins. The proposed Spin-statE Selective All Multiplicity Edited (SESAME)-HSQC combines gradient selected and sensitivity enhanced (15)N- and constant-time (13)C-HSQC experiments with the recently introduced spin-state selective method (Nolis et al., J. Magn. Reson. 180 (2006) 39-50) for measuring couplings simultaneously at amide and aliphatic regions. Excellent resolution and high sensitivity is warranted by removing all coupling interactions during the indirectly detected t(1) period, and by employing pulsed field gradients for coherence selection and utilizing coherence order selective spin-state selection. The scalar and residual dipolar couplings can be readily measured from a two-dimensional (15)N/(13)C-HSQC spectrum without additional spectral crowding. SESAME-HSQC can be used for epitope mapping by observing chemical shift changes in both amide and aliphatic regions. Simultaneously, potential conversion in protein conformation can be probed by analyzing changes in residual dipolar couplings induced by ligand binding. The pulse sequence is experimentally verified with a sample of (15)N/(13)C enriched human ubiquitin. The internuclear vector directions determined from the residual dipolar couplings are found to be in excellent correlation with those predicted from ubiquitin's refined solution structure.     

Surfaces of Demoulin: differential geometry, Bäcklund transformation and integrability

The surfaces of Demoulin constitute an important subclass of surfaces in projective differential geometry which arise in many seemingly unrelated geometric constructions. Analytically, they are described by a two-component system which coincides with the D₃⁽¹⁾ Toda lattice. We review some of the most important geometric properties of the Demoulin surfaces and construct a Bäcklund transformation which may be specialized to the well-known Bäcklund transformation for the Tzitzeica equation governing affine spheres in affine geometry.     

Anyons and the Elliptic Calogero–Sutherland Model

We obtain a second quantization of the elliptic Calogero–Sutherland (eCS) model by constructing a quantum field theory model of anyons on a circle and at a finite temperature. This yields a remarkable identity involving anyon correlation functions and providing an algorithm for solving of the eCS model. The eigenfunctions obtained define an elliptic generalization of the Jack polynomials.     

An Investigation of Poly(dimethylsiloxane) Chain Dynamics and Order in Its Inclusion Compound with γ‐Cyclodextrin by Fast‐MAS Solid‐State NMR Spectroscopy

The dynamics of poly(dimethylsiloxane) in its inclusion compound with γ‐cyclodextrin are elucidated using modern fast‐MAS solid‐state NMR techniques. Measurements of methyl ¹H–¹H and ¹H–¹³C dipolar coupling constants indicate that the polymer undergoes a uniform motion, rendering all methyl groups equivalent. The dynamics of the Si—C bond is characterized by either a dynamic order parameter of S = 0.72, or, assuming a stably rotating helical structure, an inclination angle of 73° relative to the rotation axis.