Hadron Systematics and Emergent Diquarks

We briefly review a variety of theoretical and phenomenological indications for the probable importance of powerful diquark correlations in hadronic physics. We demonstrate that the bulk of light hadron spectroscopy can be organized using three simple hypotheses: Regge-Chew-Frautschi mass formulae, feebleness of spin-orbit forces, and energetic distinctions among a few different diquark configurations. Those hypotheses can be implemented in a semi-classical model of color flux tubes, extrapolated down from large orbital angular momentum $L$. We discuss refinements of the model to include the effects of tunneling, mass loading, and internal excitations. We also discern effects of diquark correlations in observed patterns of baryon decays. Many predictions and suggestions for further work appear.Comment: 18 pages, 11 figures. Talk by FW at a workshop at Schloss Ringberg, October 2005. To appear in the Proceeding

A note on the Bethe ansatz solution of the supersymmetric t-J model

The three different sets of Bethe ansatz equations describing the Bethe ansatz solution of the supersymmetric t-J model are known to be equivalent. Here we give a new, simplified proof of this fact which relies on the properties of certain polynomials. We also show that the corresponding transfer matrix eigenvalues agree.Comment: 6 pages, Latex, contributed to the 12th Int. Colloquium on Quantum Groups and Integrable Systems, Prague, 200

The spectral density of a product of spectral projections

We consider the product of spectral projections $$\Pi_\epsilon(\lambda) = 1_{(-\infty,\lambda-\epsilon)}(H_0) 1_{(\lambda+\epsilon,\infty)}(H) 1_{(-\infty,\lambda-\epsilon)}(H_0)$$ where $H_0$ and $H$ are the free and the perturbed Schr\"odinger operators with a short range potential, $\lambda>0$ is fixed and $\epsilon\to0$. We compute the leading term of the asymptotics of $\mathrm{Tr}\ f(\Pi_\epsilon(\lambda))$ as $\epsilon\to0$ for continuous functions $f$ vanishing sufficiently fast near zero. Our construction elucidates calculations that appeared earlier in the theory of "Anderson's orthogonality catastrophe" and emphasizes the role of Hankel operators in this phenomenon.Comment: 23 pages; minor revision

Kato smoothness and functions of perturbed self-adjoint operators

We consider the difference $f(H_1)-f(H_0)$ for self-adjoint operators $H_0$ and $H_1$ acting in a Hilbert space. We establish a new class of estimates for the operator norm and the Schatten class norms of this difference. Our estimates utilise ideas of scattering theory and involve conditions on $H_0$ and $H_1$ in terms of the Kato smoothness. They allow for a much wider class of functions $f$ (including some unbounded ones) than previously available results do. As an important technical tool, we propose a new notion of Schatten class valued smoothness and develop a new framework for double operator integrals

K. Saito's Conjecture for Nonnegative Eta Products and Analogous Results for Other Infinite Products

We prove that the Fourier coefficients of a certain general eta product considered by K. Saito are nonnegative. The proof is elementary and depends on a multidimensional theta function identity. The z=1 case is an identity for the generating function for p-cores due to Klyachko [17] and Garvan, Kim and Stanton [10]. A number of other infinite products are shown to have nonnegative coefficients. In the process a new generalization of the quintuple product identity is derived.Comment: 15 pages; greatly expanded version of the earlier 8 page paper math.NT/060760