We explore a new connection between Seiberg-Witten theory and quantum statistical systems by relating the dual partition function of SU (2) Super Yang-Mills theory in a self-dual Ω-background to the spectral determinant of an ideal Fermi gas. We show that the spectrum of this gas is encoded in the zeroes of the Painlevé III 3 τ function. In addition we find that the Nekrasov partition function on this background can be expressed as an O(2) matrix model. Our construction arises as a four-dimensional limit of a recently proposed conjecture relating topological strings and spectral theory. In this limit, we provide a mathematical proof of the conjecture for the local P 1 × P 1 geometry.

We present the inverse scattering transform approach to the Cauchy problem on the line for the shortwave model for the Camassa-Holm equation utxx − 2ux + 2uxuxx + uuxxx = 0 in the form of an associated Riemann-Hilbert problem. This approach allows us to give a representation of the classical (smooth) solutions, describe their asymptotics as t → ∞, and describe cuspons-non-smooth soliton solutions with a cusp.

Abstract. We develop the inverse scattering transform method for the Novikov equation u t − u txx + 4u 2 u x = 3uu x u xx + u 2 u xxx considered on the line x ∈ (−∞, ∞) in the case of non-zero constant background. The approach is based on the analysis of an associated Riemann-Hilbert (RH) problem, which in this case is a 3 × 3 matrix problem. The structure of this RH problem shares many common features with the case of the Degasperis-Procesi

We consider a class of unbounded self-adjoint operators including the Hamiltonian of the Jaynes-Cummings model without the rotating-wave approximation (RWA). The corresponding operators are defined by infinite Jacobi matrices with discrete spectrum. Our purpose is to give the asymptotic behavior of large eigenvalues.

We consider quantum systems with variable but finite number of particles. For such systems we develop geometric and commutator techniques. We use these techniques to find the location of the spectrum, to prove absence of singular continuous spectrum and identify accumulation points of the discrete spectrum. The fact that the total number of particles is bounded allows us to give relatively elementary proofs of these basic results for an important class of many-body systems with non-conserved number of particles.