The Heat Kernel Coefficients to the Matrix Schr\"odinger Operator

The heat kernel coefficients $H_k$ to the Schr\"odinger operator with a matrix potential are investigated. We present algorithms and explicit expressions for the Taylor coefficients of the $H_k$. Special terms are discussed, and for the one-dimensional case some improved algorithms are derived.Comment: 16 pages, Plain TeX, 33 KB, no figure

The inverse resonance problem for perturbations of algebro-geometric potentials

We prove that a compactly supported perturbation of a rational or simply periodic algebro-geometric potential of the one-dimensional Schr\"odinger equation on the half line is uniquely determined by the location of its Dirichlet eigenvalues and resonances.Comment: 14 page

On Darboux-Treibich-Verdier potentials

It is shown that the four-parameter family of elliptic functions $$u_D(z)=m_0(m_0+1)\wp(z)+\sum_{i=1}^3 m_i(m_i+1)\wp(z-\omega_i)$$ introduced by Darboux and rediscovered a hundred years later by Treibich and Verdier, is the most general meromorphic family containing infinitely many finite-gap potentials.Comment: 8 page

An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy

We develop an alternative systematic approach to the AKNS hierarchy based on elementary algebraic methods. In particular, we recursively construct Lax pairs for the entire AKNS hierarchy by introducing a fundamental polynomial formalism and establish the basic algebro-geometric setting including associated Burchnall-Chaundy curves, Baker-Akhiezer functions, trace formulas, Dubrovin-type equations for analogs of Dirichlet and Neumann divisors, and theta function representations for algebro-geometric solutions.Comment: LaTeX, submitted to Reviews in Mathematical Physic

Algebras generated by two bounded holomorphic functions

We study the closure in the Hardy space or the disk algebra of algebras generated by two bounded functions, of which one is a finite Blaschke product. We give necessary and sufficient conditions for density or finite codimension of such algebras. The conditions are expressed in terms of the inner part of a function which is explicitly derived from each pair of generators. Our results are based on identifying z-invariant subspaces included in the closure of the algebra. Versions of these results for the case of the disk algebra are given.Comment: 22 pages ; a number of minor mistakes have been corrected, and some points clarified. Conditionally accepted by Journal d'Analyse Mathematiqu

Superintegrability with third order invariants in quantum and classical mechanics

We consider here the coexistence of first- and third-order integrals of motion in two dimensional classical and quantum mechanics. We find explicitly all potentials that admit such integrals, and all their integrals. Quantum superintegrable systems are found that have no classical analog, i.e. the potentials are proportional to \hbar^2, so their classical limit is free motion.Comment: 15 page

Lax matrices for Yang-Baxter maps

It is shown that for a certain class of Yang-Baxter maps (or set-theoretical solutions to the quantum Yang-Baxter equation) the Lax representation can be derived straight from the map itself. A similar phenomenon for 3D consistent equations on quad-graphs has been recently discovered by A. Bobenko and one of the authors, and by F. Nijhoff

Dispersionful analogues of Benney's equations and NN-wave systems

We recall Krichever's construction of additional flows to Benney's hierarchy, attached to poles at finite distance of the Lax operator. Then we construct a ``dispersionful'' analogue of this hierarchy, in which the role of poles at finite distance is played by Miura fields. We connect this hierarchy with $N$-wave systems, and prove several facts about the latter (Lax representation, Chern-Simons-type Lagrangian, connection with Liouville equation, $\tau$-functions).Comment: 12 pages, latex, no figure

Quantum tops as examples of commuting differential operators

We study the quantum analogs of tops on Lie algebras $so(4)$ and $e(3)$ represented by differential operators.Comment: 24 p