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We clarify the algebraic structure of continuous and discrete quasi-exactly solvable spectral problems by embedding them in the framework of the quantum inverse scattering method. The quasi-exactly solvable hamiltonians in one dimension are identified with traces of quantum monodromy matrices for specific integrable systems with nonperiodic boundary conditions. Applications to the Azbel-Hofstadter problem are outlined.

The motion of layered shell systems (in which there may be gaps) associated with the instantaneous imput of energy into the outer layers is considered in the hydrodynamic approximation. A sequence of transitions describing the principal laws of motion of the system as it converges on the center is constructed. Relations that permit an appropriate choice of initial data for the purpose of optimizing the cumulation parameters are obtained. The results of the analytic constructions are compared with direct numerical calculations of model targets for thermonuclear synthesis.

We establish a previously conjectured connection betweenp-adics and quantum groups. We find in Sklyanin's two parameter elliptic quantum algebra and its generalizations, the conceptual basis for the Macdonald polynomials, which “interpolate” between the zonal spherical functions of related real andp-adic symmetric spaces. The elliptic quantum algebras underlie theZn-Baxter models. We show that in then→∞ limit, the Jost function for the scattering offirst level excitations in the 1+1 dimensional field theory model associated to theZn-Baxter model coincides with the Harish-Chandra-likec-function constructed from the Macdonald polynomials associated to the root systemA1. The partition function of theZ2-Baxter model itself is also expressed in terms of this Macdonald-Harish-Chandrac-function, albeit in a less simple way. We relate the two parametersq andt of the Macdonald polynomials to the anisotropy and modular parameters of the Baxter model. In particular thep-adic “regimes” in the Macdonald polynomials correspond to a discrete sequence of XXZ models. We also discuss the possibility of “q-deforming” Euler products.

The spectral problem for theq-Knizhnik-Zamolodchikov equations for at arbitrary non-negative levelk is considered. The case of two-point functions in the fundamental representation is studied in detail. The scattering states are given explicitly in terms of continuousq-Jacobi polynomials, and theS-matrix is derived from their asymptotic behavior. The level zeroS-matrix is closely connected with the kink-antikinkS-matrix for the spin- XXZ antiferromagnet. An interpretation of the latter in terms of scattering on (quantum) symmetric spaces is discussed. In the limit of infinite level we observe connections with harmonic analysis onp-adic groups with the primep given byp=q−2.

It is shown that the Bruhat-Tits tree for thep-adic linear groupGL(2) is a natural non-archimedean analog of the open string world sheet. The boundary of the tree can be identified with the field ofp-adic numbers. We construct a “lattice” quantum field theory on the Bruhat-Tits tree with a simple local lagrangian and show that it leads to the Freund-Olson amplitudes for emission processes of the particle states from the boundary.

We treat the openp-adic string world sheet as a coset spaceF=T/G, whereT is the Bruhat-Tits tree for thep-adic linear groupGL(2, Qp) and G?PGL(2, Qp) is some Schottky group. The boundary of this world sheet corresponds to ap-adic Mumford curve of finite genus. The string dynamics is governed by the local gaussian action on the coset spaceF. The tachyon amplitudes expressed in terms ofp-adic ?-functions are proposed for the Mumford curve of arbitrary genus. We compare them with the corresponding usual archimedean amplitudes. The sum over moduli space of the algebraic curves is conjectured to be expressed in the arithmetic surface terms. We also give the necessary mathematical background including the Mumford approach top-adic algebraic curves. The connection of the problem of closedp-adic strings with the considered topics is discussed.