Surface instability of laminated bodies of regular structure under combination loading

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 27, No. 5, pp. 20–27, May, 1991.     

Topological expansion of the <Emphasis Type="Italic">β</Emphasis>-ensemble model and quantum algebraic geometry in the sectorwise approach

We construct the solution of the loop equations of the β-ensemble model in a form analogous to the solution in the case of the Hermitian matrices β = 1. The solution for β = 1 is expressed in terms of the algebraic spectral curve given by y² = U(x). The spectral curve for arbitrary β converts into the Schrödinger equation (??)² ? U(x) ψ(x) = 0, where ? ∝ \({{\left( {{{\sqrt \beta - 1} \mathord{\left/ {\vphantom {{\sqrt \beta - 1} {\sqrt \beta }}} \right. \kern-\nulldelimiterspace} {\sqrt \beta }}} \right)} \mathord{\left/ {\vphantom {{\left( {{{\sqrt \beta - 1} \mathord{\left/ {\vphantom {{\sqrt \beta - 1} {\sqrt \beta }}} \right. \kern-\nulldelimiterspace} {\sqrt \beta }}} \right)} N}} \right. \kern-\nulldelimiterspace} N}\). The basic ingredients of the method based on the algebraic solution retain their meaning, but we use an alternative approach to construct a solution of the loop equations in which the resolvents are given separately in each sector. Although this approach turns out to be more involved technically, it allows consistently defining the B-cycle structure for constructing the quantum algebraic curve (a D-module of the form y² ? U(x), where [y, x] = ?) and explicitly writing the correlation functions and the corresponding symplectic invariants Fh or the terms of the free energy in an 1/N²-expansion at arbitrary ?. The set of “flat” coordinates includes the potential times t_k and the occupation numbers \(\tilde \varepsilon _\alpha \). We define and investigate the properties of the A- and B-cycles, forms of the first, second, and third kinds, and the Riemann bilinear identities. These identities allow finding the singular part of \(\mathcal{F}_0 \), which depends only on \(\tilde \varepsilon _\alpha \).