Semiclassical pseudodifferential operators with discontinuous symbols and their applications to the problems of statistical physics.(English)Zbl 1142.35109

Birman, M. S. (ed.) et al., Nonlinear equations and spectral theory. Dedicated to the memory of Olga Aleksandrovna Ladyzhenskaya. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4209-6/hbk). Translations. Series 2. American Mathematical Society 220. Advances in the Mathematical Sciences 59, 45-81 (2007).
The paper is devoted to the study of the large time asymptotic behavior of solutions of some well-known equations that can be explicitly solved using spectral methods that allow to reduce nonlinear equations to certain linear ones. This can be formulated in terms of the appropriate matrix Riemann problem for analytic functions, that can be reduced to a scalar (regular or singular) integral equation. The symbol of the corresponding integral operator turns out to be discontinuous.
In previous papers the authors of this article used an approach to study the integral equation that consists of two stages. In the first stage the support of the symbol is represented as a set of disjoint subdomains such that the problems corresponding to these subdomains can be studied explicitly (usually by reduction to a scalar Riemann problem). In the second stage an appropriate form of the alternating Schwartz method is applied to combine the previous results.
The goal of this paper is to study the method to one specific problem. Let us consider the two particle correlation functions of the Bose gas with the point iteration in the dimension one. It is represented by Fredholm’s determinant $$g(t,s,\beta)= \text{det}(I-V)$$, where the operator $$V$$ has the kernel
\begin{aligned} V(x,y)&= \theta(x) i[\pi(x- y)]^{-1} [e^{it(x^2- y^2)}v_t(y)- v_t(x) e^{-it(x^2- y^2)}]\theta(y),\\ v_t(\lambda)&= i\pi^{-1} -\hskip-.9em\int_{\mathbb{R}} (\lambda-\mu)^{-1} e^{2it(\lambda^2- \mu^2)+ 2is(\lambda-\mu)} \,d\mu, \end{aligned}
and $$\theta$$ is a smooth function sufficiently rapidly decreasing at infinity and satisfying the inequality $$|\theta(x)|\leq 2^{-1/2}$$. In these formulas $$t$$ means times, $$s$$ is the space coordinate, $$\beta$$ is the reduced chemical potential. The authors discuss asymptotic behavior of $$g$$ as $$t\to\infty$$ under the additional condition $$0< c_1< {s\over t}< c_2<\infty$$. The result is up to asymptotically vanishing terms, the final answer is a linear function of the asymptotic parameter and its logarithm. This result is already known, but the approach does not use the specific character of the function $$\theta$$.
For the entire collection see [Zbl 1111.35001].

MSC:

 35S05 Pseudodifferential operators as generalizations of partial differential operators 82B10 Quantum equilibrium statistical mechanics (general) 35B40 Asymptotic behavior of solutions to PDEs