Asymptotic of a class of operator determinants with application to the cylindrical Toda equations. (English) Zbl 1167.47026
Baik, Jinho (ed.) et al., Integrable systems and random matrices. In honor of Percy Deift. Conference on integrable systems, random matrices, and applications in honor of Percy Deift’s 60th birthday, New York, NY, USA, May 22–26, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4240-9/pbk). Contemporary Mathematics 458, 31-53 (2008).
The paper is a continuation of results by the author and C.A.Tracy [Commun.Math.Phys.190, No.3, 697–721 (1998; Zbl 0907.35125)] on the asymptotics of certain solutions to the cylindrical Toda equations
\[ q_k^{\prime \prime}(t)+t^{-1}q_k^{\prime}(t)=4(\exp\{ q_k(t)-q_{k-1}(t)\} -\exp\{q_{k+1}(t)-q_k(t)\}), \]
where \(k\) runs through \(\mathbb{Z}\). These solutions are expressed in terms of determinants \(\det(I+K_k(t))\), where \(K_k(t)\) denotes certain integral operators acting on \(L^2(\mathbb{R}^+)\). Generalizations to a wider class of functions arising as operator determinants have been given in [H.Widom, Operator Theory: Advances and Applications 170, 249–256 (2006; Zbl 1123.47027)]. So far, all results were restricted to the case where the symbol of a convolution operator associated to \(K_0(t)\) does not vanish and has zero index. This condition is what the author calls the regular case and it leads to asymptotics of the form \(\det(I+K_0(t))\sim bt^a\) as \(t\rightarrow 0+\). Both constants \(a\) and \(b\) can be expressed by integral formulas.
In the present paper, singular cases are also treated. Here, the symbol has a double zero at a single point and an additional logarithmic factor appears in the asymptotic expansion: \(\det(I+K_0(t))\sim bt^a\log t^{-1}\). These results are specialized to the most interesting case of the \(n\)-periodic Toda kernels in that \(q_{k+n}=q_k\) for all \(k\). Then \(a\) and \(b\) can be calculated more explicitly in terms of the zeros of an associated function and Barnes \(G\)-function for both the regular and the singular case.
The first part of the paper reviews the regular case in a more general setting than before. As a main tool for calculating a formula for the small time asymptotics of \(\det(I+K_k(t))\), the Kac–Achieser theorem together with a clever matrix decompositions of the operators is used. Then all expressions appearing in this formula can be obtained in a more explicit way at least in the \(n\)-periodic case.
In the second part, the singular situation is analyzed. The proofs follow the same line as in the regular case, using approximations and the theory of Wiener–Hopf operators. However, a careful analysis is needed, the assumptions on the integral operators have to be strengthened, and the \(L^2\)-spaces before have to be replaced by weighted \(L^2\)-spaces.
As an application, the asymptotics of a solutions to the cylindrical \(\sinh\)-Gordon equation and the Bullough–Dodd equation are discussed in the end of the paper.
For the entire collection see [Zbl 1139.37001].
\[ q_k^{\prime \prime}(t)+t^{-1}q_k^{\prime}(t)=4(\exp\{ q_k(t)-q_{k-1}(t)\} -\exp\{q_{k+1}(t)-q_k(t)\}), \]
where \(k\) runs through \(\mathbb{Z}\). These solutions are expressed in terms of determinants \(\det(I+K_k(t))\), where \(K_k(t)\) denotes certain integral operators acting on \(L^2(\mathbb{R}^+)\). Generalizations to a wider class of functions arising as operator determinants have been given in [H.Widom, Operator Theory: Advances and Applications 170, 249–256 (2006; Zbl 1123.47027)]. So far, all results were restricted to the case where the symbol of a convolution operator associated to \(K_0(t)\) does not vanish and has zero index. This condition is what the author calls the regular case and it leads to asymptotics of the form \(\det(I+K_0(t))\sim bt^a\) as \(t\rightarrow 0+\). Both constants \(a\) and \(b\) can be expressed by integral formulas.
In the present paper, singular cases are also treated. Here, the symbol has a double zero at a single point and an additional logarithmic factor appears in the asymptotic expansion: \(\det(I+K_0(t))\sim bt^a\log t^{-1}\). These results are specialized to the most interesting case of the \(n\)-periodic Toda kernels in that \(q_{k+n}=q_k\) for all \(k\). Then \(a\) and \(b\) can be calculated more explicitly in terms of the zeros of an associated function and Barnes \(G\)-function for both the regular and the singular case.
The first part of the paper reviews the regular case in a more general setting than before. As a main tool for calculating a formula for the small time asymptotics of \(\det(I+K_k(t))\), the Kac–Achieser theorem together with a clever matrix decompositions of the operators is used. Then all expressions appearing in this formula can be obtained in a more explicit way at least in the \(n\)-periodic case.
In the second part, the singular situation is analyzed. The proofs follow the same line as in the regular case, using approximations and the theory of Wiener–Hopf operators. However, a careful analysis is needed, the assumptions on the integral operators have to be strengthened, and the \(L^2\)-spaces before have to be replaced by weighted \(L^2\)-spaces.
As an application, the asymptotics of a solutions to the cylindrical \(\sinh\)-Gordon equation and the Bullough–Dodd equation are discussed in the end of the paper.
For the entire collection see [Zbl 1139.37001].
Reviewer: Wolfram Bauer (Greifswald)
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
