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Asymptotics for a singularly perturbed GUE, Painlevé III, double-confluent Heun equations, and small eigenvalues
Jianduo Yu1; Chuanzhong Li1,2; Mengkun Zhu3,4; Yang Chen4
Source PublicationJournal of Mathematical Physics


We discuss the recurrence coefficients of the three-term recurrence relation for the orthogonal polynomials with a singularly perturbed Gaussian weight 𝑤(𝑧)=|𝑧|𝛼exp(−𝑧2−𝑡/𝑧2),𝑧∈ℝ,𝑡>0,𝛼>1w(z)=|z|α⁡exp−z2−t/z2,z∈R,t>0,α>1. Based on the ladder operator approach, two auxiliary quantities are defined. We show that the auxiliary quantities and the recurrence coefficients satisfy some equations with the aid of three compatibility conditions, which will be used to derive the Riccati equations and Painlevé III. We show that the Hankel determinant has an integral representation involving a particular σ-form of Painlevé III and to calculate the asymptotics of the Hankel determinant under a suitable double scaling, i.e., n → ∞ and t → 0 such that s = (2n + 1 + λ)t is fixed, where λ is a parameter with λ ≔ (α ∓ 1)/2. The asymptotic behaviors of the Hankel determinant for large s and small s are obtained, and Dyson’s constant is recovered here. They have generalized the results in the literature [Min et al., Nucl. Phys. B 936, 169–188 (2018)] where α = 0. By combining the Coulomb fluid method with the orthogonality principle, we obtain the asymptotic expansions of the recurrence coefficients, which are applied to derive the relationship between second order differential equations satisfied by our monic orthogonal polynomials and the double-confluent Heun equations as well as to calculate the smallest eigenvalue of the large Hankel matrices generated by the above weight. In particular, when α = t = 0, the asymptotic behavior of the smallest eigenvalue for the classical Gaussian weight exp(−z2) [Szegö, Trans. Am. Math. Soc. 40, 450–461 (1936)] is recovered.

Indexed BySCIE
WOS Research AreaPhysics
WOS SubjectPhysics, Mathematical
WOS IDWOS:000814154400001
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Document TypeJournal article
CollectionFaculty of Science and Technology
Corresponding AuthorYang Chen
Affiliation1.School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China
2.College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
3.School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China
4.Department of Mathematics, Faculty of Science and Technology, University of Macau, Avenida da Universidade, Taipa, Macau, China
Corresponding Author AffilicationFaculty of Science and Technology
Recommended Citation
GB/T 7714
Jianduo Yu,Chuanzhong Li,Mengkun Zhu,et al. Asymptotics for a singularly perturbed GUE, Painlevé III, double-confluent Heun equations, and small eigenvalues[J]. Journal of Mathematical Physics,2022,63.
APA Jianduo Yu,Chuanzhong Li,Mengkun Zhu,&Yang Chen.(2022).Asymptotics for a singularly perturbed GUE, Painlevé III, double-confluent Heun equations, and small eigenvalues.Journal of Mathematical Physics,63.
MLA Jianduo Yu,et al."Asymptotics for a singularly perturbed GUE, Painlevé III, double-confluent Heun equations, and small eigenvalues".Journal of Mathematical Physics 63(2022).
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