Affiliated with RC | false |
Status | 即將出版Forthcoming |
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 | |
2022-06-21 | |
Source Publication | Journal of Mathematical Physics |
ISSN | 0022-2488 |
Volume | 63 |
Abstract | ABSTRACTWe 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|αexp−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(−z^{2}) [Szegö, Trans. Am. Math. Soc. 40, 450–461 (1936)] is recovered. |
DOI | 10.1063/5.0062949 |
Indexed By | SCIE |
Language | 英語English |
WOS Research Area | Physics |
WOS Subject | Physics, Mathematical |
WOS ID | WOS:000814154400001 |
Fulltext Access | |
Citation statistics | |
Document Type | Journal article |
Collection | Faculty of Science and Technology DEPARTMENT OF MATHEMATICS |
Corresponding Author | Yang Chen |
Affiliation | 1.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 Affilication | Faculty 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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