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G Bessis - One of the best experts on this subject based on the ideXlab platform.
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perturbed factorization of the symmetric anharmonic oscillator eigenequation
Physical Review A, 1992Co-Authors: N Bessis, G BessisAbstract:The perturbed-Ladder-Operator method is applied to the analytical solution of the harmonic-oscillator eigenequation perturbed by a symmetric potential V(x). This method, well adapted for computer algebra, is an extension of the original Schrodinger-Infeld-Hull factorization method within the perturbative scheme and allows an analytical solution of nonfactorizable Sturm-Liouville eigenequations in almost the same way as factorizable ones
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perturbed Ladder Operator method an algebraic recursive solution of the perturbed coulomb eigenequation
Physical Review A, 1991Co-Authors: N Bessis, G BessisAbstract:The perturbed Ladder-Operator method is applied to the solution of the Coulomb eigenequation perturbed by an anharmonic potential, i.e., with a total potential U(x)=-m(m+1)/${\mathit{x}}^{2}$-2q/x +${\mathit{b}}_{1}$x+${\mathit{b}}_{2}$${\mathit{x}}^{2}$+${\mathit{b}}_{3}$${\mathit{x}}^{3}$+... . This method is an extension of the Schr\"odinger-Infeld-Hull factorization method within the perturbative scheme. The introduction of specific basis functions for the finite-difference solution of the factorazibility condition, together with the use of the symmetry properties of the Bernoulli polynomials, allows a straightforward determination of analytical expressions of the perturbed Coulomb Ladder functions and eigenvalues. Some illustrative examples showing the capabilities of the method are given; particularly, analytical expressions of the linear, quadratic, and cubic Stark shifts are quickly derived.
Chen Yang - One of the best experts on this subject based on the ideXlab platform.
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Painlev\'e V for a Jacobi unitary ensemble with random singularities
2021Co-Authors: Zhu Mengkun, Li Chuanzhong, Chen YangAbstract:In this paper, we focus on the relationship between the fifth Painlev\'{e} equation and a Jacobi weight perturbed with random singularities, \begin{equation*} w(z)=\left(1-z^2\right)^{\alpha}{\rm e}^{-\frac{t}{z^2-k^2}},~~~z,k\in[-1,1],~\alpha,t>0. \end{equation*} By using the Ladder Operator approach, we obtain that an auxiliary quantity $R_n(t)$, which is closely related to the recurrence coefficients of monic polynomials orthogonal with $w(z)$, satisfies a particular Painlev\'{e} V equation.Comment: Applied Mathematics Letters,202
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Asymptotics of the Largest Eigenvalue Distribution of the Laguerre Unitary Ensemble
2020Co-Authors: Lyu Shulin, Min Chao, Chen YangAbstract:We study the probability that all the eigenvalues of $n\times n$ Hermitian matrices, from the Laguerre unitary ensemble with the weight $x^{\gamma}\mathrm{e}^{-4nx},\;x\in[0,\infty),\;\gamma>-1$, lie in the interval $[0,\alpha]$. By using previous results for finite $n$ obtained by the Ladder Operator approach of orthogonal polynomials, we derive the large $n$ asymptotics of the largest eigenvalue distribution function with $\alpha$ ranging from 0 to the soft edge. In addition, at the soft edge, we compute the constant conjectured by Tracy and Widom [Commun. Math. Phys. 159 (1994), 151-174], later proved by Deift, Its and Krasovsky [Commun. Math. Phys. 278 (2008), 643-678]. Our results are reduced to those of Deift et al. when $\gamma=0$.Comment: 18 page
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Painlev\'{e} V and the Hankel Determinant for a Singularly Perturbed Jacobi Weight
2020Co-Authors: Min Chao, Chen YangAbstract:We study the Hankel determinant generated by a singularly perturbed Jacobi weight $$ w(x,t):=(1-x^2)^\alpha\mathrm{e}^{-\frac{t}{x^{2}}},\;\;\;\;\;\;x\in[-1,1],\;\;\alpha>0,\;\;t\geq 0. $$ If $t=0$, it is reduced to the classical symmetric Jacobi weight. For $t>0$, the factor $\mathrm{e}^{-\frac{t}{x^{2}}}$ induces an infinitely strong zero at the origin. This Hankel determinant is related to the Wigner time-delay distribution in chaotic cavities. In the finite $n$ dimensional case, we obtain two auxiliary quantities $R_n(t)$ and $r_n(t)$ by using the Ladder Operator approach. We show that the Hankel determinant has an integral representation in terms of $R_n(t)$, where $R_n(t)$ is closely related to a particular Painlev\'{e} V transcendent. Furthermore, we derive a second-order nonlinear differential equation and also a second-order difference equation for the logarithmic derivative of the Hankel determinant. This quantity can be expressed in terms of the Jimbo-Miwa-Okamoto $\sigma$-function of a particular Painlev\'{e} V. Then we consider the asymptotics of the Hankel determinant under a suitable double scaling, i.e. $n\rightarrow\infty$ and $t\rightarrow 0$ such that $s=2n^2 t$ is fixed. Based on previous results by using the Coulomb fluid method, we obtain the large $s$ and small $s$ asymptotic behaviors of the scaled Hankel determinant, including the constant term in the asymptotic expansion.Comment: 28 page
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Painlev\'{e} VI, Painlev\'{e} III and the Hankel Determinant Associated with a Degenerate Jacobi Unitary Ensemble
'Wiley', 2019Co-Authors: Min Chao, Chen YangAbstract:This paper studies the Hankel determinant generated by a perturbed Jacobi weight, which is closely related to the largest and smallest eigenvalue distribution of the degenerate Jacobi unitary ensemble. By using the Ladder Operator approach for the orthogonal polynomials, we find that the logarithmic derivative of the Hankel determinant satisfies a nonlinear second-order differential equation, which turns out to be the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlev\'{e} VI equation by a translation transformation. We also show that, after a suitable double scaling, the differential equation is reduced to the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlev\'{e} III. In the end, we obtain the asymptotic behavior of the Hankel determinant as $t\rightarrow1^{-}$ and $t\rightarrow0^{+}$ in two important cases, respectively.Comment: 21 page
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A characterization theorem for semi-classical orthogonal polynomials on non-uniform lattices
2018Co-Authors: Rebocho M. N., Chen Yang, Filipuk Galina, Branquinho A.Abstract:It is proved a characterization theorem for semi-classical orthogonal polynomials on non- uniform lattices that states the equivalence between the Pearson equation for the weight and some systems involving the orthogonal polynomials as well as the functions of the second kind. As a consequence, it is deduced the analogue of the so-called compatibility conditions in the Ladder Operator scheme. The classical orthogonal polynomials on non- uniform lattices are then recovered under such compatibility conditions, through a closed formula for the recurrence relation coefficients.info:eu-repo/semantics/publishedVersio
Yang Chen - One of the best experts on this subject based on the ideXlab platform.
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a characterization theorem for semi classical orthogonal polynomials on non uniform lattices
Applied Mathematics and Computation, 2018Co-Authors: A Branquinho, M N Rebocho, Yang Chen, Galina FilipukAbstract:Abstract It is proved a characterization theorem for semi-classical orthogonal polynomials on non-uniform lattices that states the equivalence between the Pearson equation for the weight and some systems involving the orthogonal polynomials as well as the functions of the second kind. As a consequence, it is deduced the analogue of the so-called compatibility conditions in the Ladder Operator scheme. The classical orthogonal polynomials on non-uniform lattices are then recovered under such compatibility conditions, through a closed formula for the recurrence relation coefficients.
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Painlevé III′ and the Hankel determinant generated by a singularly perturbed Gaussian weight
'Elsevier BV', 2018Co-Authors: Chao Min, Shulin Lyu, Yang ChenAbstract:In this paper, we study the Hankel determinant generated by a singularly perturbed Gaussian weightw(x,t)=e−x2−tx2,x∈(−∞,∞),t>0. By using the Ladder Operator approach associated with the orthogonal polynomials, we show that the logarithmic derivative of the Hankel determinant satisfies both a non-linear second order difference equation and a non-linear second order differential equation. The Hankel determinant also admits an integral representation involving a Painlevé III′. Furthermore, we consider the asymptotics of the Hankel determinant under a double scaling, i.e. n→∞ and t→0 such that s=(2n+1)t is fixed. The asymptotic expansions of the scaled Hankel determinant for large s and small s are established, from which Dyson's constant appears
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continuous and discrete painleve equations arising from the gap probability distribution of the finite n gaussian unitary ensembles
Journal of Statistical Physics, 2014Co-Authors: Yang Chen, James GriffinAbstract:In this paper we study the gap probability problem in the Gaussian unitary ensembles of \(n\) by \(n\) matrices : The probability that the interval \(J := (-a,a)\) is free of eigenvalues. In the works of Tracy and Widom, Adler and Van Moerbeke, and Forrester and Witte on this subject, it has been shown that two Painleve type differential equations arise in this context. The first is the Jimbo–Miwa–Okomoto \(\sigma \)-form and the second is a particular Painleve IV. Using the Ladder Operator technique of orthogonal polynomials we derive three quantities associated with the gap probability, denoted as \(\sigma _n(a)\), \(R_n(a)\) and \(r_n(a)\). We show that each one satisfies a second order Painleve type differential equation as well as a discrete Painleve type equation. In particular, in addition to providing an elementary derivation of the aforementioned \(\sigma \)-form and Painleve IV we are able to show that the quantity \(r_n(a)\) satisfies a particular case of Chazy’s second degree second order differential equation. For the discrete equations we show that the quantity \(r_n(a)\) satisfies a particular form of the modified discrete Painleve II equation obtained by Grammaticos and Ramani in the context of Backlund transformations. We also derive second order second degree difference equations for the quantities \(R_n(a)\) and \(\sigma _n(a)\).
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continuous and discrete painleve equations arising from the gap probability distribution of the finite n gaussian unitary ensembles
arXiv: Classical Analysis and ODEs, 2014Co-Authors: Yang Chen, James GriffinAbstract:In this paper we study the gap probability problem in the Gaussian Unitary Ensembles of $n$ by $n$ matrices : The probability that the interval $J := (-a,a)$ is free of eigenvalues. In the works of Tracy and Widom, Adler and Van Moerbeke and Forrester and Witte on this subject, it has been shown that two Painleve type differential equations arise in this context. The first is the Jimbo-Miwa-Okomoto $\sigma-$form and the second is a particular Painleve IV. Using the Ladder Operator technique of orthogonal polynomials we derive three quantities associated with the gap probability, denoted by $\sigma_n(a)$, $R_n(a)$ and $r_n(a)$, and show that each one satisfying a second order, non-linear, differential equation as well as a second order, non-linear difference equation. In particular, in addition to providing an elementary derivation of the aforementioned $\sigma-$form and Painleve IV we show that the quantity $r_n(a)$ satisfies a particular case of Chazy's second degree second order differential equation. For the discrete equations we show that the quantity $r_n(a)$ satisfies a particular form of the modified discrete Painleve II equation obtained by Grammaticos and Ramani in the context of Backlund transformations. We also derive second order second degree difference equations for the quantities $R_n(a)$ and $\sigma_n(a)$.
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painleve v and a pollaczek jacobi type orthogonal polynomials
Journal of Approximation Theory, 2010Co-Authors: Yang Chen, Dan DaiAbstract:We study a sequence of polynomials orthogonal with respect to a one-parameter family of weights w(x)@?w(x,t)=e^-^t^/^xx^@a(1-x)^@b,t>=0, defined for x@?[0,1]. If t=0, this reduces to a shifted Jacobi weight. Our Ladder Operator formalism and the associated compatibility conditions give an easy determination of the recurrence coefficients. For t>0, the factor e^-^t^/^x induces an infinitely strong zero at x=0. With the aid of the compatibility conditions, the recurrence coefficients are expressed in terms of a set of auxiliary quantities that satisfy a system of difference equations. These, when suitably combined with a pair of Toda-like equations derived from the orthogonality principle, show that the auxiliary quantities are particular Painleve V and/or allied functions. It is also shown that the logarithmic derivative of the Hankel determinant, D"n(t)@?det(@!"0^1x^i^+^je^-^t^/^xx^@a(1-x)^@bdx)"i","j"="0^n^-^1, satisfies the Jimbo-Miwa-Okamoto @s-form of the Painleve V equation and that the same quantity satisfies a second-order non-linear difference equation which we believe to be new.
N Bessis - One of the best experts on this subject based on the ideXlab platform.
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perturbed factorization of the symmetric anharmonic oscillator eigenequation
Physical Review A, 1992Co-Authors: N Bessis, G BessisAbstract:The perturbed-Ladder-Operator method is applied to the analytical solution of the harmonic-oscillator eigenequation perturbed by a symmetric potential V(x). This method, well adapted for computer algebra, is an extension of the original Schrodinger-Infeld-Hull factorization method within the perturbative scheme and allows an analytical solution of nonfactorizable Sturm-Liouville eigenequations in almost the same way as factorizable ones
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perturbed Ladder Operator method an algebraic recursive solution of the perturbed coulomb eigenequation
Physical Review A, 1991Co-Authors: N Bessis, G BessisAbstract:The perturbed Ladder-Operator method is applied to the solution of the Coulomb eigenequation perturbed by an anharmonic potential, i.e., with a total potential U(x)=-m(m+1)/${\mathit{x}}^{2}$-2q/x +${\mathit{b}}_{1}$x+${\mathit{b}}_{2}$${\mathit{x}}^{2}$+${\mathit{b}}_{3}$${\mathit{x}}^{3}$+... . This method is an extension of the Schr\"odinger-Infeld-Hull factorization method within the perturbative scheme. The introduction of specific basis functions for the finite-difference solution of the factorazibility condition, together with the use of the symmetry properties of the Bernoulli polynomials, allows a straightforward determination of analytical expressions of the perturbed Coulomb Ladder functions and eigenvalues. Some illustrative examples showing the capabilities of the method are given; particularly, analytical expressions of the linear, quadratic, and cubic Stark shifts are quickly derived.
James Griffin - One of the best experts on this subject based on the ideXlab platform.
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continuous and discrete painleve equations arising from the gap probability distribution of the finite n gaussian unitary ensembles
Journal of Statistical Physics, 2014Co-Authors: Yang Chen, James GriffinAbstract:In this paper we study the gap probability problem in the Gaussian unitary ensembles of \(n\) by \(n\) matrices : The probability that the interval \(J := (-a,a)\) is free of eigenvalues. In the works of Tracy and Widom, Adler and Van Moerbeke, and Forrester and Witte on this subject, it has been shown that two Painleve type differential equations arise in this context. The first is the Jimbo–Miwa–Okomoto \(\sigma \)-form and the second is a particular Painleve IV. Using the Ladder Operator technique of orthogonal polynomials we derive three quantities associated with the gap probability, denoted as \(\sigma _n(a)\), \(R_n(a)\) and \(r_n(a)\). We show that each one satisfies a second order Painleve type differential equation as well as a discrete Painleve type equation. In particular, in addition to providing an elementary derivation of the aforementioned \(\sigma \)-form and Painleve IV we are able to show that the quantity \(r_n(a)\) satisfies a particular case of Chazy’s second degree second order differential equation. For the discrete equations we show that the quantity \(r_n(a)\) satisfies a particular form of the modified discrete Painleve II equation obtained by Grammaticos and Ramani in the context of Backlund transformations. We also derive second order second degree difference equations for the quantities \(R_n(a)\) and \(\sigma _n(a)\).
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continuous and discrete painleve equations arising from the gap probability distribution of the finite n gaussian unitary ensembles
arXiv: Classical Analysis and ODEs, 2014Co-Authors: Yang Chen, James GriffinAbstract:In this paper we study the gap probability problem in the Gaussian Unitary Ensembles of $n$ by $n$ matrices : The probability that the interval $J := (-a,a)$ is free of eigenvalues. In the works of Tracy and Widom, Adler and Van Moerbeke and Forrester and Witte on this subject, it has been shown that two Painleve type differential equations arise in this context. The first is the Jimbo-Miwa-Okomoto $\sigma-$form and the second is a particular Painleve IV. Using the Ladder Operator technique of orthogonal polynomials we derive three quantities associated with the gap probability, denoted by $\sigma_n(a)$, $R_n(a)$ and $r_n(a)$, and show that each one satisfying a second order, non-linear, differential equation as well as a second order, non-linear difference equation. In particular, in addition to providing an elementary derivation of the aforementioned $\sigma-$form and Painleve IV we show that the quantity $r_n(a)$ satisfies a particular case of Chazy's second degree second order differential equation. For the discrete equations we show that the quantity $r_n(a)$ satisfies a particular form of the modified discrete Painleve II equation obtained by Grammaticos and Ramani in the context of Backlund transformations. We also derive second order second degree difference equations for the quantities $R_n(a)$ and $\sigma_n(a)$.
