Biorthogonal quantum criticality in non-Hermitian many-body systems

We develop the perturbation theory of the fidelity susceptibility in biorthogonal bases for arbitrary interacting non-Hermitian many-body systems with real eigenvalues. The quantum criticality in the non-Hermitian transverse field Ising chain is investigated by the second derivative of the ground-state energy and the ground-state fidelity susceptibility. We show that the system undergoes a second-order phase transition with the Ising universal class by numerically computing the critical points and the critical exponents from the finite-size scaling theory. Interestingly, our results indicate that the biorthogonal quantum phase transitions are described by the biorthogonal fidelity susceptibility instead of the conventional fidelity susceptibility.     

1/<Emphasis Type="Italic">N</Emphasis><Superscript>2</Superscript> Correction to Free Energy in Hermitian Two-Matrix Model

Using the loop equations we find an explicit expression for genus 1 correction in hermitian two-matrix model in terms of holomorphic objects associated to spectral curve arising in large N limit. Our result generalises known expression for F¹ in hermitian one-matrix model. We discuss the relationship between F¹, Bergman tau-function on Hurwitz spaces, G-function of Frobenius manifolds and determinant of Laplacian over spectral curve.Mathematics Subject Classifications. 15A52, 34M55.     

The Origin and Mathematical Characteristics of the Super-Universal Associated-Legendre Polynomials

We study the mathematical characteristics of the super-universal associated-Legendre polynomials arising from a kind of double ring-shaped potentials and obtain their polar angular wave functions with certain parity. We find that there exists the even or odd parity for the polar angular wave functions when the parameter η is taken to be positive integer, while there exist both even and odd parities when η is taken as positive non-integer real values. The relations among the super-universal associated-Legendre polynomials, the hypergeometric polynomials, and the Jacobi polynomials are also established.     

New Properties of the Zeros of Krall Polynomials

We identify a new class of algebraic relations satisfied by the zeros of orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two, known as Krall polynomials. Given an orthogonal polynomial family , we relate the zeros of the polynomial p_N with the zeros of p_m for each m ≤ N (the case m = N corresponding to the relations that involve the zeros of p_N only). These identities are obtained by finding exact expressions for the similarity transformation that relates the spectral and the (interpolatory) pseudospectral matrix representations of linear differential operators, while using the zeros of the polynomial p_N as the interpolation nodes. The proposed framework generalizes known properties of classical orthogonal polynomials to the case of nonclassical polynomial families of Krall type. We illustrate the general result by proving new identities satisfied by the Krall-Legendre, the Krall-Laguerre and the Krall-Jacobi orthogonal polynomials.     

The Origin and Mathematical Characteristics of the Super-Universal Associated-Legendre Polynomials

We study the mathematical characteristics of the super-universal associated-Legendre polynomials arising from a kind of double ring-shaped potentials and obtain their polar angular wave functions with certain parity. We find that there exists the even or odd parity for the polar angular wave functions when the parameter η is taken to be positive integer, while there exist both even and odd parities when η is taken as positive non-integer real values. The relations among the super-universal associated-Legendre polynomials, the hypergeometric polynomials, and the Jacobi polynomials are also established.     

基于人的视觉特征的双正交小波基的构造

构造了一个与人的视觉系统模型的调制转移函数相近似的紧支撑双正交小波基。文中采用遗传算法，寻找一个紧支撑双正交小波函数，使它与调制转移函数在频域中相近似。同时将所构造的双正交小波基用于图像压缩，并将它与其它双正交小波进行比较。实验结果表明，所构造的紧支撑双正交小波基与Daubechies的9／3小波相比较有更高的压缩比，与9／7小波相比压缩性能相当，但计算量更小，它更适合于纹理图像的压缩。     

Creation Operators for the Macdonald and Jack Polynomials

Formulas of Rodrigues-type for the Macdonald polynomials are presented. They involve creation operators, certain properties of which are proved and other conjectured. The limiting case of the Jack polynomials is discussed.     

On the Recurrence Coefficients for Generalized q-Laguerre Polynomials

In this paper we consider a semi-classical variation of the weight related to the q-Laguerre polynomials and study their recurrence coefficients. In particular, we obtain a second degree second order discrete equation which in particular cases can be reduced to either the qPV or the qPIII equation.     

Properties of the Zeros of the Sum of two Polynomials

Some properties—including relations having a Diophantine character—of the zeros of the sum of two polynomials are reported.     

Duality of Orthogonal Polynomials on a Finite Set

We prove a certain duality relation for orthogonal polynomials defined on a finite set. The result is used in a direct proof of the equivalence of two different ways (using particles or holes) of computing the correlation functions of a discrete orthogonal polynomial ensemble.     

Linear Statistics of Point Processes via Orthogonal Polynomials

For arbitrary β>0, we use the orthogonal polynomials techniques developed in (Killip and Nenciu in , 2005; Killip and Nenciu in Int. Math. Res. Not. 50: 2665–2701, 2004) to study certain linear statistics associated with the circular and Jacobi β ensembles. We identify the distribution of these statistics then prove a joint central limit theorem. In the circular case, similar statements have been proved using different methods by a number of authors. In the Jacobi case these results are new.     

Weight Function Approach to q-Difference Equations for the q-Hypergeometric Polynomials

In this paper we define a new algebra generated by the difference operators D _q and D _q-1 with two analytic functions (x) and (x). Also, we define an operator M = J ₁ J ₂–J ₃ J ₄ s.t. all q-hypergeometric orthogonal polynomials Y _n(x), x cos(), are eigenfunctions of the operator M with eigenvalues _q [n] _q . The choice of (x) and (x) depend on the weight function of Y _n (x).     

量子光学理论中双模厄米多项式的新应用

强调双模厄米多项式在量子光学理论中的地位,认为它是研究连续变量纠缠态和压缩态的必要函数,具有明确的物理意义。利用双模厄米多项式,结合有序算符内的积分技术,给出了若干新的算符恒等式和互逆的积分变换公式,证明了压缩双模粒子数态恰好是双变量厄米多项式激发压缩真空态。     

Pair Correlation Statistics for the Zeros of Lamé Polynomials

The joint eigenfunctions of a quantum completely integrable system can naturally be described in terms of products of Lamé polynomials. In this paper, we compute the limiting pair correlation distribution for the zeros of Lamé polynomials in various thermodynamic, asymptotic regimes. We give results both in the mean and pointwise, for an asymptotically full set of values of the parameters α₀,. . .,α_N. Mathematics Subject Classifications (2000) 81R12, 53A55.     

A Maximum Entropy Method Based on Orthogonal Polynomials for Frobenius-Perron Operators

Let $S$: [0, 1]→[0, 1] be a chaotic map and let $f^∗$ be a stationary density of the Frobenius-Perron operator $P_S$: $L^1$→$L^1$ associated with $S$. We develop a numerical algorithm for approximating $f^∗$, using the maximum entropy approach to an under-determined moment problem and the Chebyshev polynomials for the stability consideration. Numerical experiments show considerable improvements to both the original maximum entropy method and the discrete maximum entropy method.     

Factorizations of Polynomials over Noncommutative Algebras and Sufficient Sets of Edges in Directed Graphs

To directed graphs with unique sink and source we associate a noncommutative associative algebra and a polynomial over this algebra. Edges of the graph correspond to pseudo-roots of the polynomial. We give a sufficient condition when coefficients of the polynomial can be rationally expressed via elements of a given set of pseudo-roots (edges). Our results are based on a new theorem for directed graphs also proved in this paper. To the memory of Felix Alexandrovich Berezin. Vladimir Retakh was partially supported by NSA     

New Operator Ordering Formulas Related to Hermite Polynomials Derived by Virtue of IWOP Technique

By virtue of the technique of integration within an ordered product of operators and the fundamentaloperator identity Hn(X) = 2n : Xn :, where X is the coordinate operator and Hn is the n-order Hermite polynomials,:: is the normal ordering symbol, we not only simplify the derivation of the main properties of Hermite polynomials,but also directly derive some new operator identities regarding to Hn(X). Operation for transforming f(X) → :f(X) :is also discussed.