Moments of the logarithmic derivative of characteristic polynomials from $SO(N)$ and $USp(2N)$

Quantum Anomalies and Logarithmic Derivatives

This papers deals with connections between quantum anomalies and transformations of Feynman pseudo-measures. Mathematical objects related to the notion of the volume element in an infinite-dimensional space considered in the physics literature are considered and disagreement in the related literature regarding the origin of quantum anomalies is explained.

随机酉矩阵特征多项式的对数导数

On the logarithmic derivative of characteristic polynomials for random unitary matrices

本文研究 Haar 随机酉矩阵特征多项式的对数导数，证明其在单位圆附近可由邻近零点逼近，并在远离单位圆处建立介观中心极限定理，分别类比 Riemann zeta 函数的相关结果。

Symmetric logarithmic derivative for general n-level systems and the quantum Fisher information tensor for three-level systems

Within a geometrical context, we derive an explicit formula for the computation of the symmetric logarithmic derivative for arbitrarily mixed quantum systems, provided that the structure constants of the associated unitary Lie algebra are known. To give examples of this procedure, we first recover the known formulae for two-level mixed and three-level pure state systems and then apply it to the novel case of U(3), that is for arbitrarily mixed three-level systems (q-trits). Exploiting the latter result, we finally calculate an expression for the Fisher tensor for a q-trit considering also all …

Tau-functions for the Ablowitz--Ladik hierarchy: the matrix-resolvent method

We extend the matrix-resolvent method for computing logarithmic derivatives of tau-functions to the Ablowitz--Ladik hierarchy. In particular, we derive a formula for the generating series of the logarithmic derivatives of an arbitrary tau-function in terms of matrix resolvents. As an application, we provide a way of computing certain integrals over the unitary group.

An explicit bound on the Logarithmic Sobolev constant of weakly dependent random variables

We prove logarithmic Sobolev inequality for measures $$ q^n(x^n)=\text{dist}(X^n)=\exp\bigl(-V(x^n)\bigr), \quad x^n\in \Bbb R^n, $$ under the assumptions that: (i) the conditional distributions $$ Q_i(\cdot| x_j, j\neq i)=\text{dist}(X_i| X_j= x_j, j\neq i) $$ satisfy a logarithmic Sobolev inequality with a common constant $ρ$, and (ii) they also satisfy some condition expressing that the mixed partial derivatives of the Hamiltonian $V$ are not too large relative to $ρ$. \bigskip Condition (ii) has the form that the norms of some matrices defined in terms of the mixed partial derivatives of $…

Calculation of the Characteristic Functions of Anharmonic Oscillators

The energy levels of quantum systems are determined by quantization conditions. For one-dimensional anharmonic oscillators, one can transform the Schrodinger equation into a Riccati form, i.e., in terms of the logarithmic derivative of the wave function. A perturbative expansion of the logarithmic derivative of the wave function can easily be obtained. The Bohr-Sommerfeld quantization condition can be expressed in terms of a contour integral around the poles of the logarithmic derivative. Its functional form is B_m(E,g) = n + 1/2, where B is a characteristic function of the anharmonic oscillat…

复速度场与对称对数导数的规范不变丛同构

A Gauge-Invariant Bundle Isomorphism Between Complex Velocity Fields and Symmetric Logarithmic Derivatives

本文建立复速度场与量子估计理论中对称对数导数运算符的规范不变同构关系，揭示其在概率密度中的内在联系，并推导出福斯特度量的表达式。