Liouville transformations and quantum reflection

量子力学自举法中的一维反射

One-dimensional reflection in the quantum mechanical bootstrap

本研究将量子力学自举法应用于一维散射问题，通过调节边界Robin参数提取反射系数，结合半定规划求解半线薛定谔问题，并用WKB近似确定Liouville理论指数势的散射行为。

达布可积非线性Liouville-von Neumann方程

Darboux-integrable nonlinear Liouville-von Neumann equation

本文利用一种新的二元达布变换来产生非线性Liouville-von Neumann方程的解析解。一般理论是由明确的例子说明。

On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators

We consider one-dimensional Schroedinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations in detail. We show that they can be expressed as the sum of the identity and an integral Hilbert-Schmidt operator. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for th…

实势与复势之间的不可约二阶SUSY变换

Irreducible second order SUSY transformations between real and complex potentials

从物理角度分析了三个重要的Sturm-Liouville问题，即有限区间Dirichlet边界条件问题、半轴Dirichlet边界条件问题和整条实直线Dirichlet边界条件问题的实势和复势之间的二阶SUSY变换.对于每一个问题的条件，变换函数制定时，变换是不可约的，即当中间哈密顿量是不是很好地定义在同一个希尔伯特空间的初始和最终的哈密顿量或其本征函数不能通过应用变换算子的本征函数的初始哈密顿量或这些最终的哈密顿量。得到的结果说明了许多简单的例子。

Partition-free approach to open quantum systems in harmonic environments: an exact stochastic Liouville equation

We present a partition-free approach to the evolution of density matrices for open quantum systems coupled to a harmonic environment. The influence functional formalism combined with a two-time Hubbard-Stratonovich transformation allows us to derive a set of exact differential equations for the reduced density matrix of an open system, termed the Extended Stochastic Liouville-von Neumann equation. Our approach generalises previous work based on Caldeira-Leggett models and a partitioned initial density matrix. This provides a simple, yet exact, closed-form description for the evolution of open …

Weyl Quantization of Fractional Derivatives

The quantum analogs of the derivatives with respect to coordinates q_k and momenta p_k are commutators with operators P_k and $Q_k. We consider quantum analogs of fractional Riemann-Liouville and Liouville derivatives. To obtain the quantum analogs of fractional Riemann-Liouville derivatives, which are defined on a finite interval of the real axis, we use a representation of these derivatives for analytic functions. To define a quantum analog of the fractional Liouville derivative, which is defined on the real axis, we can use the representation of the Weyl quantization by the Fourier transfor…

The Canonical Coset Decomposition of Unitary Matrices Through Householder Transformations

This paper reveals the relation between the canonical coset decomposition of unitary matrices and the corresponding decomposition via Householder reflections. These results can be used to parametrize unitary matrices via Householder reflections.

拟精确可解周期势的对偶性和自对偶性（能量反射对称性）

Duality and Self-Duality (Energy Reflection Symmetry) of Quasi-Exactly Solvable Periodic Potentials

研究了一类具有隐李代数结构的谱问题。我们定义了一个对偶变换，它将一个准精确可解（QES）周期势的谱映射到另一个QES周期势的谱。这种变换的自对偶点对应于先前在某些QES系统中发现的能量反射对称性。对偶变换将一个势的谱的底部（顶部）的带与另一个对偶势的谱的顶部（底部）的带隙互换。因此，对偶变换提供了弱耦合（微扰）和半经典（非微扰）扇区之间的精确映射。

Analysis of geometric phase effects in the quantum-classical Liouville formalism

We analyze two approaches to the quantum-classical Liouville (QCL) formalism that differ in the order of two operations: Wigner transformation and projection onto adiabatic electronic states. The analysis is carried out on a two-dimensional linear vibronic model where geometric phase (GP) effects arising from a conical intersection profoundly affect nuclear dynamics. We find that the Wigner-then-Adiabatic (WA) QCL approach captures GP effects, whereas the Adiabatic-then-Wigner (AW) QCL approach does not. Moreover, the Wigner transform in AW-QCL leads to an ill-defined Fourier transform of doub…

玻色化超对称性的三个方面与带反射的线性微分场方程

Three aspects of bosonized supersymmetry and linear differential field equation with reflection

最近，一位作者观察到，超对称量子力学（SUSYQM）只允许一个玻色子自由度。这样的结构，称为最小玻色化的SUSYQM，出现在可积系统和动力学对称的背景下。我们表明，最小玻色化的SUSYQM可以从威滕的SUSYQM通过应用到它的非局部酉变换与随后减少的全反射运营商的本征空间之一。这种变换依赖于宇称算符，而带有反射的变形海森堡代数，与抛物子和仿费米子密切相关，在这里以一种自然的方式出现。结果表明，最小玻色化的SUSYQM也可以理解为超对称双费米子系统。在这种解释下，玻色化结构被推广到二维N=1超对称的情形。同样特殊的么正变换对角化了二维有质量自由狄拉克理论的哈密顿算符。由此产生的哈密顿量不是像在Foldy-Wouthuysen情况下的平方根，而是空间导数的线性。随后减少到'上'或'下'场分量产生一个线性微分方程的反射，其'平方'是大量的克莱因-戈登方程。在无质量极限下，这就变成了自对偶外尔方程。具有反射的线性微分方程可以推广到高维，并且可以一致地耦合到规范场。玻色化的超对称量子力学也可以在非线性标量场的扭结位形背景下，对狄拉克场进行非定域么正变换而得到。

Infinitely many inequivalent field theories from one Lagrangian

Logarithmic time-like Liouville quantum field theory has a generalized PT invariance, where T is the time-reversal operator and P stands for an S-duality reflection of the Liouville field $ϕ$. In Euclidean space the Lagrangian of such a theory, $L=\frac{1}{2}(\nablaϕ)^2-igϕ\exp(iaϕ)$, is analyzed using the techniques of PT-symmetric quantum theory. It is shown that L defines an infinite number of unitarily inequivalent sectors of the theory labeled by the integer n. In one-dimensional space (quantum mechanics) the energy spectrum is calculated in the semiclassical limit and the mth energy leve…