在Schroedinger和Hermite之间：q-变形非局部算子的超对称对

Hermite polynomials and Fibonacci Oscillators

We compute the ($q_1,q_2$)-deformed Hermite polynomials by replacing the quantum harmonic oscillator problem to Fibonacci oscillators. We do this by applying the ($q_1, q_2$)-extension of Jackson derivative. The deformed energy spectrum is also found in terms of these parameters. We conclude that the deformation is more effective in higher excited states. We conjecture that this achievement may find applications in the inclusion of disorder and impurity in quantum systems. The ordinary quantum mechanics is easily recovered as $q_1 = 1$ and $q_2\to1$ or vice versa.

基本形变量子力学

Basic-deformed quantum mechanics

本文从$q$ -对称振子代数及其相关的$q$ -演算性质出发，研究了定义在基本平方可积波函数空间框架下的变形量子力学.在这方面，我们介绍了一个变形薛定谔方程，它满足主要的量子力学假设，并承认，在自由的情况下，平面波函数，可以表示在q-变形指数，最初引入的基本超几何函数的框架。

位置不确定性最小的动量空间中的精确可解问题

Exactly solvable problems in the momentum space with a minimum uncertainty in position

本文提出了一种求解最小长度变形空间中简单量子力学问题的新方法。在最小长度变形海森堡代数的情形下，给出了动量表象下薛定谔方程的推广。在假定势能算符的核在形变情况下不变的条件下，得到了粒子在δ势和双δ势中本征问题的精确解。重新讨论了类库仑势中的粒子，解决了逆坐标算符的可逆性和赫密性问题。

Quantum models with energy-dependent potentials solvable in terms of exceptional orthogonal polynomials

We construct energy-dependent potentials for which the Schroedinger equations admit solu- tions in terms of exceptional orthogonal polynomials. Our method of construction is based on certain point transformations, applied to the equations of exceptional Hermite, Jacobi and Laguerre polynomials. We present several examples of boundary-value problems with energy-dependent potentials that admit a discrete spectrum and the corresponding normalizable solutions in closed form.

矩阵D-膜的非线性Schroedinger动力学

Non-linear Schroedinger Dynamics of Matrix D-branes

通过将Wilson重整化群方程应用于描述系统的变形σ模型的世界面配分函数，我们建立了描述D 0-膜系统量子动力学的有效Schroedinger波动方程，其中包括由于单个D-粒子之间弦态交换而产生的量子反冲。我们得到一个有效的Fokker-Planck方程的概率密度与扩散系数由反冲系统的总动能。我们使用伽利略不变性的系统，有三个可能的解决方案的相关的非线性薛定谔方程依赖于D粒子之间的开弦相互作用的强度。当开弦能量与系统总动能相比很小时，解由自由传播的孤立波控制。当弦的耦合常数达到一个动态确定的临界值时，系统可以用最小不确定性波包来描述，它描述了由于弦的相互作用导致周围时空的扭曲而导致的D粒子坐标的模糊。对于强的弦相互作用，存在束缚态解，其有效质量由D 0-膜的静态BPS质量的能量依赖的位移决定。

精确可解的量子力学

Exactly Solvable Quantum Mechanics

本文对精确可解量子力学作了一个全面的综述，重点介绍了最近发现的多指标正交多项式。主要讨论的问题是因式分解的哈密顿量，薛定谔方程解空间的一般结构（Crum定理及其修正）、形状不变性、薛定谔图像和海森堡图像中的精确可解性、产生/湮灭算子和动力学对称代数、相干态、各种变形方案（多重达布变换）和多指标正交多项式的无限族、例外正交多项式和变形的精确可解散射问题。

Polynomial solution of quantum Grassmann matrices

We study a model of quantum mechanical fermions with matrix-like index structure (with indices $N$ and $L$) and quartic interactions, recently introduced by Anninos and Silva. We compute the partition function exactly with $q$-deformed orthogonal polynomials (Stieltjes-Wigert polynomials), for different values of $L$ and arbitrary $N$. From the explicit evaluation of the thermal partition function, the energy levels and degeneracies are determined. For a given $L$, the number of states of different energy is quadratic in $N$, which implies an exponential degeneracy of the energy levels. We als…