Dirac Equation on the Torus and Rationally Extended Trigonometric Potentials within Supersymmetric QM

Strong uniqueness for Dirichlet operators related to stochastic quantization under exponential/trigonometric interactions on the two-dimensional torus

We consider space-time quantum fields with exponential/trigonometric interactions. In the context of Euclidean quantum field theory, the former and the latter are called the Hoegh-Krohn model and the Sine-Gordon model, respectively. The main objective of the present paper is to construct infinite dimensional diffusion processes which solve modified stochastic quantization equations for these quantum fields on the two-dimensional torus by the Dirichlet form approach and to prove strong uniqueness of the corresponding Dirichlet operators.

The bound states of the massless Dirac equation on the torus

The Dirac equation on the toroidal surface is studied. The non-constant Fermi velocity functions are used in the Dirac Hamiltonian to get trigonometric Scarf and Rosen-Morse potentials using the method given in [19], then, the exact solutions are written. On the other hand, consecutive mappings are used to get a trigonometric Scarf I-like potential model and the solutions are obtained in the second part of the paper.

与例外正交多项式相关的(1+1)维Dirac方程的解及参数对称性

Solutions of (1+1)-dimensional Dirac equation associated with exceptional orthogonal polynomials and the parametric symmetry

本文研究三类有理扩展标量势下(1+1)维Dirac方程的求解，利用例外正交多项式给出解，并通过参数对称性构造新的Dirac标量势族，进一步用常规及例外正交多项式求得其解。

$\nmathbb{T}^d$ 上三角多项式平移的丢番图估计

Diophantine estimates on shifts of trigonometric polynomials on $\mathbb{T}^d$

本文在环面上建立三角多项式平移及其平方根的丢番图估计，源于拟周期算子谱分析，可推广Bourgain结果至非线性方程。

Infinite families of (non)-Hermitian Hamiltonians associated with exceptional $X_m$ Jacobi polynomials

Using an appropriate change of variable, the Schrödinger equation is transformed into a second-order differential equation satisfied by recently discovered Jacobi type $X_m$ exceptional orthogonal polynomials. This facilitates the derivation of infinite families of exactly solvable Hermitian as well as non-Hermitian trigonometric Scarf potentials and finite number of Hermitian and infinite number of non-Hermitian PT-symmetric hyperbolic Scarf potentials. The bound state solutions of all these potentials are associated with the aforesaid exceptional orthogonal polynomials. These infinite famili…

Exact solution of an su(n) spin torus

The trigonometric su(n) spin chain with anti-periodic boundary condition (su(n) spin torus) is demonstrated to be Yang-Baxter integrable. Based on some intrinsic properties of the R-matrix, certain operator product identities of the transfer matrix are derived. These identities and the asymptotic behavior of the transfer matrix together allow us to obtain the exact eigenvalues in terms of an inhomogeneous T-Q relation via the off-diagonal Bethe Ansatz.