Polynomial deformations of sl(2,R) in a three-dimensional invariant subspace of monomials

Quasi-exact solvability and intertwining relations

We emphasize intertwining relations as a universal tool in constructing one-dimensional quasi-exactly solvable operators and offer their possible generalization to the multidimensional case. Considered examples include all quasi-exactly solvable operators with invariant subspaces of monomials. We show that the simplest case of generalized intertwining relations allows to naturally relate quasi-exactly solvable operators with two invariant monomial subspaces to a nonlinear parasupersymmetry of second order. Quantum-mechanical systems with linear and nonlinear supersymmetry are discussed from th…

多量子比特系统对称子空间的变形

Deformations of the symmetric subspace of qubit chains

本文研究了多量子比特系统对称子空间的变形，通过量子群U_q(su(2))的结构变形，实现了对称态的局部内积变形。

Krylov复杂度在哈密顿变形和Toda流中的表现

Krylov Complexity Under Hamiltonian Deformations and Toda Flows

本文研究了哈密顿变形和Toda流对Krylov复杂度的影响，通过构造可解模型分析变形前后理论的复杂度变化，并探讨了相干Gibbs态及随机矩阵的统计特性。

The Darboux transformation and algebraic deformations of shape-invariant potentials

We investigate the backward Darboux transformations (addition of a lowest bound state) of shape-invariant potentials on the line, and classify the subclass of algebraic deformations, those for which the potential and the bound states are simple elementary functions. A countable family, $m=0,1,2,...$, of deformations exists for each family of shape-invariant potentials. We prove that the $m$-th deformation is exactly solvable by polynomials, meaning that it leaves invariant an infinite flag of polynomial modules $\mathcal{P}^{(m)}_m\subset\mathcal{P}^{(m)}_{m+1}\subset...$, where $\mathcal{P}^{…

Quasi-Exact Solvability and the direct approach to invariant subspaces

We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of sl(2). This approach is used here to construct new exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line which are not Lie-algebraic. It is also applied to generate potentials with multiple algebraic sectors. We discuss two illustrative examples of these two applications: an interesting generalization of the Lamé potential which posses four algebraic sectors, and a quasi-exactly solvable deform…