Matching method and exact solvability of discrete PT-symmetric square wells

Discrete PT-symmetric square-well oscillators

Exact solvability of the discretized N-point version of the PT-symmetric square-well model is pointed out. Its wave functions are found proportional to the classical Tshebyshev polynomials of a complex argument. At all N a compact secular equation is derived giving the real spectrum of energies at any non-Hermiticity strength Z below its finite and weakly N-dependent critical value. In the limit of vanishing Z the model degenerates to a Hermitian Hueckel Hamiltonian.

受一般势场约束的束缚态Gross-Pitaevskii方程的精确可解性

Exact solvability of the Gross-Pitaevskii equation for bound states subjected to general potentials

本文通过将一维Gross-Pitaevskii方程映射为一阶非自治动力系统，并利用Glauber-Trotter公式和有序指数级数，给出了一般势场下束缚态及稳定孤子的解析求解方法。

Solvable quantum lattices with nonlocal non-Hermitian endpoint interactions

Discrete multiparametric 1D quantum well with PT-symmetric long-range boundary conditions is proposed and studied. As a nonlocal descendant of the square well families endowed with Dirac (i.e., Hermitian) and with complex Robin (i.e., non-Hermitian but still local) boundary conditions, the model is shown characterized by the survival of solvability in combination with an enhanced spectral-design flexibility. The solvability incorporates also the feasibility of closed-form constructions of the physical Hilbert-space inner products rendering the time-evolution unitary.

Minimum weight perfect matching of fault-tolerant topological quantum error correction in average $O(1)$ parallel time

Consider a 2-D square array of qubits of extent $L\times L$. We provide a proof that the minimum weight perfect matching problem associated with running a particular class of topological quantum error correction codes on this array can be exactly solved with a 2-D square array of classical computing devices, each of which is nominally associated with a fixed number $N$ of qubits, in constant average time per round of error detection independent of $L$ provided physical error rates are below fixed nonzero values, and other physically reasonable assumptions. This proof is applicable to the fully…

PT symmetric square well

In a PT symmetrically complexified square well, bound states are constructed by the matching technique. Their energies prove real in a domain of weak non-Hermiticity, and continuous in the Hermitian limit. At a sequence of certain strong-Hermiticity thresholds, the lowest two energies merge and disappear.

A study of the bound states for square potential wells with position-dependent mass

A square potential well with position-dependent mass is studied for bound states. Applying appropriate matching conditions, a transcendental equation is derived for the energy eigenvalues. Numerical results are presented graphically and the variation of the energy of the bound states are calculated as a function of the well-width and mass.