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首页> 外文期刊>Journal of physics, A. Mathematical and theoretical >Quantum models with spectrum generated by the flows of polynomial zeros
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Quantum models with spectrum generated by the flows of polynomial zeros

机译：由多项式零流产生的具有光谱的量子模型

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摘要

A class R of purely bosonic models is characterized having the following properties in a Hilbert space of analytic functions: (i) wave function psi(is an element of,z) = Sigma(infinity)(n=0)phi(n)(is an element of)z(n) is the generating function for orthogonal polynomials phi(n)(is an element of) of a discrete energy variable is an element of, (ii) any Hamiltonian (H) over cap (b) is an element of R has nondegenerate purely point spectrum that corresponds to infinite discrete support of measure dv(x) in the orthogonality relation of the polynomials phi(n), (iii) the support is determined exclusively by the points of discontinuity of v(x), (iv) the spectrum of (H) over cap (b) is an element of R can be numerically determined as fixed points of monotonic flows of the zeros of orthogonal polynomials phi(n)(is an element of), (v) one can compute practically an unlimited number of energy levels (e.g. 2(53) in double precision). If a model of R is exactly solvable, its spectrum can only assume one of four qualitatively different types. The results are applied to spin-boson quantum models that are, at least partially, diagonalizable and have at least single one-dimensional irreducible component in the spin subspace. Examples include the Rabi model and its various generalizations.
机译：纯Bosonic模型的R类在解析函数的希尔伯特空间中具有以下特性：（i）波函数psi（是z的元素）= Sigma（无限大）（n = 0）phi（n）（是（z）的元素z（n）是正交多项式的生成函数phi（n）（是的元素）的离散能量变量是（ii）上限（b）上的任何哈密顿量（H）是R的元素具有非简并的纯点谱，其在多项式phi（n）的正交关系中对应于量度dv（x）的无限离散支持，（iii）支持仅由v（x）的不连续点确定），（iv）上限（b）上的（H）的频谱是R的元素，可以通过数值确定为正交多项式零的单调流的不动点phi（n）（是的元素），（v ）实际上可以计算出无限数量的能级（例如，双精度的2（53））。如果R的模型是完全可解的，则它的光谱只能假设四个定性不同的类型之一。将结果应用于自旋玻色子量子模型，该模型至少部分可对角化，并且在自旋子空间中至少具有一个一维不可约分量。示例包括Rabi模型及其各种概括。

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