关于谱与素数集重合的哈密顿量
On the Hamiltonian whose spectrum coincides with the set of primes
摘要：
考虑了构造一个简单的一维哈密顿量的问题，该哈密顿量的谱与素数集重合。我们注意到，准经典地，一个汉密尔顿算子的谱具有与首序中的素数相同的计数函数（即。e.积分对数）可以用函数V（x）构造，其逆函数由x=li（V^（1/2））渐近给出。数值构造了其谱与前1 - 1000个素数重合的哈密顿量，揭示了其分形结构。使用这样的“素生成”的哈密顿量的调查某些数论问题的可能性进行了讨论。
英文摘要：
The problem of construction of a simple one - dimensional Hamiltonian whose spectrum coincides with the set of primes is considered. We note that quasiclassically a Hamiltonian whose spectrum has the same counting function as that of the primes in the leading order (i. e. integral logarithm) can be constructed with the function V(x) whose inverse is asymptotically given by x=li(V^(1/2)). Hamiltonians, whose spectra coincide with the first 1 - 1000 primes are constructed numerically and their fractal structure is revealed. The possibility of using such a "prime-generating" Hamiltonian for investigation of certain number-theoretical problems is discussed.
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