PT-对称量子系统的完备性和正交正规性
Completeness and Orthonormality in PT-symmetric Quantum Systems
摘要：
一些PT对称的非厄米哈密顿算子只有实本征值。有数值证据表明，相关的PT不变的能量本征态满足一个非常规的完整性关系。一个特设标量积之间的状态是正定的，只有当最近推出的“收费运营商”是包括在其定义。给出了证明完备性和规范正交关系的一个简单推导。它利用了这样一个事实，即PT对称性提供了一个额外的联系之间的本征态的哈密顿量和它的伴随，形成一个对偶对基地。“电荷算符”在仅用一个基来表示对偶基的性质时自然出现。
英文摘要：
Some PT-symmetric non-hermitean Hamiltonians have only real eigenvalues. There is numerical evidence that the associated PT-invariant energy eigenstates satisfy an unconventional completeness relation. An ad hoc scalar product among the states is positive definite only if a recently introduced `charge operator' is included in its definition. A simple derivation of the conjectured completeness and orthonormality relations is given. It exploits the fact that PT-symmetry provides an additional link between the eigenstates of the Hamiltonian and those of its adjoint, which form a dual pair of bases. The `charge operator' emerges naturally upon expressing the properties of the dual bases in terms of one basis only.
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