PT-Symmetric Schrödinger Operators with Unbounded Potentials
PT-Symmetric Schrödinger Operators with Unbounded Potentials
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具有无限势的 PT 对称薛定谔算子
DOI：
10.1007/978-3-8348-8327-8
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
Nesemann
中科院分区：
文献类型：
--
作者：
Nesemann
在量子力学理论中，哈密顿量H通常是自伴的，即H= H。自伴性确保了表示H的能谱的哈密顿量的谱是真实的，但它不是必要条件。在所谓的PT对称量子力学的文献中（参见[BB 98]，[BBM 99]，[BBJ 03]，[Ben 04 b]和[Ben 07]），人们认为自伴性是一个数学要求而不是物理上建立的事实。因此，它被认为是一个惊喜，运营商存在的不是自共轭在给定的量子力学希尔伯特空间，但有真实的频谱，如果复本征值存在，他们只出现在复共轭对。然而，从数学的角度来看，这一点也不奇怪--只要熟悉具有不定内积的空间（克莱因空间）中的自伴算子理论。物理结构被发现是光谱真实性的原因是PT对称性（时空反射对称性），这相当于在某些克莱因空间中的自伴性。一个哈密顿量H是PT对称的，如果它与PT交换，即PTH= HPT，比较，例如，[Ben 07]和[AT 10]。这里P表示空间反射（奇偶）算子，T表示时间反射算子。P和T满足关系P2= T2=（PT）2= I和PT= TP。如果p= id/dx和x是动量和位置算子，则P具有效应p→− p，x→− x，T具有效应p→− p，x→ x，i→− i，比较，例如，[BB 98]，[BBM 99]，[BBJ 03]，[Ben 04 b]和[Ben 07]。与希尔伯特空间中的自伴性相反，PT-对称性不一定导致完全真实的谱。例如，Hamilton
In the theory of quantum mechanics the Hamiltonian H is typically selfadjoint, ie, H= H∗. The self-adjointness ensures that the spectrum of the Hamiltonian, representing the energy spectrum of H, is real but it is not a necessary condition. In the literature on so-called PT-symmetric quantum mechanics (see, eg,[BB98],[BBM99],[BBJ03],[Ben04b] and [Ben07]), it is believed that self-adjointness is rather a mathematical requirement than a physically established fact. Therefore, it was considered a surprise that operators exist which are not self-adjoint in the given quantum mechanical Hilbert space, but have real spectrum and that–if eg complex eigenvalues were present–they occurred only in complex conjugate pairs. From a mathematical point of view, however, this is no surprise at all–provided one is familiar with the theory of self-adjoint operators in spaces with indefinite inner product (Krein spaces). The physical structure found to be the reason for the reality of the spectrum is PT-symmetry (spacetime reflection symmetry), which amounts to self-adjointness in some Krein space. A Hamiltonian H is PT-symmetric if it commutes with PT, that is PTH= HPT, compare, eg,[Ben07] and [AT10]. Here P denotes the space reflection (parity) operator and T the time reflection operator. P and T satisfy the relations P2= T2=(PT) 2= I and PT= TP. If p= id/dx and x are the momentum and position operators, then P has the effect p→− p, x→− x and T has the effect p→− p, x→ x, i→− i, compare, eg,[BB98],[BBM99],[BBJ03],[Ben04b] and [Ben07]. In contrast to self-adjointness in Hilbert spaces, PT-symmetry does not necessarily lead to a completely real spectrum. For example, the Hamiltonian