Rigged Hilbert Space Approach for Non-Hermite Systems with Positive Definite Metric

• URL: http://arxiv.org/abs/2209.01598v4
• Date: Mon, 15 May 2023 14:58:22 GMT
• Title: Rigged Hilbert Space Approach for Non-Hermite Systems with Positive Definite Metric
• Abstract summary: Dirac's bra-ket formalism is based on a rigged Hilbert space for a non-Hermite quantum system with a positive-definite metric. spectral expansions are shown for the bra-kets by the generalized eigenvectors of a quasi-Hermite operator. We show a specific description of our rigged Hilbert space treatment for some parity-time symmetrical quantum systems.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We investigate Dirac's bra-ket formalism based on a rigged Hilbert space for a non-Hermite quantum system with a positive-definite metric. First, the rigged Hilbert space, characterized by positive-definite metric, is established. With the aid of the nuclear spectral theorem for the obtained rigged Hilbert space, spectral expansions are shown for the bra-kets by the generalized eigenvectors of a quasi-Hermite operator. The spectral expansions are utilized to endow the complete bi-orthogonal system and the transformation theory between the Hermite and non-Hermite systems. As an example of application, we show a specific description of our rigged Hilbert space treatment for some parity-time symmetrical quantum systems.

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