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Pseudo-Hermitian quantum mechanics with unbounded metric operators
1Department of Mathematics, Koç University, 34450 Sariyer, Istanbul, Turkey. amostafazadeh@ku.edu.tr
Summary
This study extends pseudo-Hermitian quantum mechanics to Hamiltonians with unbounded metric operators. It details constructing the physical Hilbert space and observables for systems with real, discrete spectra.
Area of Science:
- Quantum Mechanics
- Theoretical Physics
- Mathematical Physics
Background:
- Pseudo-Hermitian quantum mechanics offers an alternative framework to standard quantum theory.
- Unbounded metric operators present unique challenges in constructing physical Hilbert spaces.
Purpose of the Study:
- To extend the formulation of pseudo-Hermitian quantum mechanics to include unbounded metric operators (η(+)).
- To detail the construction of the physical Hilbert space, observables, and equivalent Hermitian Hamiltonians.
Main Methods:
- The study focuses on Hamiltonian operators (H) with an unbounded metric operator (η(+)).
- It considers cases where H has a real and discrete spectrum.
- Eigenvectors are analyzed within the domain of η(+) and its square root (√η(+)).
Main Results:
- A detailed construction of the physical Hilbert space for η(+)-pseudo-Hermitian Hamiltonians with unbounded metrics is provided.
- The framework allows for the definition of observables and an equivalent Hermitian Hamiltonian.
- The analysis is specifically tailored for Hamiltonians with real, discrete spectra.
Conclusions:
- The extended formulation provides a rigorous mathematical framework for dealing with unbounded metric operators in pseudo-Hermitian quantum mechanics.
- This work facilitates the study of physical systems described by such operators, ensuring a well-defined Hilbert space and observable algebra.
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