PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues - High Energy Physics - TheoryReport as inadecuate




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Abstract: Despite its common use in quantum theory, the mathematical requirement ofDirac Hermiticity of a Hamiltonian is sufficient to guarantee the reality ofenergy eigenvalues but not necessary. By establishing three theorems, thispaper gives physical conditions that are both necessary and sufficient. First,it is shown that if the secular equation is real, the Hamiltonian isnecessarily PT symmetric. Second, if a linear operator C that obeys the twoequations C,H=0 and C^2=1 is introduced, then the energy eigenvalues of aPT-symmetric Hamiltonian that is diagonalizable are real only if this Coperator commutes with PT. Third, the energy eigenvalues of PT-symmetricHamiltonians having a nondiagonalizable, Jordan-block form are real. Thesetheorems hold for matrix Hamiltonians of any dimensionality.



Author: Carl M. Bender, Philip D. Mannheim

Source: https://arxiv.org/







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