| In the complex extension of ordinary Quantum Mechanics the complex(non-Hermitian)Hamiltonians with real spectrum play the key rule. It is possible to restore the Hermiticity of non-Hermitian Hamiltonians by redefining the inner product of the corresponding Hilbert space (pseudo-Hermiticity). The existence of a way to handle a non-Hermitian Hamiltonians does not provide an ultimate framework for the Quantum-Mechanical consideration since constructing a unique and unitary Quantum system for non-Hermitian Hamiltonians is not always possible due to the presence of exceptional points (for a discrete spectrum) and spectral singularities (for a continuous spectrum). Mathematically, spectral singularities are energies for which the Reflection and Transmission coefficients of the scattered wave diverge after scattering off a potential therefore they can be associated with a type of resonance states in ordinary quantum mechanics but with a real energies and zero width. The problem of Hamiltonians with a coupled complex (PT-symmetric) potential is relatively the same problem when trying to solve the Maxwell's wave equation of EM beams traveling through a waveguide filled with a dielectric material having a complex refractive index. After theoretically identifying the location of spectral singularities, as points on the continuous spectrum of complex Hamiltonians which spoils the construction of a unitary quantum system, experimental observations can confirm the concept of spectral singularities as a new type of resonance effect and give approval to the possible implications in the related fields, specially in laser physics. |
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