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I was reading this review paper by Bender, in particular section VI where they show that, despite $\mathcal{PT}$-symmetric Hamiltonians not being hermitian, they can have a real spectra. They go on to show that one can cook a proper inner product by defining an operator $\mathcal{C}$ such that the evolution is unitary.

My question is very simple:

Is it always possible to find a Hermitian Hamiltonian dual to any given $\mathcal{PT}$-symmetric Hamiltonian whose evolution is unitary?

By dual here I mean that it reproduces the same physics, i.e. it can reproduce any observables at any time $t$ identically to the $\mathcal{PT}$-symmetric Hamiltonian.

My motivation is simple: For any unitary matrix $U$ of finite size, $U$ can be written as $U = e^{iH}$, where $H$ is a Hermitian matrix. Now, I don't know if this is sufficient to show the existence of this duality.

I mean, if this is true, I can't help the feeling that this $\mathcal{PT}$-symmetric Hamiltonians research business is a bit misleading in making the community believe there is more than Hermitian $H$'s. So what am I missing?

I was reading this review paper by Bender, in particular section VI where they show that, despite $\mathcal{PT}$-symmetric Hamiltonians not being hermitian, they can have a real spectra. They go on to show that one can cook a proper inner product by defining an operator $\mathcal{C}$ such that the evolution is unitary.

My question is very simple:

Is it always possible to find a Hermitian Hamiltonian equivalent to any given $\mathcal{PT}$-symmetric Hamiltonian whose evolution is unitary?

By dual here I mean that it reproduces the same physics, i.e. it can reproduce any observables at any time $t$ identically to the $\mathcal{PT}$-symmetric Hamiltonian.

My motivation is simple: For any unitary matrix $U$ of finite size, $U$ can be written as $U = e^{iH}$, where $H$ is a Hermitian matrix. Now, I don't know if this is sufficient to show the existence of this equivalence.

I mean, if this is true, I can't help the feeling that this $\mathcal{PT}$-symmetric Hamiltonians research business is a bit misleading in making the community believe there is more than Hermitian $H$'s. So what am I missing?

I was reading this review paper by Bender, in particular section VI where they show that, despite $\mathcal{PT}$-symmetric Hamiltonians not being hermitian, they can have a real spectra. They go on to show that one can cook a proper inner product by defining an operator $\mathcal{C}$ such that the evolution is unitary.

My question is very simple:

Is it always possible to find a Hermitian Hamiltonian dual to any given $\mathcal{PT}$-symmetric Hamiltonian whose evolution is unitary?

By dual here I mean that it reproduces the same physics, i.e. it can reproduce any observables at any time $t$ identically to the $\mathcal{PT}$-symmetric Hamiltonian.

My motivation is simple: For any unitary matrix $U$ of finite size, $U$ can be written as $U = e^{iH}$, where $H$ is a Hermitian matrix. Now, I don't know if this is sufficient to show the existence of this duality.

I mean, if this is true, I can't help the feeling that this $\mathcal{PT}$-symmetric Hamiltonians research business is a bit misleading in making the community believe there is more than Hermitian $H$'s. So what am I missing?

I was reading this review paper by Bender, in particular section VI where they show that, despite $\mathcal{PT}$-symmetric Hamiltonians not being hermitian, they can have a real spectra. They go on to show that one can cook a proper inner product by defining an operator $\mathcal{C}$ such that the evolution is unitary.

My question is very simple:

Is it always possible to find a Hermitian Hamiltonian dual to any given $\mathcal{PT}$-symmetric Hamiltonian whose evolution is unitary?

By dual here I mean that it reproduces the same physics, i.e. it can reproduce any observables at any time $t$ identically to the $\mathcal{PT}$-symmetric Hamiltonian.

My motivation is simple: For any unitary matrix $U$ of finite size, $U$ can be written as $U = e^{iH}$, where $H$ is a Hermitian matrix. Now, I don't know if this is sufficient to show the existence of this duality.

I mean, if this is true, I can't help the feeling that this $\mathcal{PT}$-symmetric Hamiltonians research business is a bit misleading in making the community believe there is more than Hermitian $H$'s. So what am I missing?