This is a quadratic Hamiltonian, so there's no need to use perturbation theory. It can be diagonalized exactly. The statement about zero diagonal terms is strange. 

Of course it's unbounded from below and it's not Hermitian, so just what it's useful for isn't obvious. 

hello see this

@Nicolis

The Hamiltonian can never be diagonalised . However if you feel  , you can diagonalise the same then pl find some timlto do this and give your calculational details in pdf form. Further the basic point in this question is that the diagonal term is zero. How to apply perturbation theory  ?

@Osamah

The suggested PT diverges in view of zero diagonal terms . The wave function also diverges . The basic problem lies with denominatorm . All corrections beyond first diverges . However if you do not accept this then send your calculations with details

Of course it can-quadratic Hamiltonians can always be diagonalized. This is in all the textbooks, since it can be studied exactly like the usual harmonic oscillator.  The fact that the coefficients here are complex doesn't affect the method.

The problem is not formulated very precisely, so I will only make the QM assumption that [p, x] = -i and [p, p] = [x,x] = 0. The "Hamiltonian" can be written

H = (p+ix)^2 + ix = P^2 + ix,

where also [P, x] = -i and [P,  P] = [x, x] = 0. To fulfil the commutation relation I use a representation where P is a multiplication operator and x = i d/dP. Then the eigenvalue problem becomes

(-d/dP + P^2) f(P) = E f(P)

with solutions f(P) = exp(P^3/3 -E*P), where E can take any value in the complex plane. To proceed one must think through which restrictions one wants to impose on the solution.

@ Nicolis

The problem is like HO but all principles fails here. As you can diagonalise , please do the same. The suggested operator is a modified form of h=p^2 + ix .I want to address the problem h=p^2 -x^2 +i(xp + px) + ix^3. If you find some time see my paper uploaded in RG on 4.4.17.

@Olaussen

Your approach is correct. However the solution is not physical as it diverves in momentum space . The original problem when shited to h=p^2 - x^2 + i(xp+px)+ ix^3

then suggest how to address perturbation theory ?

Biswanath> However the solution is not physical as it diverges in momentum space .

Not necessarily, because P may be defined on curves such that P^3  behaves like -|P|^3 as |P| -> infinity. Just look up how the Airy function and friends are defined through Fourier integrals over curves in "momentum" space.

The original problem and my representation are mathematically equivalent. It is just a matter of multiplying the wave-function with a suitable factor, or making a Fourier type integral representation of the solution (to reduce the equation to first order).  By this procedure your new "Hamiltonian" is equivalent (by a similarity transform) to the differential operator

L = (d^2/dz^2 + z^3)

acting on functions defined some appropriate curve in the complex plane. Which curve may depend on the physics of the problem; the latter has not been specified.

In common formulations it is automatically assumed that p and q are hermitian operators (by physicists standards), and hence have real spectra. The problem in this case is that the time evolution defined by H is not unitary under the condition that p and q are hermitian, so the latter condition will not be preserved by time evolution. So we have to give up the condition that p and q have real spectra, since this is not a self-consistent assumption. Which opens for a much wider range of possibilities.

I have absolutely no idea of what you mean by perturbation theory in this case, and why you want to perform such a thing. How do you want to split H into perturbed and un-perturbed parts? What is your expansion parameter? What type of perturbation theory do you have in mind?

But above all, what kind of physics is your Hamiltonian supposed to model?  

it's not Hermitian，hamiltonian need Hermitian，itsatisfy group， It can be diagonalized exactly for some conditon.

@Olaussen

In momentum space , your  analysis is correct  . But what the eigenvalue(s) of the system .Secondly , how to apply pertubation thoery for this ?

Give you views. 

Biswanath> Give you views.

My view is that, since you do not make any attempts to answer my questions -- only repeating meaningless questions already brought forth, you do not have any serious interests in the problem.

A priori, the spectrum is the whole complex plane.

I think all these people do not understand what a perturbation theory is. Neither do I.

@Olaussen

I stand on my question . I feel by virtue of your transformation my original question has lost its charm. Any way I am trying to write  a short paper for AJP . I am willing to send the same to you for your valuable suggestion for rectification/modification. If you give your mail address it will be easier .

@Igor

• I accept your remarks. In my opinion people are moving on the surface of water and giving information on depth without a measuring stick. Even though I worked for ten years in PT and developed a series , but after my own question , I feel I have not understood PT . Any way I am trying to address my question.
• Many thanks for remarks.

@Zhu

One can apply PT to non- hermitian operator .

Biswanath> One can apply PT

And perhaps suitable generalisations. But in this case I don't think one can avoid obtaining a spectrum which is continuous and unbounded in both directions. I will not rule out the possibility of imposing conditions which makes the spectrum real.

Note added: I probably misread your use of the PT acronym; there is an interesting extension of hermitian quantum mechanics to PT-symmetric quantum mechanics.

So: The first-and-formost fact you should remember about perturbation theory is to never use PT as an acronym for it. Because that refers to an entirely different,  well-established notion.

https://arxiv.org/abs/quant-ph/0501052

See, go don't know where, bring me don't know what. Kåre continues to speak in terms of spectrum which was not the task. Try to formulate the problem in a way everybody understands, first of all you. That is 90% of solution. maybe 100%

@Olaussen

@Goliney

After the work , I will contact through RG.