It is known that spectra of H = p^2 - x^4 is positive.My question is that
can we construct some more H reflecting iso-spectra ? If yes , suggest those Hamiltonians in complex space.
B.Rath
@Biswanath Rath,
Would you please clarify what you mean by your statement: "It is known that spectra of H = p^2 - x^4 is positive" ?
The Hamiltonian, as it stands, has no stable ground state and therefore, no stationary point-spectrum. From that point of view, the study of the Hamiltonian is merely of academic-interest, possibly in the context of non-hermitian Hamiltonians. (There are some studies pertaining to computation of resonance-properties and also its use in obtaining large-order-behavior of perturbation theory for the "physical" -Hamiltonian obtained by reversing the sign of the "x^4"-term.)
As to the iso-spectral behavior, let us note that the potential is not derivable from any super-potential because of the instability of the ground state and, therefore, not relevant in SUSYQM. Hence, the question of partner-Hamiltonian(s) and iso-spectrality etc are also not relevant in any physical-context.
@Bimal Mahapatra
As I know the spectrum of H=p^2 -x^4 is positive . This has been verified using many methods . My question relating to this refers in finding equivalent Hamiltonian in complex space reflecting iso'spectral behaviour . One can suggest H(1),H(2) , ...........etc .
As such H=p^2 + x^4 can not be generated using SUSY . However H=p^2 + x^4 - or + zx can be generated using SUSY .In the same context
using SUSY .It is not difficult to generate
H=p^2 -x^4 - 2x or H=p^2 -x^4 + 2x
but verification of result is not that easy .In both the cases spectrum is real and having well defined groundstate energy .
Prior to work of Bender ,people are aware of --x^4 potential . In my understanding it is a bounded operator .
I hope to give more elaborate answer in future as my paper is under press and it is a combined paper with outside authors.So I am restricted .
B.Rath
.
@Biswanath Rath,
With reference to your response, it is standard text-book knowledge that a potential (in one dimension) without a global/local minimum is unbounded and gives rise to instability, thus preventing meaningful spectrum.In contrast, your emphatic statement that the system has 'positive'-spectrum and the Hamiltonian is bounded is rather surprising. Hence, it would be very interesting if you could forward some proof/argument in support of your statement-and your paper ' in press' need not prevent you from this demonstration.
@Bimal Mahapatra
The groundstate of the system is 0 (zero) .
The Hamiltonians are H= p^2 - x^4 /pm 2x .
In my own limit , I am restricted to move further with your request . Practically in that paper I have commented ' author of highly read book in quantum mechanics " .
My simple suggestion is that why many hermitian operators do not give real spectra ?
The answer of bounded operator in the context of - x^4 lies with that.
B.Rath
@Bimal Mahapatra
In order to understand the real positive nature of spectrum of V(x)= -x^4 one should read this article first:
Z.Ahmed , C.M.Bender and M.V.Berry .J.Phys.A 38;L627(2005).
. B.Rath
@Biswanath Rath
You are right-one should read the relevant literature in the field, but what is more important, is to understand them first before referring these to others. In the context of your suggestion, I tend to believe that you have not perhaps understood the content and results of the suggested paper.
While it is possible to generate a positive acceptable spectrum of Hamiltonians (which are proven unphysical in the real-world context,) via a suitably chosen complex/non hermitian deformations, one must also clearly understand the boundary conditions necessary to achieve this. For the example under discussion, the wave function has to satisfy typical boundary conditions in some complex-domain of the coordinate ‘x’, which have little in common to physical boundary conditions for point spectrum in the physical space with real coordinate. It is no wonder that a non-hermitian Hamiltonian can possess real bounded spectrum when, simultaneously, ‘x’ is also analytically continued to complex-domain. But all these analytic- continuations are of mere academic value since they do not change the results/conclusions of quantum theory established in the physical world in real space (where experiments are done) described by Hermitian-Hamiltonians.
@Bimal Mohapatra
Spectral reality is not confined hermiticity only . As long as one thinks hermiticity , he can not well come the door for other symmetry
In my small understanding real spectra is reflected in SUSY which can be (1) hermiticity (2) PT symmetry (3) T symmetry
(4) PT ~T symmetry . One should not give emphasis on Dirac BRA or KET . However importance should be given BRAKET i.e < > . Secondly one must yake some time a paper in nature physics to understand PT material . I hope this will help somebody
to find some time before propagating hermiticity . When one wants to know the depth of sea at a place , he never goes to bottom but look at the surface .
My simple suggestion is that gather some knowledge in quantum system of real spectra and be ready to write a comment in my paper and publish the same , so that people will know the correct thing . Within 15 days paper will be available in Proc.Ind.Nat.Sci.Aca (A) .
@Bimal Mahapatra
My simple suggestion is that read my paper in RG on the p-pseudo-hermiticity work Mostfazadeh (having 5 PRL papers) and
Zafar Ahmed (having 50 PLA papers).So far they are not able to give a rebutal.Hope you can help them in getting a suitable rebutal , so that not only me but also all interested people will know the correct answer. I have controlled my self not to write further comment on the work of Zafar Ahmed because people (foreign countries) have advised me to not to point out mistakes openly. The PAPER which I have commented on Zafar was published in PRAMANA .
Hope you can write rebutals on their behalf,
so that I will have oppertunity to understand your understanding in real spectra in QM.
B.Rath
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