Pub Date : 2018-11-21DOI: 10.1142/9781786345967_0004
C. Bender, Daniel W. Hook
This paper reports the results of an ongoing in-depth analysis of the classical trajectories of the class of non-Hermitian PT -symmetric Hamiltonians H = p+x(ix) (ε ≥ 0). A variety of phenomena, heretofore overlooked, have been discovered such as the existence of infinitely many separatrix trajectories, sequences of critical initial values associated with limiting classical orbits, regions of broken PT -symmetric classical trajectories, and a remarkable topological transition at ε = 2. This investigation is a work in progress and it is not complete; many features of complex trajectories are still under study.
{"title":"PT-Symmetric Classical Mechanics","authors":"C. Bender, Daniel W. Hook","doi":"10.1142/9781786345967_0004","DOIUrl":"https://doi.org/10.1142/9781786345967_0004","url":null,"abstract":"This paper reports the results of an ongoing in-depth analysis of the classical trajectories of the class of non-Hermitian PT -symmetric Hamiltonians H = p+x(ix) (ε ≥ 0). A variety of phenomena, heretofore overlooked, have been discovered such as the existence of infinitely many separatrix trajectories, sequences of critical initial values associated with limiting classical orbits, regions of broken PT -symmetric classical trajectories, and a remarkable topological transition at ε = 2. This investigation is a work in progress and it is not complete; many features of complex trajectories are still under study.","PeriodicalId":139017,"journal":{"name":"PT Symmetry","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115491022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-11-21DOI: 10.1142/9781786345967_0003
B. Rath
We propose a model CCS(complex-conjugate-space) to understand the inner and outer product nature of wave functions in non-hermitian PT-symmetry model in quantum mechanics considering (N × N) matrix model . Further we reflect the correct nature of C-symmetry ,P-parity , T-time reversal in matrix form and the orignal Hamiltonian matrix for any arbitrary values of N. Interestingly the present result on C and P for N=2 , remains the same reported earlier by Bender,Brody and Jones model PT-symmetry operaor . In non-conventional way one can notice that wave functions in a PT-symmetry model satifies similar relations as in hermitian operator . PACS no-03.65.Db Key wordsPT-symmetry ,Hermiticity ,wave functions , properties ,C-symmetry 1.Introduction Since the development of quantum mechanics it is well known that Wave functions in Hermitian operator satisfies the following relations[1,2]
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Pub Date : 2018-11-21DOI: 10.1142/9781786345967_0008
{"title":"Kreĭn-Space Theory and PTQM","authors":"","doi":"10.1142/9781786345967_0008","DOIUrl":"https://doi.org/10.1142/9781786345967_0008","url":null,"abstract":"","PeriodicalId":139017,"journal":{"name":"PT Symmetry","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129759540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-11-21DOI: 10.1142/9781786345967_0006
{"title":"Proof of Reality for Some Simple Examples","authors":"","doi":"10.1142/9781786345967_0006","DOIUrl":"https://doi.org/10.1142/9781786345967_0006","url":null,"abstract":"","PeriodicalId":139017,"journal":{"name":"PT Symmetry","volume":"2628 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129624887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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