{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:43:56Z","timestamp":1760150636918,"version":"build-2065373602"},"reference-count":54,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,12,24]],"date-time":"2023-12-24T00:00:00Z","timestamp":1703376000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Reinterpretation of mathematics behind the exactly solvable Calogero\u2019s A-particle quantum model is used to propose its generalization. Firstly, it is argued that the strongly singular nature of Calogero\u2019s particle\u2013particle interactions makes the original permutation-invariant Hamiltonian tractable as a direct sum H=\u2a01Ha of isospectral components, which are mutually independent. Secondly, after the elimination of the center-of-mass motion, the system is reconsidered as existing in the reduced Euclidean space RA\u22121 of relative coordinates and decaying into a union of subsets Wa called Weyl chambers. The mutual independence of the related reduced forms of operators Ha enables us to makes them nonisospectral. This breaks the symmetry and unfolds the spectral degeneracy of H. A new multiparametric generalization of the conventional A-body Calogero model is obtained. Its detailed description is provided up to A=4.<\/jats:p>","DOI":"10.3390\/sym16010027","type":"journal-article","created":{"date-parts":[[2023,12,24]],"date-time":"2023-12-24T20:44:15Z","timestamp":1703450655000},"page":"27","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Calogero-like Model without Rearrangement Symmetry"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6076-0093","authenticated-orcid":false,"given":"Miloslav","family":"Znojil","sequence":"first","affiliation":[{"name":"The Czech Academy of Sciences, Nuclear Physics Institute, Hlavn\u00ed 130, 250 68 \u0158e\u017e, Czech Republic"},{"name":"Department of Physics, Faculty of Science, University of Hradec Kr\u00e1lov\u00e9, Rokitansk\u00e9ho 62, 500 03 Hradec Kr\u00e1lov\u00e9, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1007\/BF00420664","article-title":"Quantum completely integrable systems connected with semi-simple Lie algebras","volume":"2","author":"Olshanetsky","year":"1977","journal-title":"Lett. Math. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/0370-1573(83)90018-2","article-title":"Quantum integrable systems related to Lie algebras","volume":"94","author":"Olshanetsky","year":"1983","journal-title":"Phys. Rept."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/0370-2693(94)90657-2","article-title":"Hidden algebra of the N-body Calogero problem","volume":"320","author":"Turbiner","year":"1994","journal-title":"Phys. Lett. B"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2191","DOI":"10.1063\/1.1664820","article-title":"Solution of a three-body problem in one dimension","volume":"10","author":"Calogero","year":"1969","journal-title":"J. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2197","DOI":"10.1063\/1.1664821","article-title":"Ground state of a one-dimensional N-body problem","volume":"10","author":"Calogero","year":"1969","journal-title":"J. Math. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1063\/1.1665604","article-title":"Solution of the one-dimensional N-body problem with quadratic and\/or inversely quadratic pair potentials","volume":"12","author":"Calogero","year":"1971","journal-title":"J. Math. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Van Diejen, J.F., and Vinet, L. (2000). Calogero-Moser- Sutherland Models (CRM Series in Mathematical Physics), Springer.","DOI":"10.1007\/978-1-4612-1206-5"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2213","DOI":"10.1142\/S0217732395002374","article-title":"Exact solvability of the Calogero and Sutherland models","volume":"10","author":"Turbiner","year":"1995","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Sutherland, B. (2004). Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems, World Scientific.","DOI":"10.1142\/5552"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"12793","DOI":"10.1088\/0305-4470\/39\/41\/S07","article-title":"Physics and mathematics of Calogero particles","volume":"39","author":"Polychronakos","year":"2006","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"7216","DOI":"10.4249\/scholarpedia.7216","article-title":"Calogero-Moser system","volume":"3","author":"Calogero","year":"2008","journal-title":"Scholarpedia"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1103\/PhysRevA.58.821","article-title":"Supersymmetry, shape invariance, and solvability of AN-1 and BCN Calogero-Sutherland model","volume":"58","author":"Ghosh","year":"1998","journal-title":"Phys. Rev. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"9033","DOI":"10.1088\/0305-4470\/33\/49\/303","article-title":"Quantum Calogero-Moser Models: Integrability for all Root Systems","volume":"33","author":"Khastgir","year":"2000","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s00220-005-1401-y","article-title":"Solvability of the Hamiltonians related to exceptional root spaces: Rational case","volume":"260","author":"Boreskov","year":"2005","journal-title":"Commun. Math. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Dunkl, C.F., and Hu, Y. (2001). Orthogonal Polynomials of Several Variables, Cambridge University Press.","DOI":"10.1017\/CBO9780511565717"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1007\/s002200050174","article-title":"Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement","volume":"188","year":"1997","journal-title":"Commun. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1950190","DOI":"10.1142\/S0217732319501906","article-title":"One-dimensional quantum mechanics with Dunkl derivative","volume":"34","author":"Chung","year":"2019","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"169445","DOI":"10.1016\/j.aop.2023.169445","article-title":"Investigation of the generalised Wigner-Dunkl harmonic oscillator and its coherent states","volume":"458","author":"Sedaghatnia","year":"2023","journal-title":"Ann. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/0003-4916(92)90284-S","article-title":"Quasi-Hermitian Operators in Quantum Mechanics and the Variational Principle","volume":"213","author":"Scholtz","year":"1992","journal-title":"Ann. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1793","DOI":"10.1088\/0305-4470\/34\/8\/321","article-title":"Complex Calogero model with real energies","volume":"34","author":"Znojil","year":"2001","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/S0375-9601(01)00299-7","article-title":"Exactly solvable three-body Calogero-type model with translucent two-body barriers","volume":"284","author":"Znojil","year":"2001","journal-title":"Phys. Lett. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.physleta.2004.01.020","article-title":"On the real spectra of Calogero model with complex coupling","volume":"323","author":"Ghosh","year":"2004","journal-title":"Phys. Lett. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"425206","DOI":"10.1088\/1751-8113\/42\/42\/425206","article-title":"From real fields to complex Calogero particles","volume":"42","author":"Assis","year":"2009","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"325201","DOI":"10.1088\/1751-8113\/43\/32\/325201","article-title":"Antilinear deformations of Coxeter groups, an application to Calogero models","volume":"43","author":"Fring","year":"2010","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Bagarello, F., Gazeau, J.-P., Szafraniec, F., and Znojil, M. (2015). Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects, Wiley.","DOI":"10.1002\/9781118855300"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/S0375-9601(99)00429-6","article-title":"PT symmetric harmonic oscillators","volume":"259","author":"Znojil","year":"1999","journal-title":"Phys. Lett. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1023\/B:CJOP.0000014369.89531.94","article-title":"PT-symmetric Calogero-type model","volume":"54","author":"Jakubsky","year":"2004","journal-title":"Czechosl. J. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4391","DOI":"10.1142\/S0217751X04019858","article-title":"On PT symmetric extensions of the Calogero model","volume":"19","author":"Brihaye","year":"2004","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"194010","DOI":"10.1088\/1751-8113\/41\/19\/194010","article-title":"PT-symmetric deformations of Calogero models","volume":"41","author":"Fring","year":"2008","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_30","first-page":"20120046","article-title":"PT-symmetric deformations of integrable models","volume":"371","author":"Fring","year":"2013","journal-title":"Phil. Trans. Roy. Soc. Lond. A"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"012007","DOI":"10.1088\/1742-6596\/2038\/1\/012007","article-title":"Algebraic integrability of PT-deformed Calogero models","volume":"2038","author":"Correa","year":"2021","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.physleta.2004.11.020","article-title":"Trigonometric identities, angular Schr\u00f6dinger equations and a new family of solvable models","volume":"334","author":"Jakubsky","year":"2005","journal-title":"Phys. Lett. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"205206","DOI":"10.1088\/1751-8113\/42\/20\/205206","article-title":"The cuboctahedric Higgs oscillator from the rational Calogero model","volume":"42","author":"Hakobyan","year":"2009","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1023\/A:1023231402145","article-title":"Intertwining relations for the spherical parts of generalized Calogero operators","volume":"135","author":"Feigin","year":"2003","journal-title":"Theor. Math. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1016\/j.nuclphysb.2012.01.002","article-title":"The spherical sector of the Calogero model as a reduced matrix model","volume":"858","author":"Hakobyan","year":"2012","journal-title":"Nucl. Phys. B"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1007\/JHEP07(2013)162","article-title":"The quantum angular Calogero-Moser model","volume":"1307","author":"Feigin","year":"2013","journal-title":"J. High Energy Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/JHEP10(2015)191","article-title":"The tetrahexahedric angular Calogero model","volume":"1510","author":"Correa","year":"2015","journal-title":"J. High Energy Phys."},{"key":"ref_38","unstructured":"Landau, L.D., and Lifshitz, E.M. (1977). Quantum Mechanics: Non-Relativistic Theory, Pergamon."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1007\/BF02099456","article-title":"Exact Operator Solution of the Calogero-Sutherland Model","volume":"178","author":"Lapointe","year":"1996","journal-title":"Commun. Math. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"066101","DOI":"10.1103\/PhysRevA.61.066101","article-title":"Comment on conditionally exactly soluble class of quantum potentials","volume":"61","author":"Znojil","year":"2000","journal-title":"Phys. Rev. A"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Ishkhanyan, A.M. (2023, January 28\u201331). Conditionally exactly solvable Dirac potential, including x1\/3 pseudoscalar interaction. Proceedings of the International Conference of Analytic and Algebraic Methods in Physics XX (FNSPE), Prague, Czech Republic.","DOI":"10.1088\/1402-4896\/acdf2a"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"13482","DOI":"10.1038\/s41598-023-40604-9","article-title":"Exact solution of the 1D Dirac equaiton for a pseudoscalar interaction potential with the inverse-square-root variation law","volume":"13","author":"Ishkhanyan","year":"2023","journal-title":"Sci. Rep."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"075229","DOI":"10.1088\/1402-4896\/acdf2a","article-title":"Conditionally exactly solvable Dirac potential, including x1\/3 pseudoscalar interaction","volume":"98","author":"Ishkhanyan","year":"2023","journal-title":"Phys. Scr."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Fl\u00fcgge, S. (1971). Practical Quantum Mechanics I, Springer.","DOI":"10.1007\/978-3-642-61995-3"},{"key":"ref_45","first-page":"113","article-title":"Quasi-solvability of Calogero-Sutherland model","volume":"3","author":"Post","year":"1995","journal-title":"Russ. J. Math. Phys."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"660","DOI":"10.2991\/jnmp.2005.12.s1.51","article-title":"Quasi-Exactly Solvable Hamiltonians related to Root Spaces","volume":"12","author":"Turbiner","year":"2005","journal-title":"J. Nonlin. Math. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1285","DOI":"10.1063\/1.532380","article-title":"Hidden algebras of the (super) Calogero and Sutherland models","volume":"39","author":"Brink","year":"1998","journal-title":"J. Mathe. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"4769","DOI":"10.1142\/S0217751X0100550X","article-title":"Solvability of the F4 integrable system","volume":"16","author":"Boreskov","year":"2001","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Humphreys, J.E. (1990). Reflection Groups and Coxeter Groups, Cambridge University Press.","DOI":"10.1017\/CBO9780511623646"},{"key":"ref_50","unstructured":"Coxeter, H.S.M. (1973). Regular Polytopes, Dover Publications. [3rd ed.]."},{"key":"ref_51","unstructured":"(2023, November 27). Available online: https:\/\/commons.wikimedia.org\/wiki\/File:Disdyakis_6_spherical.png."},{"key":"ref_52","unstructured":"(2023, November 27). Available online: https:\/\/en.wikipedia.org\/wiki\/File:Tetrakishexahedron.jpg."},{"key":"ref_53","unstructured":"(2023, November 27). Available online: https:\/\/commons.wikimedia.org\/wiki\/File:Disdyakis_6_in_rhombic_6.png."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/0370-2693(92)90166-2","article-title":"Explicit solution to the N body Calogero problem","volume":"286","author":"Brink","year":"1992","journal-title":"Phys. Lett. B"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/1\/27\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:41:21Z","timestamp":1760132481000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/1\/27"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,24]]},"references-count":54,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1]]}},"alternative-id":["sym16010027"],"URL":"https:\/\/doi.org\/10.3390\/sym16010027","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,12,24]]}}}