baldassarri baronchelli barrat caglioti castellano cattuto complexity dattoli evolutionary_dynamics granular_media granular_media herrmann information_theory innovation_dynamics kreyon language_dynamics loreto mari mukherjee petri pietronero puglisi quantum_optics self_organization servedio statistical_physics techno_social_systems tria zapperi zippers
1991 |
Dattoli, Giuseppe; Loreto, Vittorio; Mari, Carlo; Richetta, Maria; Torre, Amalia Biunitary transformations and ordinary differential equations - I Journal Article LA RIVISTA DEL NUOVO CIMENTO DELLA SOCIETÀ ITALIANA DI FISICA, 106B , pp. 1357–1374, 1991. Abstract | Links | BibTeX | Tags: dattoli, loreto, mari, quantum_optics, richetta, torre @article{b, title = {Biunitary transformations and ordinary differential equations - I}, author = {Giuseppe Dattoli and Vittorio Loreto and Carlo Mari and Maria Richetta and Amalia Torre}, url = {http://dx.doi.org/10.1007/BF02728366, http://www.scopus.com/inward/record.url?eid=2-s2.0-51249174048&partnerID=65&md5=71e6cba395d9f74b6cdd9da971d3c6e0 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=A1991HE48800002&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=0c7ff228ccbaaa74236f48834a34396a}, year = {1991}, date = {1991-01-01}, journal = {LA RIVISTA DEL NUOVO CIMENTO DELLA SOCIETÀ ITALIANA DI FISICA}, volume = {106B}, pages = {1357--1374}, publisher = {Italian Physical Society}, abstract = {We reformulate the theory of ordinary differential equations of arbitrary order with nonconstant coefficients, using the formalism of non-Hermitian operators. In particular, exploiting the technique of dissipative quantum mechanics, we show that the solution of the equations can be written in terms of a nonunitary evolution operator. Furthermore, we point out that the solution of the adjoint equations can be derived from an associated biunitary operator. We show that a number of invariants, not previously discussed, exhists. Finally, we prove that the method allows the search for approximate solutions that can be used in many physical problems.}, keywords = {dattoli, loreto, mari, quantum_optics, richetta, torre}, pubstate = {published}, tppubtype = {article} } We reformulate the theory of ordinary differential equations of arbitrary order with nonconstant coefficients, using the formalism of non-Hermitian operators. In particular, exploiting the technique of dissipative quantum mechanics, we show that the solution of the equations can be written in terms of a nonunitary evolution operator. Furthermore, we point out that the solution of the adjoint equations can be derived from an associated biunitary operator. We show that a number of invariants, not previously discussed, exhists. Finally, we prove that the method allows the search for approximate solutions that can be used in many physical problems. |
Dattoli, Giuseppe; Loreto, Vittorio; Mari, Carlo; Richetta, Maria; Torre, Amalia Biunitary transformations and ordinary differential equations - II Journal Article LA RIVISTA DEL NUOVO CIMENTO DELLA SOCIETÀ ITALIANA DI FISICA, 106 , pp. 1375–1390, 1991. Abstract | Links | BibTeX | Tags: dattoli, loreto, mari, quantum_optics, richetta, torre @article{b, title = {Biunitary transformations and ordinary differential equations - II}, author = {Giuseppe Dattoli and Vittorio Loreto and Carlo Mari and Maria Richetta and Amalia Torre}, url = {http://rd.springer.com/article/10.1007/BF02728367, http://www.scopus.com/inward/record.url?eid=2-s2.0-51249177614&partnerID=65&md5=2b85bcbf36039a33533f9e968c2e627e http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=A1991HE48800003&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=0c7ff228ccbaaa74236f48834a34396a}, year = {1991}, date = {1991-01-01}, journal = {LA RIVISTA DEL NUOVO CIMENTO DELLA SOCIETÀ ITALIANA DI FISICA}, volume = {106}, pages = {1375--1390}, publisher = {EDITRICE COMPOSITORI BOLOGNA, VIA STALINGRADO 97/2, I-40128 BOLOGNA, ITALY}, abstract = {We use the results of a recent reformulation of the theory of arbitrary-order differential equations in terms of non-Hermitian operators to show that the invariant binorm is associated to a generalized Courant-Snyder invariant. Furthermore, we indicate the existence of higher-order invariants associated to the Casimir operators of the group, utilized to treat higher-order equations. We also discuss the intrinsic supersymmetric nature of the theory developed. Finally, we show the relevance of the proposed mathematical technique to the design of fiber-optics transport systems.}, keywords = {dattoli, loreto, mari, quantum_optics, richetta, torre}, pubstate = {published}, tppubtype = {article} } We use the results of a recent reformulation of the theory of arbitrary-order differential equations in terms of non-Hermitian operators to show that the invariant binorm is associated to a generalized Courant-Snyder invariant. Furthermore, we indicate the existence of higher-order invariants associated to the Casimir operators of the group, utilized to treat higher-order equations. We also discuss the intrinsic supersymmetric nature of the theory developed. Finally, we show the relevance of the proposed mathematical technique to the design of fiber-optics transport systems. |
Dattoli, Giuseppe; Loreto, Vittorio; Mari, Carlo; Richetta, Maria; Torre, Amalia Biunitary transformations and ordinary differential equations - III Journal Article LA RIVISTA DEL NUOVO CIMENTO DELLA SOCIETÀ ITALIANA DI FISICA, 106 B , pp. 1391–1399, 1991. Abstract | Links | BibTeX | Tags: dattoli, loreto, mari, quantum_optics, richetta, torre @article{b, title = {Biunitary transformations and ordinary differential equations - III}, author = {Giuseppe Dattoli and Vittorio Loreto and Carlo Mari and Maria Richetta and Amalia Torre}, url = {http://rd.springer.com/article/10.1007/BF02728368, http://www.scopus.com/inward/record.url?eid=2-s2.0-51249177364&partnerID=65&md5=84b216f43701ba80b3ce08d77b98f277 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=A1991HE48800004&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=0c7ff228ccbaaa74236f48834a34396a}, year = {1991}, date = {1991-01-01}, journal = {LA RIVISTA DEL NUOVO CIMENTO DELLA SOCIETÀ ITALIANA DI FISICA}, volume = {106 B}, pages = {1391--1399}, publisher = {EDITRICE COMPOSITORI BOLOGNA}, abstract = {In two previous papers, the authors have introduced the concept of binormal differential equations. It has been shown that invariants of the Courant-Snyder type are associated to the scalar products of the column vector associated to an ordinary differential equation and to its binormal. In this paper, we show the equivalence of the above invariant and the Lewis form. We also introduce a density matrix for a second-order differential equation and clarify the geometrical meaning of the Twiss parameters. The importance of the above results in the analysis of quantum problems such as, e.g., the evolution of squeezed states is finally stressed.}, keywords = {dattoli, loreto, mari, quantum_optics, richetta, torre}, pubstate = {published}, tppubtype = {article} } In two previous papers, the authors have introduced the concept of binormal differential equations. It has been shown that invariants of the Courant-Snyder type are associated to the scalar products of the column vector associated to an ordinary differential equation and to its binormal. In this paper, we show the equivalence of the above invariant and the Lewis form. We also introduce a density matrix for a second-order differential equation and clarify the geometrical meaning of the Twiss parameters. The importance of the above results in the analysis of quantum problems such as, e.g., the evolution of squeezed states is finally stressed. |
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