Symmetry, Integrability and Geometry: Methods and Applications, 2009, том 5, випуск за цей рік
http://dspace.nbuv.gov.ua:80/handle/123456789/145988
2020-04-05T14:36:59ZThree-Hilbert-Space Formulation of Quantum Mechanics
http://dspace.nbuv.gov.ua:80/handle/123456789/149281
Three-Hilbert-Space Formulation of Quantum Mechanics
Znojil, M.
In paper [Znojil M., Phys. Rev. D 78 (2008), 085003, 5 pages, arXiv:0809.2874] the two-Hilbert-space (2HS, a.k.a. cryptohermitian) formulation of Quantum Mechanics has been revisited. In the present continuation of this study (with the spaces in question denoted as H(auxiliary) and H(standard)) we spot a weak point of the 2HS formalism which lies in the double role played by H(auxiliary). As long as this confluence of roles may (and did!) lead to confusion in the literature, we propose an amended, three-Hilbert-space (3HS) reformulation of the same theory. As a byproduct of our analysis of the formalism we offer an amendment of the Dirac's bra-ket notation and we also show how its use clarifies the concept of covariance in time-dependent cases. Via an elementary example we finally explain why in certain quantum systems the generator H(gen) of the time-evolution of the wave functions may differ from their Hamiltonian H.
2009-01-01T00:00:00ZRemarks on Multi-Dimensional Conformal Mechanics
http://dspace.nbuv.gov.ua:80/handle/123456789/149262
Remarks on Multi-Dimensional Conformal Mechanics
Burdík, C.; Nersessian, A.
Recently, Galajinsky, Lechtenfeld and Polovnikov proposed an elegant group-theoretical transformation of the generic conformal-invariant mechanics to the free one. Considering the classical counterpart of this transformation, we relate this transformation with the Weil model of Lobachewsky space.
2009-01-01T00:00:00ZMulticomponent Burgers and KP Hierarchies, and Solutions from a Matrix Linear System
http://dspace.nbuv.gov.ua:80/handle/123456789/149261
Multicomponent Burgers and KP Hierarchies, and Solutions from a Matrix Linear System
Dimakis, A.; Müller-Hoissen, F.
Via a Cole-Hopf transformation, the multicomponent linear heat hierarchy leads to a multicomponent Burgers hierarchy. We show in particular that any solution of the latter also solves a corresponding multicomponent (potential) KP hierarchy. A generalization of the Cole-Hopf transformation leads to a more general relation between the multicomponent linear heat hierarchy and the multicomponent KP hierarchy. From this results a construction of exact solutions of the latter via a matrix linear system.
2009-01-01T00:00:00ZQuiver Varieties and Branching
http://dspace.nbuv.gov.ua:80/handle/123456789/149260
Quiver Varieties and Branching
Nakajima, H.
Braverman and Finkelberg recently proposed the geometric Satake correspondence for the affine Kac-Moody group Gaff [Braverman A., Finkelberg M., arXiv:0711.2083]. They conjecture that intersection cohomology sheaves on the Uhlenbeck compactification of the framed moduli space of Gcpt-instantons on R4/Zr correspond to weight spaces of representations of the Langlands dual group GaffÚ at level r. When G = SL(l), the Uhlenbeck compactification is the quiver variety of type sl(r)aff, and their conjecture follows from the author's earlier result and I. Frenkel's level-rank duality. They further introduce a convolution diagram which conjecturally gives the tensor product multiplicity [Braverman A., Finkelberg M., Private communication, 2008]. In this paper, we develop the theory for the branching in quiver varieties and check this conjecture for G = SL(l).
2009-01-01T00:00:00Z