http://dspace.nbuv.gov.ua:80/handle/123456789/145987 2026-04-06T10:05:19Z 2026-04-06T10:05:19Z Znojil, M. http://dspace.nbuv.gov.ua:80/handle/123456789/149281 2019-02-19T23:27:23Z 2009-01-01T00:00:00Z

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.

2009-01-01T00:00:00Z Burdík, C. Nersessian, A. http://dspace.nbuv.gov.ua:80/handle/123456789/149262 2019-02-19T23:28:06Z 2009-01-01T00:00:00Z

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.

2009-01-01T00:00:00Z Dimakis, A. Müller-Hoissen, F. http://dspace.nbuv.gov.ua:80/handle/123456789/149261 2019-02-19T23:28:04Z 2009-01-01T00:00:00Z

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.

2009-01-01T00:00:00Z Nakajima, H. http://dspace.nbuv.gov.ua:80/handle/123456789/149260 2019-02-19T23:26:37Z 2009-01-01T00:00:00Z

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