The generalized Kähler structures were introduced and studied by M. Gualtieri in his PhD thesis [16] in the more general context of generalized geometry started by N. Hitchin in [20]. There are many explicit constructions of non-trivial generalized-Kähler structures [1, 2, 21, 24, 25, 4, 7]. For instance Gualtieri proved that all compact-even dimensional semisimple Lie groups are generalized Kä...

In quantum mechanics, any quantum state ‘vector’ | i may be expanded as a linear superposition of the eigenvectors of any Hermitian operator | i = P n an|ni. This is the Expansion Principle of quantum mechanics. For most problems, the Hermitian operator of choice is the Hamiltonian operator Ĥ = (p̂/2m0) + V (r), but it need not be. We choose the Hamiltonian operator since there exist a few probl...

Recently, Bender et al. have considered the quantum brachistochrone problem for the non-Hermitian PT -symmetric quantum system and have shown that the optimal time evolution required to transform a given initial state |ψi〉 into a specific final state |ψf 〉 can be made arbitrarily small. Additionally, it has been shown that finding the shortest possible time requires only the solution of the two...

In this paper, we first construct a preconditioned two-parameter generalized Hermitian and skew-Hermitian splitting (PTGHSS) iteration method based on the two-parameter generalized Hermitian and skew-Hermitian splitting (TGHSS) iteration method for non-Hermitian positive definite linear systems. Then a class of PTGHSSbased iteration methods are proposed for solving weakly nonlinear systems base...

Motivated by the existence of bi-Hamiltonian classical systems and the correspondence principle, in this paper we analyze the problem of finding Hermitian scalar products which turn a given flow on a Hilbert space into a unitary one. We show how different invariant Hermitian scalar products give rise to different descriptions of a quantum system in the Ehrenfest and Heisenberg picture.

A harmonic oscillator Hamiltonian augmented by a non-Hermitian PT -symmetric part and its su(1,1) generalizations, for which a family of positive-definite metric operators was recently constructed, are re-examined in a supersymmetric context. QuasiHermitian supersymmetric extensions of such Hamiltonians are proposed by enlarging su(1,1) to a su(1, 1/1) ∼ osp(2/2,R) superalgebra. This allows the...

Consider the Lie algebras Lr,t : [K1,K2] = sK3, [K3,K1] = rK1, [K3,K2] = −rK2, [K3,K4] = 0, [K4,K1] = −tK1, and [K4,K2] = tK2, subject to the physical conditions, K3 and K4 are real diagonal operators representing energy, K2 = K † 1, and the Hamiltonian H = ω1K3 + (ω1 + ω2)K4 + λ(t)(K1e−iφ +K2e) is a Hermitian operator. Matrix representations are discussed and faithful representations of least ...

Any non-Hermitian PT −symmetric quantum Hamiltonian H remains physical in a domain of parameters D where the spectrum is real and, hence, measurable. It has been conjectured and verified, recently, that the postulate of a self-duality of the spectrum Ej = −EN+1−j could simplify the structure of the manifold D. An independent constructive support of the plausibility of such a relationship is giv...

The need to compute expressions of the form uf(A)v, where A is a large square matrix, u and v are vectors, and f is a function, arises in many applications, including network analysis, quantum chromodynamics, and the solution of linear discrete ill-posed problems. Commonly used approaches first reduce A to a small matrix by a few steps of the Hermitian or non-Hermitian Lanczos processes and the...

A structure preserving Sort-Jacobi algorithm for computing eigenvalues or singular values is presented. The proposed method applies to an arbitrary semisimple Lie algebra on its (−1)-eigenspace of the Cartan involution. Local quadratic convergence for arbitrary cyclic schemes is shown for the regular case. The proposed method is independent of the representation of the underlying Lie algebra an...