Let J be a unitary almost complex structure on a Riemannian manifold (M, g). If x is a unit tangent vector, let π := Span{x, Jx} be the associated complex line in the tangent bundle of M . The complex Jacobi operator and the complex curvature operators are defined, respectively, by J (π) := J (x) + J (Jx) and R(π) := R(x, Jx). We show that if (M, g) is Hermitian or if (M,g) is nearly Kähler, th...

Weyl Anomaly in the dilaton-scalar system in 2 dimensional gravity is examined. Discrepancy in recent results by some authors is clarified. We take the heat-kernel regularization for the ultraviolet divergences. In the present approach, the Weyl anomaly depends on the choice of an 2nd order differetial operator for the regularization. We have the freedom of its choice due to the arbitrariness o...

Among P-pseudo-Hermitian Hamiltonians H = P−1H† P with real spectra, the “weakly pseudo-Hermitian” ones (i.e., those employing non-self-adjoint P 6 = P†) form a remarkable subfamily. We list some reasons why it deserves a special attention. In particular we show that whenever P 6 = P†, the current involutive operator of charge C gets complemented by a nonequivalent alternative involutive quasip...

We list some reasons why among P−pseudo-Hermitian Hamiltonians H = P HP with real spectra, the “weakly pseudo-Hermitian” ones (i.e., those employing P 6 = P, even when belonging to the former class) form such an interesting special family that they deserve a special name and attention. In particular we show that for P 6 = P, the current auxiliary involutive operator of charge C gets complemente...

A complexified von Roos Hamiltonian is considered and a Hermitian first-order intertwining differential operator is used to obtain the related position dependent mass η-weak-pseudo-Hermitian Hamiltonians. Two ”user -friendly” reference-target maps are introduced to serve for exactsolvability of some non-Hermitian η-weak-pseudo-Hermitian position dependent mass Hamiltonians. A non-Hermitian PT -...

We investigate chiral properties of the domain-wall fermion (DWF) system by using the four-dimensional hermitian Wilson-Dirac operator. We first derive a formula which connects a chiral symmetry breaking term in the five dimensional DWF Ward-Takahashi identity with the four dimensional Wilson-Dirac operator, and simplify the formula in terms of only the eigenvalues of the operator , using an an...

An index relation dim ker a†a− dim ker aa† = 1 is satisfied by the creation and annihilation operators a† and a of a harmonic oscillator. A hermitian phase operator, which inevitably leads to dim ker a†a − dim ker aa† = 0, cannot be consistently defined. If one considers an s + 1 dimensional truncated theory, a hermitian phase operator of Pegg and Barnett which carries a vanishing index can be ...

Let X = G/P be a homogeneous space of a complex semisimple Lie group G equipped with a hermitian metric. We study the action of the Hodge star operator on the space of harmonic differential forms on X. We obtain explicit combinatorial formulas for this action when X is an irreducible hermitian symmetric space of compact type.

A transformation of the form x → ±iy ∈ iR; x, y ∈ R, or an equivalent similarity transformation with a metric operator η are shown to map non-Hermitian PT -symmetric Hamiltonians into Hermitian partner Hamiltonians in Hilbert space. Isospectrality and mass signature are also discussed. PACS codes: 03.65.Ge, 03.65.Ca