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We use the shear construction to construct and classify a wide range of two-step solvable Lie groups admitting a left-invariant SKT structure. We reduce this to a specification of SKT shear data on Abelian Lie algebras, and which then is studied more deeply in different cases. We obtain classifications and structure results for $mathfrak{g}$ almost Abelian, for derived algebra $mathfrak{g}$ of codimension 2 and not $J$-invariant, for $mathfrak{g}$ totally real, and for $mathfrak{g}$ of dimension at most 2. This leads to a large part of the full classification for two-step solvable SKT algebras of dimension six.
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Motivated by the interest in non-relativistic quantum mechanics for determining exact solutions to the Schrodinger equation we give two potentials that are conditionally exactly solvable. The two potentials are partner potentials and we obtain that each linearly independent solution of the Schrodinger equation includes two hypergeometric functions. Furthermore we calculate their reflection and transmission amplitudes. Finally we discuss some additional properties of these potentials.
We answer in the affirmative a question posed by Ivanov and Vassilev on the existence of a seven dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the classification of seven dimensional solvable Lie groups having an integrable left invariant quaternionic contact structure. In particular, we prove that the unique seven dimensional nilpotent Lie group admitting such a structure is the quaternionic Heisenberg group.
A compact solvmanifold of completely solvable type, i.e. a compact quotient of a completely solvable Lie group by a lattice, has a Kahler structure if and only if it is a complex torus. We show more in general that a compact solvmanifold $M$ of completely solvable type endowed with an invariant complex structure $J$ admits a symplectic form taming J if and only if $M$ is a complex torus. This result generalizes the one obtained in [7] for nilmanifolds.
A dipole sequence has been observed and investigated in the 143 Sm nucleus populated through the heavy-ion induced fusion-evaporation reaction and studied using the Indian National Gamma Array (INGA) as the detection system. The sequence has been established as a Magnetic Rotation (MR) band primarily from lifetime measurements of the band members using the Doppler Shift Attenuation Method (DSAM). A configuration based on nine quasiparticles, with highly asymmetric angular momentum blades, has been assigned to the shears band in the light of the theoretical calculations within the framework of Shears mechanism with the Principal Axis Cranking (SPAC) model. This is hitherto the maximum number of quasiparticles along with the highest asymmetricity associated with a MR band. Further, as it has followed from the SPAC calculations, the contribution of the core rotation to the angular momentum of this shears band is substantial and greater than in any other similar sequence, at least in the neighbouring nuclei. This band can thus be perceived as a unique phenomenon of shears mechanism in operation at the limits of quasiparticle excitations, as manifested in MR band-like phenomena, evolving into collectivity.
The paper is a continuation of the authors et al.s work in the first half of the year 2021. It has classified a special class of 7-dimensional real solvable Lie algebras such that the nilradical of each from them is well-known 5-dimensional nilpotent Lie algebra in that work. In this paper, we will consider exponential, connected and simply connected Lie groups which are corresponding to these Lie algebras. Namely, we will describe the geometry of generic (i.e. 6-dimensional) orbits in coadjoint representation of considered Lie groups. Next, we will prove that for each considered group, the family of generic coadjoint orbits forms a measurable foliation in the sense of Connes and give the topological classification of these foliations.
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