The electronic structure of few-electron atoms, in particular near their critical charge (below which the outermost electron is not bound): In most of their electronic states, atoms become “simple” in this limit (in a technically precise sense) but qualitatively very different from the corresponding high nuclear charge isoelectronic analogues, which become asymptotically hydrogen-like. The behavior of neutral atoms appears to be closer to that in the vicinity of the critical charge. These observations allow a refinement of my work on the dynamical origin (“interpretation”) of Hund’s rule ordering in open-shell atoms, going back to the early seventies of the previous century [that, though sometimes misrepresented, affected the textbook presentation of this topic]. Some atomic electronic states (including the ground states of the rare-gases) remain “complex” upon approaching the critical charge (these two types of behavior have technically been referred to as “expanding” and “absorbing”, respectively). These issues involve several open problems that I am currently dealing with, and a promise of further qualitative insight.
Efficient combinatorial algorithms for the characters of the symmetric group and for the structure constants in its group-algebra: Though this is a mathematically solved problem, it becomes computationally intractable even in applications to systems consisting of a moderate number of identical particles. Some powerful simplifications (some of which also apply to the unitary groups, the quantum unitary groups, and to the Hecke algebra of the symmetric group) were elucidated in a series of articles that I wrote since the late eighties [becoming textbook items, cf. T. Ceccherini-Silberstein et al., Representation Theory of the Symmetric Group, 2010]. However, further elucidation of certain rather challenging “details” is definitely called for, and I intend to keep trying for a while.
Permutational symmetry classification of identical higher-spin particles: My original interest in this issue was motivated by the fact that the simplicity of the corresponding problem for spin-½ particles is not shared by higher spins. Specifically, the one-to-one correspondence between the total spin and the label of the irreducible representation does not hold for n identical particles with elementary spins s≥1 The current interest in higher spin Bose-Einstein condensates provides a motivation for further refinement and extension of results that I published concerning the classification of such systems about a decade ago.
Is there a bound (1σg,1σu) 3∑(zero bond order) state of a (heavy hole) biexciton? Such a state was predicted to be feasible in a polar crystal [thesis by Kamer Murat, that I supervised in the late seventies] but so far never observed. Recent developments in very low temperature solid state spectroscopy suggest that a more up-to-date treatment of this problem may be of interest. I may get to doing something about this one of these days.
1. J. Katriel, An interpretation of Hund’s rule, Theoret. Chim. Acta 26, 163-170 (1972).
2. K. Murat and J. Katriel, The lowest triplet state of the biexciton may be bound, Phys. Lett. 71A, 143-145 (1979).
3. J. Katriel and E.R. Davidson, Shellwise virial scaling: An approximation for atomic hole states, Intern. J. Quantum Chem. 18, 1049-1055 (1980).
4. J. Katriel and D.G. Hummer, Analytic solutions for three- and four-wave mixing via generalised Bose operators, J. Phys. A 14, 1211-1224 (1981). MR0611983.
5. J. Katriel, Reduction of the excited state into the ground state of a super-Hamiltonian, Intern. J. Quantum Chem. 23, 1767-1780 (1983).
6. M. Orenstein, S. Speiser and J. Katriel, An eikonal approximation for non linear resonators exhibiting bistability, Opt. Comm.48, 367-373 (1984).
7. J. Katriel and A. Rosenhouse, The classical limit of the Korteweg deVries hierarchy of isospectral transformations, Phys. Rev. D 32, 884-890 (1985). MR0803860.
8. J. Katriel, M. Rasetti and A. I. Solomon, Generalized Holestein-Primakoff squeezed states for SU(n), Phys. Rev. D 35, 2601-2602 (1987).
9. A. Novoselsky, J. Katriel and R. Gilmore, Coefficients of fractional parentage in the L-S coupling scheme, J. Math. Phys. 29, 1368-1388 (1988). MR0944451.
10. J. Katriel and A. Novoselsky, Multi-cluster wavefunctions with arbitrary permutational symmetry, Ann. Phys. (N.Y.) 211, 1-23 (1991). MR1128181.
11. J. Katriel and M. Kibler, Normal ordering for deformed boson operators and operator-valued deformed Stirling numbers, J. Phys. A. 25, 2683-2691 (1992) MR1164371.
12. A. Govari, A. Mann and J. Katriel, Fractional quantum Hall effect studied by means of a nonspurious basis set, Phys. Rev. B48, 11404-11407 (1993).
13. J. Katriel, Permutations as minimal powers of a single-cycle class-sum, Europ. J. Combinatorics 16, 633-637 (1995).MR1356852.
14. J. Katriel, B. Abdesselam and A. Chakrabarti, The fundamental invariant of the Hecke algebra Hn(q) characterizes the representations of Hn(q), Sn, SUq(N) and SU(N), J. Math. Phys. 36, 5139-5158 (1995). MR1347137, Zbl 0847.20041.
15. J. Katriel, Explicit expressions for the central characters of the symmetric group, Discrete Appl. Math. 67, 149-156 (1996).MR1393301.
16. J. Katriel and C. Quesne, Recursively minimally-deformed oscillators, J. Math. Phys. 37, 1650-1661 (1996). MR1380866, Zbl 0879.47048.
17. J. Katriel, Minimal set of class-sums characterizing the ordinary irreducible representations of the symmetric group, and the Tarry-Escott problem, Discrete Math. 173, 91-95 (1997). MR1468843.
18. J. Katriel, Products of arbitrary class-sums in the symmetric group, Intern. J. Quantum Chem. 70, 429-440 (1998).
19. J. Katriel, F. Zahariev and K. Burke, Symmetry and degeneracy in density functional theory, Int. J. Quant. Chem. 85, 432-435 (2001).
20. J. Katriel, Refined Stirling numbers: Enumeration of special sets of permutations and set-partitions, J. Combinatorial Theory A, 99, 85-94 (2002). MR1911458, Zbl 1013.05009.
21. J. Katriel, Coherent states and combinatorics, J. Opt. B: Quantum Semiclass. Opt. 4, S200-S203 (2002). MR1920218.
22. J. Katriel, Weights of spin and permutational-symmetry adapted states for arbitrary elementary spins, in Fundamental World of Quantum Chemistry, Vol III, E. Brandas and E. Kryachko, Eds. (2004).
23. J. Katriel, S. Roy and M. Springborg, Non-universality of commonly used correlation-energy density-functionals, J. Chem. Phys. 124, 234111-(1-11) (2006).
24. J. Katriel and V. Buˇzek, Multipartite EPR states, Mol. Phys. 106, 497-508 (2008).
25. Z. Li, C. Bao and J. Katriel, Equilibrium population of spin components in a trapped spin-1 Bose gas, Phys. Rev. A 77, 023614-(1-2) (2008).
26. J. Katriel, A multitude of expressions for the Stirling numbers of the first kind, Integers: Electronic J. Comb. Number Theory (#A22) 10, 273-297 (2010). MR2652565.
27. J. Katriel and R. Gilmore, Entropy of bounding tori, Entropy 12, 953-960 (2010). MR2644520.
28. J. Katriel, The splitting of atomic orbitals with a common principal quantum number revisited: np vs. ns, J. Chem. Phys. 136, 144112-(1-8) (2012).
29. J. Katriel and H. E. Montgomery, Jr., The virial theorem for the smoothly and sharply, penetrably and impenetrably confined hydrogen atom, J. Chem. Phys.
30. J. Katriel, M. Puchalski and K. Pachucki, Binding energies of the lithium isoelectronic sequence approaching the critical charge, Phys. Rev. A, 86, 042508-(1-5) (2012).
31. J. Katriel and H. E. Montgomery, Jr., Hund’s rule in the two-electron quantum dot, Eur. Phys. J. B 85, 394-(1-4) (2012).