B. Zhilinskii : scientific work

Scientific work

Qualitative analysis of excited quantum systems is my global direction of researches. Quantum-classical correspondence, symmetry and topological effects are studied on different examples of molecular systems. Effective Hamiltonians describing rotational, vibrational, rovibrational, electronic (Rydberg) states are analysed through the qualitative methods originating from the dynamical theory of Hamiltonian systems.

Publications

The list of the most important publications is given in a chronological order.

See also the complete list of publications.

D. Sadovskii, B. Zhilinskii,
Qualitative models of intramolecular dynamics of acetylene: relation between the bending polyads of acetylene and perturbed Keplerian systems.
Mol. Phys. 116, 23-24, 3564-3601 (2018).

G. Dhont, T. Iwai, B. Zhilinskii,
Topological phase transition in molecular Hamiltonian with symmetry and pseudo-symmetry, studied through quantum, semi-quantum and classical models
arXive quant-ph. 1703.04472, (2017) 37 pages

T. Iwai, B. Zhilinskii,
Chern number modification in crossing the boundary between different band structures: Three-band model with cubic symmetry
Rev. Math. Phys. 29, 1750004 (2017) [91 pages] DOI: http://dx.doi.org/10.1142/S0129055X17500040

A.V. Borisov, A.V. Bolsinov, A.I. Nejshtadt, D.A. Sadovskii, B. Zhilinskii,
Nikolai N. Nekhoroshev
Regular and Chaotic Dinamics, 21, No 6, 593-598 (2016)

T. Iwai, B. Zhilinskii,
Band rearrangement through the 2D-Dirac equation: comparing the APS and the chiral bag boundary conditions
Indag. Math. 27, 1081-1106 (2016)

T. Iwai, B. Zhilinskii,
Change in energy eigenvalues against parameters
Geometric Methods in Physics. XXXIV Workshop, Bialoweza, Poland, June 28 - July 4 2015, Springer, 2016, pp. 233-253

G. Dhont, B. Zhilinskii,
Symbolic Interpretation of the Molien Function: Free and Non-free Modules of Covariants
Geometric Methods in Physics. XXXIV Workshop, Bialoweza, Poland, June 28 - July 4 2015, Springer, 2016, pp. 105-114

B. Zhilinskií,
Introduction to Louis Michel's lattice geometry through group action.

EDP Sciences, CNRS Editions, 2015, 270 pp. ISBN EDP Sciences: 978-2-7598-1738-2 . draft

B. Zhilinskií,
On the number of k-faces of primitive parallelohedra
Acta Crystallographica Section A: Foundations and Advances, (2015), Volume A71, pages 212-215

T. Iwai, B. Zhilinskii,
Local description of band rearrangements. Comparison of semi-quantum and full quantum approach
arXiv:1407.4892 (2014); Acta Appl. Math, (2015) 137, 97-121; DOI:10.1007/s10440-014-9991-y

T. Iwai, B. Zhilinskii,
Topological phase transitions in the vibration-rotation dynamics of an isolated molecule
Theoret. Chem. Accounts, 133, 1501 (2014)

T. Damour, I. Todorov, B. Zhilinskii, edts.
Symmetries in Nature. The Scientific heritage of Louis Michel.
World Scientific, 2014, 400pp. ISBN: 978-981-4551-36-6

G.Dhont, F. Patras, B. Zhilinskii,
The action of the special orthogonal group on planar vectors: integrity basis via a generalilzation of the symbolic interpretation of Molien functions
J. Phys. A 48 035201 (19 pages) (2015)

T. Iwai, B. Zhilinskii,
Qualitative features of the rearrangement of molecular energy spectra from a ``wall-crossing'' perspective.
Phys. Lett. A 377 (2013) 2481-2486 DOI: 10.1016/j.physleta.2013.07.043

G.Dhont, B. Zhilinskii,
The action of the orthogonal group on planar vectors: invariants, covariants and syzygies.
J. Phys. A 46 (2013) 455202

T. Iwai, B. Zhilinskii,
Rearrangement of Energy Bands: Chern numbers in the Presence of Cubic symmetry.
Acta Appl. Math. 120, 153-175 (2012)

T. Iwai, B. I. Zhilinskii,
Energy bands: Chern numbers and symmetry.
Ann. Phys. (N.Y) 326, 3013-3066 (2011).

G. Dhont, B. I. Zhilinskií,
Symetrie dans la nature,.
PUG, Grenoble, 2011.

B. I. Zhilinskií,
Monodromy and complexity of quantum systems
in: The Complexity of Dynamical Systems: A Multi-disciplinary Perspective. Eds. J. Dubbeldam, K. Green, and D. Lenstra, Wiley, Singapore, 2011, p.159-181.

B. I. Zhilinskií,
Generating functions for effective Hamiltonians. Symmetry, topology, combinatorics
Physics of Atomic Nuclei, 75, 109-117 (2012).

B. I. Zhilinskií,
Reflextions on university education: soviet and french organization of teaching.
Ross. Khim. Zhurnal (Zhurnal Ross. Khim. ob-va im. D.I. Mendeleeva) 55, N 4, 97--105 (2011). In Russian. English translation: Russian Journal of General Chemistry, 83, No.3, 604-613 (2013).

B. I. Zhilinskií,
Quantum monodromy and pattern formation,
J.Phys. A Math. Gen. 43, 434033 (2010).

B. I. Zhilinskií,
Hamiltonian monodromy, its manifestations and generalizations
Proceedings ``Geometric mechanics`` workshop, Kyoto december 2009, RIMS Kokyuroku 1692, 57--77 (2010).

J.B.Delos, G. Dhont, D.A. Sadovskii, and B. I. Zhilinskií,
Dynamical manifestations of Hamiltonian monodromy
Ann. Phys. (N.Y.) 324, 1953-1982 (2009)

B. I. Zhilinskií,
Quantum Bifurcations.
In Meyers, Robert (Ed.) Encyclopedia of Complexity and Systems Science, Springer New York 2009, Part 17, Pages 7135-7154; In Mathematics of complexity and dynamical systems, 2011, 1438 - 1456. DOI: 10.1007/978-1-4614-1806-1_91

A.M. Abramov, V.I. Arnol'd, A.V. Bolsinov, A.N. Varchenko, L. Galgani, B. I. Zhilinskii, Yu.S. Il'yashenko, V.V. Kozlov, A.I.Neishtadt, V.I. Piterbarg, A.G. Khovanskii, V.V. Yashchenko
Nikolai Nikolaevich Nekhoroshev. Obituary
UMN, 64, 174-178 (2009)

G. Dhont, B. I. Zhilinskií,
Classical and quantum fold catastrophe in the presence of axial symmetry
Phys. Rev A, 78, 052117 (2008)

B. I. Zhilinskií,
Generating functions for effective Hamiltonians via the symmetrised Hadamard product
J.Phys. A: Math.Theor. 41, 382004-1-9 (2008)

B. I. Zhilinskií,
Rearrangement of energy bands. Topological aspects..
J. Math. Chem. 44 (4) , 1009-1022 (2008), DOI: 10.1007/s10910-008-9359-6

J.B.Delos, G. Dhont, D.A. Sadovskii, and B. I. Zhilinskií,
Dynamical manifestation of Hamiltonian monodromy
Europhys. Letters, 83, 24003-1-6 (2008)

B. I. Zhilinskií,
Generalization of Hamiltonian monodromy. Quantum manifestations
Symmetry and Perturbation Theory. Proceedings of the International Conference, Otranto, Italy 2-9 June 2007, ed. G. Gaeta, R. Vitolo, S. Walcher, World Scientifique, (2007)

M.S. Hansen, F. Faure, and B. I. Zhilinskií,
Fractional monodromy in systems with coupled angular momenta
J.Phys. A: Math.Theor. 40, 13075--13089 (2007)

K. Efstathiou, D.A. Sadovskii B. I. Zhilinskií,
Classification of perturbations of the hydrogen atom by small static electric and magnetic fields
Proc. R. Soc. A 463 (2083) 1771--90 (2007)

D.A. Sadovskii B. I. Zhilinskií,
Hamiltonian systems with detuned 1:1:2 resonance. Manifestation of bidromy
Ann.Phys. (N.Y) 322, 164--200 (2007)

E. Sinitsyn, B. Zhilinskii,
Qualitative Analysis of the Classical and Quantum Manakov Top .
SIGMA 3, (2007) 046, 23 pages

N. N. Nekhoroshev, D.A. Sadovskii B. I. Zhilinskií,
Fractional Hamiltonian monodromy.
Ann. Henri Poincare. 7, 1099--1211 (2006).

B. I. Zhilinskií,
Hamiltonian monodromy as lattice defect.
in: Topology in Condensed Matter, (Springer Series in Solid-State Sciences, Vol. 150), 2006, pp. 165-186.

D.A. Sadovskii B. I. Zhilinskií,
Quantum monodromy, its generalizations and molecular manifestations..
Mol. Phys. 104(16-17), 2595-2615 (2006)

B. I. Zhilinskií,
Interpretation of quantum Hamiltonian monodromy in terms of lattice defects.
Acta Appl. Math. 87, 281-307 (2005)

B. I. Zhilinskií,
Defects of quantum state lattice and assignment of spectra.
Proceedings of ICGTMP 25, Mexico 2004 , Ed. G.S.Pogosyan, L.E.Vicent, K.B.Wolf, Institute of Physics, Conference Series Number 185, pp. 575-580 (2005).

I.N. Kozin, D. Sadovskii, B. I. Zhilinskií,
Assigning vibrational polyads using relative equilibria. Application to ozone.
Spectrochim. Acta A, 61, 2867-2885 (2005).

A. Giacobbe, R. Cushman, D. Sadovskii, B. I. Zhilinskií,
Monodromy of the quantum 1:1:2 resonant swing spring.
J. Math. Phys., 45, 5076-5100 (2004).

R.H. Cushman, H. R. Dullin, A. Giacobbe, D.D. Holm, M. Joyeux, P. Lynch, D.A. Sadovskii, B. I. Zhilinskií,
CO2 molecule as a quantum realization of the 1:1:2 resonant swing-spring with monodromy.
Phys. Rev. Lett. 93, 024302-1-4 (2004)

K. Efstathiou, D. A. Sadovskii, B. I. Zhilinskií,
Analysis of rotation-vibration relative equilibria on example of a tetrahedral four atom molecule.
SIAM Journal of Dynamical Systems, 3, 261-351 (2004).

M.I. El Idrissi, B. I. Zhilinskií, P.Gaspard, and M. Herman;,
The integrated number of vibrational states in acetylene (12C2H2,13C2H2, 12C2D2)
Mol. Phys. , 101, 595-601 (2003).

B. I. Zhilinskií,
Reorganization of energy bands in quantum finite particle systems.
Proceedings of ICGTMP 2002, Paris (2002), p. 625-632

N.N. Nekhoroshev, D.A. Sadovskii, B. I. Zhilinskií,
Fractional monodromy of resonant classical and quantum oscillators.
C.R.Acad.Sci. Paris Ser. I, 335, 985-988 (2002).

F. Faure and B. I. Zhilinskií,
Topologically coupled energy bands in molecules.
Phys. Lett. A 302, 242-252 (2002); ArXive: quant-ph/0204100.

R. Cushman and B. I. Zhilinskií,
Monodromy of a two degree of freedom Liouville integrable system with many focus-focus singular points.
J.Phys. A Math. Gen. 35 L415-L419 (2002).

L. Grondin, D. Sadovskii and B. Zhilinskií,
Monodromy in systems with coupled angular momenta and rearrangement of bands in quantum spectra.
Phys. Rev. A., 65, 012105 (2002).

F. Faure and B. Zhilinskií,
Qualitative features of intra-molecular dynamics. What can be learned from symmetry and topology.
Acta Appl. Math. 70, 265-282 (2002)

B. I. Zhilinskií,
Qualitative features of quantum finite particle systems.
Proceedings of Andronov memorial conference, vol.II. Frontiers of Nonlinear Science, pp 614-619, Nozhnii Novgorod Russia (2002).

Ch. Van Hecke, D. Sadovskii, B. I. Zhilinskií, V. Boudon,
Rotational-vibrational relative equilibria and the structure of quantum energy spectrum of the tetrahedral molecule P$_4$.
Europ. Phys.J. D 17, 13-35 (2001).

J. S. Kim, L. Michel, and B. I. Zhilinskií,
Invariant theory in crystal symmetry.
Symmetry and Structural Properties of Condensed Matter. Ed. T. Lulek, B. Lulek, A. Wal. Proceedings of 6-th International School of Theoretical Physics. 31 August - 6 September 2000. World Scientific, 2001, pp. 346-357

F. Faure and B. I. Zhilinskií,
Topological properties of the Born-Oppenheimer approximation and implications for the exact spectrum.
Let. Math. Phys. 55, 239-247 (2001).

L. Michel and B. I. Zhilinskii
Symmetry, Invariants, and Topology. I. Basic Tools.
Phys. Rep. 341, 11-84 (2001).

B. I. Zhilinskii
Symmetry, Invariants, and Topology. II Symmetry, invariants, and topology in moleculsr models.
Phys. Rep. 341, 85-171 (2001).

L. Michel and B. I. Zhilinskii
Symmetry, Invariants, and Topology. III. Rydberg states of atoms and molecules. Basic group theoretical and topological analysis.
Phys. Rep. 341, 175-264 (2001).

J. S. Kim, L. Michel and B. I. Zhilinskii
Symmetry, Invariants, and Topology. V. The ring of invariant real functions on the Brillouin zone.
Phys. Rep. 341, 337-376 (2001).

Frederic Faure and Boris I. Zhilinskií,
Topological Chern indices in molecular spectra
Phys. Rev. Lett. , 85, 960-963 (2000). (quant-ph/9912091).

B. I. Zhilinskií, M.I. El Idrissi, and M. Herman;,
The vibrational energy pattern in acetylene (VI): Inter and intrapolyad structures
J.Chem. Phys. , 113, 7885-7890 (2000).

G. Dhont, D. A. Sadovskii, B. I. Zhilinskii, and V. Boudon;
Analysis of the ``Unusual'' Vibrational Components of Triply Degenerate Vibrational Mode nu_6 of Mo(CO)_6 Based on the Classical Interpretation of the Effective Rotation-Vibration Hamiltonian
J. Mol. Spectrosc. 201, 95-108 (2000).

D. A. Sadovskií and Boris I. Zhilinskií,
Monodromy, diabolic points, and angular momentum coupling
Physics Letters 256(4), 235-44 (1999).

Christophe Van Hecke, D. A. Sadovskií, and B. I. Zhilinskií,
Qualitative analysis of molecular rotation starting from inter-nuclear potential
European Phys. J. D7, 199-209 (1999).

B. Zhilinskií, I. N. Kozin, and S. Petrov
Correlation between asymmetric and spherical top: imperfect quantum bifurcations
Spectrochim. acta A 55, 1471-1484 (1999).

D. A. Sadovskií and Boris I. Zhilinskií,
Tuning the hydrogen atom in crossed fields between the Zeeman and Stark limits,
Phys. Rev. A 57(4), 2867-84 (1998).

Boris Zhilinskií,
Topological and symmetry features of intramolecular dynamics through high resolution molecular spectroscopy,
Spectrochim. Acta A 52, 881-900 (1996).

P. Soldan, B. Zhilinskií,
Density of vibrational states of a given symmetry type for octahedral AB6 molecules,
Chem. Phys. Lett. 258, 25-29 (1996).

D. A. Sadovskií, Boris I. Zhilinskií and Louis Michel,
Collapse of the Zeeman structure of the hydrogen atom in the external electric field.
preprint IHES P/95/86, Bures-sur-Yvette, 1995;
Phys. Rev. A 53(6) 4064-7 (1996)

D. A. Sadovskií and Boris I. Zhilinskií
Counting levels within vibrational polyads. Generating function approach,
J. Chem. Phys. 103(24), 10520-36 (1995).

S. Brodersen and Boris I. Zhilinskií,,
The rotational structure of the vibrational states and substates of symmetry $E$ in CF$_4$.
J. Mol. Spectrosc. 172, 303-318 (1995).

S. Brodersen and Boris I. Zhilinskií,,
Transfer of clusters between the vibrational components of CF$_4$.
J. Mol. Spectrosc. 169, 1-17 (1995).

Boris I. Zhilinskií and S. Brodersen,
The symmetry of the vibrational components in T$_d$ molecules.
J. Mol. Spectrosc. 163, 326-338 (1994).

D. A. Sadovskií, N. G. Fulton, J. R. Henderson, J. Tennyson, and B. I. Zhilinskií
Nonlinear normal modes and local bending vibrations of H₃⁺ and D₃⁺,
J. Chem. Phys. 99(2), 906-18 (1993).

Boris I. Zhilinskií, S. Brodersen, M. Madsen
The pattern of clusters in isolated vibrational components of CF$_4$ and the semiclassical model.
J. Mol. Spectrosc. 160, 192-216 (1993).

D. A. Sadovskií and Boris I. Zhilinskií
Qualitative study of a model three level Hamiltonian with SU(3) dynamical symmetry,
Phys. Rev. A 48(2), 1035-44 (1993).

D. A. Sadovskií and Boris I. Zhilinskií
Group theoretical and topological analysis of localized vibration-rotation states,
Phys. Rev. A 47(4), 2653-71 (1993).

Vl. M. Krivtsun, D. A. Sadovskií and B. I. Zhilinskií
Critical Phenomena and Diabolic Points in Rovibrational Energy Spectra of Spherical Top Molecules,
J. Molec. Spectr. 139, 126-46 (1990).

D. A. Sadovskií, Boris I. Zhilinskií, Jean-Paul Champion and Gérard Pierre
Manifestations of bifurcations and diabolic points in molecular energy spectra,
J. Chem. Phys. 92, 1523-37 (1990).

Boris Zhilinskií,
Qualitative analysis of vibrational polyads: N-mode case. ,
Chem. Phys. 137, 1-13 (1989).

D. A. Sadovskií and B. I. Zhilinskií
Contact transformations in tensor formalism. Effective Hamiltonian and dipole moment for the $\nu_2 , \nu_4$ dyad of tetrahedral molecules.
JQSRT, 42, 575-583 (1989).

Gérard Pierre, D. A. Sadovskií and B. I. Zhilinskií
Organization of quantum bifurcations: Crossover of rovibrational bands in spherical top molecules, Europhys. Lett. 10, 409-14 (1989).

Victor E. Pavlov-Verevkin, D. A. Sadovskií and B. I. Zhilinskií
On the dynamical meaning of diabolic points, Europhys. Lett. 6, 573-78 (1988).

D. A. Sadovskií and Boris I. Zhilinskií
Qualitative analysis of vibration-rotation Hamiltonians for spherical top molecules, Molec. Phys. 65 109-28 (1988).

I.M. Pavlichenkov; and Boris I. Zhilinskií
Critical phenomena in rotational spectra
Ann. Phys. (N.Y.) 184, 1-32 (1988).

V. B. Pavlov-Verevkin and Boris I. Zhilinskií
Effective Hamiltonians for vibrational polyads: Integrity basis approach.
Chem. Phys. 126, 243-253 (1988).

V. B. Pavlov-Verevkin and Boris I. Zhilinskií
Rearrangements of the vibrational po\-ly\-adic spectra with excitation: two-mode case.
Chem. Phys. 128, 429-437 (1988).

Tyuterev V.G., Pierre G., Champion J.P., Perevalov V.I., and B. I. Zhilinskií
Further analysis of effective Hamiltonians for triply degenerate fundamental bands of tetrahedral molecules.
J. Mol. Spectrosc. 117, 102-118 (1986).

D.A. Sadovskii and Boris I. Zhilinskií
Theoretical analysis of spectroscopic constants for spherical tops. $\nu_2, \nu_4$ bands of AB$_4$ molecules.
J. Mol. Spectrosc. 115, 235-257 (1986).

I.M. Pavlichenkov; and Boris I. Zhilinskií
Rotation of molecules around specific axes: axes reorientation under rotational excitation.
Chem. Phys. 100, 339-354 (1985).

Perevalov V.I., Tyuterev V.G., and Boris I. Zhilinskií
Reduced Hamiltonian for 0100 and 0001 interacting states of tetrahedral XY$_4$ molecules. Calculated $q^2J^2 , q^2J^3$ type parameters
J. Mol. Spectrosc. 111, 1-19 (1985).

Perevalov V.I., Tyuterev V.G., and Boris I. Zhilinskií
Reduced effective Hamiltonian for degenerate vibrational states of methane- type molecules.
J. Mol. Spectrosc. 103, 147-159 (1984).

Perevalov V.I., Tyuterev V.G., and Boris I. Zhilinskií
Ambiguity of spectroscopic parameters in the cases of accidental vibration-rotation resonances in tetrahedral molecules.
Chem. Phys. Lett. 104, 455-461 (1984).

Pablov-Verevkin V. B. and Boris I. Zhilinskií
The hydrogen atom in a superstrong magnetic field.
Phys.Lett. A. 75, 279-281 (1980).

Pablov-Verevkin V. B. and Boris I. Zhilinskií
Neutral hydrogen-like system in a magnetic field.
Phys.Lett. A. 78, 244-245 (1980).

Dement'ev A.I., Rambidi N.G., Simkin V.Ya., Topol' I.A., Stepanov N.F., and Zhilinskií B.I.;
Molecular structure of the boron (III) oxide molecule in the SCF approximation.
J. Mol. Structure 68, 199-202 (1980).

B. I. Zhilinskií
Spectra of tensor operators adapted to nonstandard basis. Qualitative features of clustering.
J. Mol. Spectros 78, 203-228 (1979).

Istomin V. A., Stepanov N.F. and Boris I. Zhilinskií
Vibration-rotation Hamiltonian for nonrigid triatomic molecules with diatomic rigid core.
Chem. Phys. 31, 413-423 (1978).

B. I. Zhilinskií
Nonrigid molecules with several large-amplitude coordinates.
J. Mol. Structure 46, 183-188 (1978).

Stepanov N.F. and Boris I. Zhilinskií
Nuclear charge changes: influence on the energy of highly symmetrical molecules.
Chem. Phys. Lett. 45, 589-591 (1977).

Istomin V. A., Stepanov N.F. and Boris I. Zhilinskií
Vibration-rotation problem for triatomic molecules with two large-amplitude coordinates. Spherical model.
J. Mol. Spectrosc. 67, 265-282 (1977).

Stepanov N.F. and Boris I. Zhilinskií
When and why Hund's cases arise.
J. Mol. Spectros 52, 277-286 (1974).

B. I. Zhilinskií and Stepanov N.F.
Minimization of the energy functional on the special class of unitary operators.
Teor. Mat. Fiz. 15, 146-150 (1973); Translated in : Teor. Math. Phys. (USA) . 15, 422-425 (1973).

Stepanov N.F. and B. I. Zhilinskií
Separation of variables illustrated by the example of the three-body problem
Teor. Eksp. Khim. 8, 672-676 (1972); Translated in : Teor. Eks. Chem.. (USA) . 8, 555-558 (1972).

B. I. Zhilinskií,
Method of Irreducible Tensor operators in molecular spectroscopy part 1. Basic theory. (In russian)
Chemistry department, Moscow State University, Moscow 1981.

B. I. Zhilinskií,
Method of Irreducible Tensor operators in molecular spectroscopy part 2. Tables and formulae. (In russian)
Chemistry department, Moscow State University, Moscow 1981.

Pre-publications

T. Iwai, B. I. Zhilinskií,
Rearrangement of energy bands: Chern numbers in the presence of symmetry.
Acta Appl. Math. submitted 2011.

B. I. Zhilinskií,
Symmetry, invariants, topology in molecules.
Proceedings of ICGTMP 2000, Dubna (2000).

B. I. Zhilinskií,
Rearrangement of energy bands : Quantum, semi-quantum, and classical models.
Proceedings ... , 67 RCP, Strasbourg (2000).

Preparation for special issue of Physics Reports
``Symmetry, Invariants, and Topology'' (Ed. L. Michel). submitted to Physics Reports, october 1999. [published Phys. Rep. 341, (2001))

Avant-Propos

Obituary.

L. Michel
Fundamental concepts for the study of crystal symmetry.
Chapter 4.

L. Michel and J. Zak
Elementary energy bands in crystals are connected.
Chapter 6.

References to the whole volume.

Scientific work

Publications

D. Sadovskii, B. Zhilinskii, Qualitative models of intramolecular dynamics of acetylene: relation between the bending polyads of acetylene and perturbed Keplerian systems. Mol. Phys. 116, 23-24, 3564-3601 (2018).

G. Dhont, T. Iwai, B. Zhilinskii, Topological phase transition in molecular Hamiltonian with symmetry and pseudo-symmetry, studied through quantum, semi-quantum and classical models arXive quant-ph. 1703.04472, (2017) 37 pages

T. Iwai, B. Zhilinskii, Chern number modification in crossing the boundary between different band structures: Three-band model with cubic symmetry Rev. Math. Phys. 29, 1750004 (2017) [91 pages] DOI: http://dx.doi.org/10.1142/S0129055X17500040

A.V. Borisov, A.V. Bolsinov, A.I. Nejshtadt, D.A. Sadovskii, B. Zhilinskii, Nikolai N. Nekhoroshev Regular and Chaotic Dinamics, 21, No 6, 593-598 (2016)

T. Iwai, B. Zhilinskii, Band rearrangement through the 2D-Dirac equation: comparing the APS and the chiral bag boundary conditions Indag. Math. 27, 1081-1106 (2016)

T. Iwai, B. Zhilinskii, Change in energy eigenvalues against parameters Geometric Methods in Physics. XXXIV Workshop, Bialoweza, Poland, June 28 - July 4 2015, Springer, 2016, pp. 233-253

G. Dhont, B. Zhilinskii, Symbolic Interpretation of the Molien Function: Free and Non-free Modules of Covariants Geometric Methods in Physics. XXXIV Workshop, Bialoweza, Poland, June 28 - July 4 2015, Springer, 2016, pp. 105-114

B. Zhilinskií, Introduction to Louis Michel's lattice geometry through group action. EDP Sciences, CNRS Editions, 2015, 270 pp. ISBN EDP Sciences: 978-2-7598-1738-2 . draft

B. Zhilinskií, On the number of k-faces of primitive parallelohedra Acta Crystallographica Section A: Foundations and Advances, (2015), Volume A71, pages 212-215

T. Iwai, B. Zhilinskii, Local description of band rearrangements. Comparison of semi-quantum and full quantum approach arXiv:1407.4892 (2014); Acta Appl. Math, (2015) 137, 97-121; DOI:10.1007/s10440-014-9991-y

T. Iwai, B. Zhilinskii, Topological phase transitions in the vibration-rotation dynamics of an isolated molecule Theoret. Chem. Accounts, 133, 1501 (2014)

T. Damour, I. Todorov, B. Zhilinskii, edts. Symmetries in Nature. The Scientific heritage of Louis Michel. World Scientific, 2014, 400pp. ISBN: 978-981-4551-36-6

G.Dhont, F. Patras, B. Zhilinskii, The action of the special orthogonal group on planar vectors: integrity basis via a generalilzation of the symbolic interpretation of Molien functions J. Phys. A 48 035201 (19 pages) (2015)

T. Iwai, B. Zhilinskii, Qualitative features of the rearrangement of molecular energy spectra from a ``wall-crossing'' perspective. Phys. Lett. A 377 (2013) 2481-2486 DOI: 10.1016/j.physleta.2013.07.043

G.Dhont, B. Zhilinskii, The action of the orthogonal group on planar vectors: invariants, covariants and syzygies. J. Phys. A 46 (2013) 455202

T. Iwai, B. Zhilinskii, Rearrangement of Energy Bands: Chern numbers in the Presence of Cubic symmetry. Acta Appl. Math. 120, 153-175 (2012)

T. Iwai, B. I. Zhilinskii, Energy bands: Chern numbers and symmetry. Ann. Phys. (N.Y) 326, 3013-3066 (2011).

G. Dhont, B. I. Zhilinskií, Symetrie dans la nature,. PUG, Grenoble, 2011.

B. I. Zhilinskií, Monodromy and complexity of quantum systems in: The Complexity of Dynamical Systems: A Multi-disciplinary Perspective. Eds. J. Dubbeldam, K. Green, and D. Lenstra, Wiley, Singapore, 2011, p.159-181.

B. I. Zhilinskií, Generating functions for effective Hamiltonians. Symmetry, topology, combinatorics Physics of Atomic Nuclei, 75, 109-117 (2012).

B. I. Zhilinskií, Reflextions on university education: soviet and french organization of teaching. Ross. Khim. Zhurnal (Zhurnal Ross. Khim. ob-va im. D.I. Mendeleeva) 55, N 4, 97--105 (2011). In Russian. English translation: Russian Journal of General Chemistry, 83, No.3, 604-613 (2013).

B. I. Zhilinskií, Quantum monodromy and pattern formation, J.Phys. A Math. Gen. 43, 434033 (2010).

B. I. Zhilinskií, Hamiltonian monodromy, its manifestations and generalizations Proceedings ``Geometric mechanics`` workshop, Kyoto december 2009, RIMS Kokyuroku 1692, 57--77 (2010).

J.B.Delos, G. Dhont, D.A. Sadovskii, and B. I. Zhilinskií, Dynamical manifestations of Hamiltonian monodromy Ann. Phys. (N.Y.) 324, 1953-1982 (2009)

B. I. Zhilinskií, Quantum Bifurcations. In Meyers, Robert (Ed.) Encyclopedia of Complexity and Systems Science, Springer New York 2009, Part 17, Pages 7135-7154; In Mathematics of complexity and dynamical systems, 2011, 1438 - 1456. DOI: 10.1007/978-1-4614-1806-1_91

A.M. Abramov, V.I. Arnol'd, A.V. Bolsinov, A.N. Varchenko, L. Galgani, B. I. Zhilinskii, Yu.S. Il'yashenko, V.V. Kozlov, A.I.Neishtadt, V.I. Piterbarg, A.G. Khovanskii, V.V. Yashchenko Nikolai Nikolaevich Nekhoroshev. Obituary UMN, 64, 174-178 (2009)

G. Dhont, B. I. Zhilinskií, Classical and quantum fold catastrophe in the presence of axial symmetry Phys. Rev A, 78, 052117 (2008)

B. I. Zhilinskií, Generating functions for effective Hamiltonians via the symmetrised Hadamard product J.Phys. A: Math.Theor. 41, 382004-1-9 (2008)

B. I. Zhilinskií, Rearrangement of energy bands. Topological aspects.. J. Math. Chem. 44 (4) , 1009-1022 (2008), DOI: 10.1007/s10910-008-9359-6

J.B.Delos, G. Dhont, D.A. Sadovskii, and B. I. Zhilinskií, Dynamical manifestation of Hamiltonian monodromy Europhys. Letters, 83, 24003-1-6 (2008)

B. I. Zhilinskií, Generalization of Hamiltonian monodromy. Quantum manifestations Symmetry and Perturbation Theory. Proceedings of the International Conference, Otranto, Italy 2-9 June 2007, ed. G. Gaeta, R. Vitolo, S. Walcher, World Scientifique, (2007)

M.S. Hansen, F. Faure, and B. I. Zhilinskií, Fractional monodromy in systems with coupled angular momenta J.Phys. A: Math.Theor. 40, 13075--13089 (2007)

K. Efstathiou, D.A. Sadovskii B. I. Zhilinskií, Classification of perturbations of the hydrogen atom by small static electric and magnetic fields Proc. R. Soc. A 463 (2083) 1771--90 (2007)

D.A. Sadovskii B. I. Zhilinskií, Hamiltonian systems with detuned 1:1:2 resonance. Manifestation of bidromy Ann.Phys. (N.Y) 322, 164--200 (2007)

E. Sinitsyn, B. Zhilinskii, Qualitative Analysis of the Classical and Quantum Manakov Top . SIGMA 3, (2007) 046, 23 pages

N. N. Nekhoroshev, D.A. Sadovskii B. I. Zhilinskií, Fractional Hamiltonian monodromy. Ann. Henri Poincare. 7, 1099--1211 (2006).

B. I. Zhilinskií, Hamiltonian monodromy as lattice defect. in: Topology in Condensed Matter, (Springer Series in Solid-State Sciences, Vol. 150), 2006, pp. 165-186.

D.A. Sadovskii B. I. Zhilinskií, Quantum monodromy, its generalizations and molecular manifestations.. Mol. Phys. 104(16-17), 2595-2615 (2006)

B. I. Zhilinskií, Interpretation of quantum Hamiltonian monodromy in terms of lattice defects. Acta Appl. Math. 87, 281-307 (2005)

B. I. Zhilinskií, Defects of quantum state lattice and assignment of spectra. Proceedings of ICGTMP 25, Mexico 2004 , Ed. G.S.Pogosyan, L.E.Vicent, K.B.Wolf, Institute of Physics, Conference Series Number 185, pp. 575-580 (2005).

I.N. Kozin, D. Sadovskii, B. I. Zhilinskií, Assigning vibrational polyads using relative equilibria. Application to ozone. Spectrochim. Acta A, 61, 2867-2885 (2005).

A. Giacobbe, R. Cushman, D. Sadovskii, B. I. Zhilinskií, Monodromy of the quantum 1:1:2 resonant swing spring. J. Math. Phys., 45, 5076-5100 (2004).

R.H. Cushman, H. R. Dullin, A. Giacobbe, D.D. Holm, M. Joyeux, P. Lynch, D.A. Sadovskii, B. I. Zhilinskií, CO2 molecule as a quantum realization of the 1:1:2 resonant swing-spring with monodromy. Phys. Rev. Lett. 93, 024302-1-4 (2004)

K. Efstathiou, D. A. Sadovskii, B. I. Zhilinskií, Analysis of rotation-vibration relative equilibria on example of a tetrahedral four atom molecule. SIAM Journal of Dynamical Systems, 3, 261-351 (2004).

M.I. El Idrissi, B. I. Zhilinskií, P.Gaspard, and M. Herman;, The integrated number of vibrational states in acetylene (12C2H2,13C2H2, 12C2D2) Mol. Phys. , 101, 595-601 (2003).

B. I. Zhilinskií, Reorganization of energy bands in quantum finite particle systems. Proceedings of ICGTMP 2002, Paris (2002), p. 625-632

N.N. Nekhoroshev, D.A. Sadovskii, B. I. Zhilinskií, Fractional monodromy of resonant classical and quantum oscillators. C.R.Acad.Sci. Paris Ser. I, 335, 985-988 (2002).

F. Faure and B. I. Zhilinskií, Topologically coupled energy bands in molecules. Phys. Lett. A 302, 242-252 (2002); ArXive: quant-ph/0204100.

R. Cushman and B. I. Zhilinskií, Monodromy of a two degree of freedom Liouville integrable system with many focus-focus singular points. J.Phys. A Math. Gen. 35 L415-L419 (2002).

L. Grondin, D. Sadovskii and B. Zhilinskií, Monodromy in systems with coupled angular momenta and rearrangement of bands in quantum spectra. Phys. Rev. A., 65, 012105 (2002).

F. Faure and B. Zhilinskií, Qualitative features of intra-molecular dynamics. What can be learned from symmetry and topology. Acta Appl. Math. 70, 265-282 (2002)

B. I. Zhilinskií, Qualitative features of quantum finite particle systems. Proceedings of Andronov memorial conference, vol.II. Frontiers of Nonlinear Science, pp 614-619, Nozhnii Novgorod Russia (2002).

Ch. Van Hecke, D. Sadovskii, B. I. Zhilinskií, V. Boudon, Rotational-vibrational relative equilibria and the structure of quantum energy spectrum of the tetrahedral molecule P$_4$. Europ. Phys.J. D 17, 13-35 (2001).

F. Faure and B. I. Zhilinskií, Topological properties of the Born-Oppenheimer approximation and implications for the exact spectrum. Let. Math. Phys. 55, 239-247 (2001).

L. Michel and B. I. Zhilinskii Symmetry, Invariants, and Topology. I. Basic Tools. Phys. Rep. 341, 11-84 (2001).

B. I. Zhilinskii Symmetry, Invariants, and Topology. II Symmetry, invariants, and topology in moleculsr models. Phys. Rep. 341, 85-171 (2001).

L. Michel and B. I. Zhilinskii Symmetry, Invariants, and Topology. III. Rydberg states of atoms and molecules. Basic group theoretical and topological analysis. Phys. Rep. 341, 175-264 (2001).

J. S. Kim, L. Michel and B. I. Zhilinskii Symmetry, Invariants, and Topology. V. The ring of invariant real functions on the Brillouin zone. Phys. Rep. 341, 337-376 (2001).

Frederic Faure and Boris I. Zhilinskií, Topological Chern indices in molecular spectra Phys. Rev. Lett. , 85, 960-963 (2000). (quant-ph/9912091).

B. I. Zhilinskií, M.I. El Idrissi, and M. Herman;, The vibrational energy pattern in acetylene (VI): Inter and intrapolyad structures J.Chem. Phys. , 113, 7885-7890 (2000).

B. I. Zhilinskií, Method of Irreducible Tensor operators in molecular spectroscopy part 1. Basic theory. (In russian) Chemistry department, Moscow State University, Moscow 1981.

B. I. Zhilinskií, Method of Irreducible Tensor operators in molecular spectroscopy part 2. Tables and formulae. (In russian) Chemistry department, Moscow State University, Moscow 1981.