Abstract
This paper is mainly meant to be a survey on the state-of-the-art of the understanding we have of a family of systems with polynomial coefficients, called non-commutative harmonic oscillators (NCHOs), that has shown itself to be very rich in structure. The study of this family has required, and is requiring, the study of problems arising from different parts of Mathematics, from spectral theory to the theory of modular forms, just to mention a few of them. On the side of new results, they will be concerned with the creation-annihilation relations for NCHOs and with Fredholm properties of operators belonging to certain global Weyl-Hörmander classes, of which NCHOs are a particular case.
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References
Beals R.: Weighted distribution spaces and pseudodifferential operators. J. Analyse Math. 39, 131–187 (1981)
Bony J.-M., Chemin J.-Y.: Espaces fonctionnels associés au calcul de Weyl-Hörmander. Bull. Soc. Math. France 122, 77–118 (1994)
Braak D.: Integrability of the Rabi model. Phys. Rev. Letter 107, 10040–110040 4 (2011)
Brummelhuis R.: A counterexample to the Fefferman-Phong inequality for systems. C. R. Acad. Sci. Paris 310, 95–98 (1990)
Brummelhuis R.: On Melin’s inequality for systems. Comm. Partial Differential Equations 26, 1559–1606 (2001)
J. Chazarain, Fomule de Poisson pour les variétes riemanniennes. Invent. Math. 24 (1974), 65–82.
Colinde Verdière Y.: Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes periodiques. Comment. Math. Helvetici 54, 508–522 (1979)
M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit. London Math. Soc. Lecture Note Ser. 268. Cambridge University Press, 1999.
Duistermaat J. J., Guillemin V.: The Spectrum of Positive Elliptic Operators and Periodic Bicharacteristics. Invent. Math. 29, 39–79 (1975)
Fefferman C.: The uncertainty principle. Bull. Amer. Math. Soc. (N.S.) 9, 129–206 (1983)
B. Helffer, Théorie spectrale pour des opérateurs globalement elliptiques. Astérisque 112, Soc. Math. de France, Paris, 1984.
Hirokawa M.: The Dicke-type transition for non-commutative harmonic oscillator in the light of cavity QED. Sūrikaisekikenkyūsho Kōkyūroku 1607, 93–112 (2008)
Hirokawa M.: The Dicke-type crossings among eigenvalues of differential operators in a class of non-commutative harmonic oscillators. Indiana Univ. Math. J. 58, 1493–1535 (2009)
Hiroshima F., Sasaki I.: Multiplicity of the lowest eigenvalue of non-commutative harmonic oscillators. Kyushu J. Math. 67, 355–366 (2013)
F. Hiroshima and I. Sasaki, Spectral analysis of non-commutative harmonic oscillators: the lowest eigenvalue and no crossing. Preprint 2013. To appear in J. Math. Anal. Appl.
L. Hörmander, The Cauchy problem for differential equations with double characteristics. J. Anal. Math. 32 (1977), 118–196.
L. Hörmander, The Weyl calculus of pseudodifferential operators. Comm. Pure Appl. Math. 32 (1979), 360–444.
L. Hörmander, On the asymptotic distribution of the eigenvalues of pseudodifferential operators in \({\mathbb{R}^n}\). Ark. Mat. 17 (1979), 297-313.
L. Hörmander, The analysis of partial differential operators. III. Pseudodifferential operators. Grundlehren der Mathematischen Wissenschaften, 274. Springer-Verlag, Berlin, 1985. viii+525 pp.
Hörmander L.: Symplectic classification of quadratic forms, and general Mehler formulas. Math. Z. 219, 413–449 (1995)
Ichinose T., Wakayama M.: Zeta functions for the spectrum of the non-commutative harmonic oscillators. Comm. in Math. Phys. 258, 697–739 (2005)
T. Ichinose and M. Wakayama, Special values of the spectral zeta function of the noncommutative harmonic oscillator and confluent Heun equations. Kyushu J. Math. 59 No. 1 (2005), 39–100.
Ichinose T., Wakayama M.: On the spectral zeta function for the noncommutative harmonic oscillator. Rep. Math. Phys. 59, 421–432 (2007)
V. Ivrii, Microlocal analysis and precise spectral asymptotics. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. xvi+731 pp.
I. Kamotski and M. Ruzhansky, Regularity properties, representation of solutions, and spectral asymptotics of systems with multiplicities. Comm. Partial Differential Equations 32 (2007), 1–35.
T. Kato, Perturbation theory for linear operators. Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. xxii+619 pp.
K. Kimoto and M.Wakayama, Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators. Kyushu J. Math. 60 (2006), 383–404
K. Kimoto and M. Wakayama, Elliptic curves arising from the spectral zeta function for non-commutative harmonic oscillators and Γ0(4)-modular forms. The Conference on L-Functions, 201218, World Sci. Publ., Hackensack, NJ, 2007.
K. Kimoto and M. Wakayama, Spectrum of non-commutative harmonic oscillators and residual modular forms. Noncommutative Geometry and Physics 3. World Scientific 2013; 237–267.
B. V. Lange and V. S. Rabinovich, Pseudodifferential operators on \({\mathbb{R}^n}\) and limit operators. Mat. Sb. 129 (1986), 175–185; English translation: Math. USSR-Sb. 57 (1987), 183–194.
N. Lerner, Metrics on the phase space and non-selfadjoint pseudo-differential operators. Pseudo-Differential Operators. Theory and Applications, 3. Birkhäuser Verlag, Basel, 2010. xii+397 pp.
K. Nagatou, M. T. Nakao and M.Wakayama, Verified numerical computations for eigenvalues of non-commutative harmonic oscillators. Numerical Funct. Analysis and Opt. 23 (2002), 633–650.
Ochiai H.: Non-commutative harmonic oscillators and Fuchsian ordinary differential operators. Comm. in Math. Phys. 217, 357–373 (2001)
Ochiai H.: Non-commutative harmonic oscillators and the connection problem for the Heun differential equation. Letters in Math. Phys. 70, 133–139 (2004)
Ochiai H.: A special value of the spectral zeta function of the non-commutative harmonic oscillators. Ramanujan J. 15, 31–36 (2008)
C. Parenti, Sistemi iperbolici e relazioni di Poisson. Seminario di Analisi Matematica, Dipartimento di Matematica dell’Unviersità di Bologna, A.A. 1986–87, Bologna, XVII. 1– XVII 12.
Parenti C., Parmeggiani A.: Lower bounds for systems with double characteristics. J. Analyse Math. 86, 49–91 (2002)
A. Parmeggiani, On lower bounds of pseudodifferential systems. Hyperbolic problems and related topics, 269–293, Grad. Ser. Anal., Int. Press, Somerville, MA, 2003.
Parmeggiani A.: A class of counterexamples to the Fefferman-Phong inequality for systems. Comm. Partial Differential Equations 29, 1281–1303 (2004)
Parmeggiani A.: On the spectrum and the lowest eigenvalue of certain non-commutative harmonic oscillators. Kyushu Journal of Mathematics 58, 277–322 (2004)
Parmeggiani A.: On the spectrum of certain noncommutative harmonic oscillators. Annali dell’Università di Ferrara 52, 431–456 (2006)
Parmeggiani A.: On positivity of certain systems of partial differential equations. Proc. Natl. Acad. Sci. USA 104, 723–726 (2007)
A. Parmeggiani, On the Fefferman-Phong inequality for systems of PDEs. A. Bove- F. Colombini-D. Del Santo Editors. Progress in Nonlinear Differential Equations and Their Applications 69, Birkäuser-Verlag Boston (2006), 247–266.
Parmeggiani A.: On the spectrum of certain non-commutative harmonic oscillators and semiclassical analysis. Comm. in Math. Phys. 279, 285–308 (2008)
A. Parmeggiani, Spectral theory of non-commutative harmonic oscillators: an introduction. Lecture Notes in Mathematics, 1992. Springer-Verlag, Berlin, 2010. xii+254 pp.
A. Parmeggiani, A remark on the Fefferman-Phong inequality for 2 × 2 systems. Pure Appl. Math. Q. 6 (2010), Special Issue: In honor of Joseph J. Kohn. Part 2, 1081–1103.
A. Parmeggiani, On the problem of positivity of pseudodifferential systems. M. Cicognani- F. Colombini-D. Del Santo Editors. Progress in Nonlinear Differential Equations and Their Applications 84, Birkäuser Springer Science+Business Media New York (2013), 313–335.
Parmeggiani A., Venni A.: On the essential spectrum of certain non-commutative oscillators. J. Math. Phys. 52((12), 121507–112150710 (2013)
Parmeggiani A., Wakayama M.: Oscillator representations and systems of ordinary differential equations. Proceedings of the National Academy of Sciences U.S.A. 98, 26–30 (2001)
A. Parmeggiani and M.Wakayama, Non-commutative harmonic oscillators-I, -II. Forum Mathematicum 14 (2002), 539–604 ibid. 669–690.
D. Robert, Propriétés spectrales d’opérateurs pseudodifferentiels. Comm. Partial Differrential Equations 3 (1978), 755–826.
M. Shubin, Pseudodifferential operators and spectral theory. Second edition. Springer- Verlag, Berlin, 2001. xii+288 pp.
L.-Y. Sung, Positivity of a system of differential operators. J. Differential Equations 66 (1987), 71–89.
Taniguchi S.: The heat semigroup and kernel associated with certain non-commutative harmonic oscillators. Kyushu Journal of Mathematics 62, 63–68 (2008)
Taylor M.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)
M. Wakayama, Simplicity of the lowest eigenvalue of non-commutative harmonic oscillators and the Riemann scheme of a certain Heun’s differential equation. Proceedings of the Japan Academy of Sciences, 89. Ser. A, Mathematical Sciences (2013), 69–73.
M. Wakayama, Equivalence between the eigenfunction problem of non-commutative harmonic oscillators and existence of holomorphic solutions of Heun’s differential equations, the eigenstates degeneration and Rabi’s model. Preprint 2013.
Weinstein A.: Asymptotics of eigenvalue clusters for the Laplacian plus a potential. Duke Math. J. 44, 883–892 (1977)
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Lecture given in the Seminario Matematico e Fisico di Milano on May 7, 2012
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Parmeggiani, A. Non-Commutative Harmonic Oscillators and Related Problems. Milan J. Math. 82, 343–387 (2014). https://doi.org/10.1007/s00032-014-0220-z
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DOI: https://doi.org/10.1007/s00032-014-0220-z