Artículos científicos
Una descripción breve de operadores asociados al oscilador armónico cuántico sobre las clases de Schatten-von Neumann
Publicado 2018-07-22
Palabras clave
- Oscilador armónico,
- multiplicador de Fourier,
- multiplicadores de Hermite,
- operador nuclear,
- trazas
Cómo citar
Cardona, D. (2018). Una descripción breve de operadores asociados al oscilador armónico cuántico sobre las clases de Schatten-von Neumann. Revista Integración, Temas De matemáticas, 36(1), 49–57. https://doi.org/10.18273/revint.v36n1-2018004
Resumen
En esta nota se estudia una clase de operadores definidos a través del espectro del oscilador armónico y conocidos en la literatura como pseudo multiplicadores (pseudo multiplicadores de Hermite). Se analizan criterios óptimos para clasificar estos operadores en las clases de Schatten-von Neumann sobre L2(Rn). El trabajo culmina con una investigación sobre la traza espectral y/o nuclear de tales operadores.
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Referencias
[1] Bagchi S. and Thangavelu S., “On Hermite pseudo-multipliers”, J. Funct. Anal. 268 (2015), No. 1, 140–170,
[2] Barraza E.S. and Cardona D., “On nuclear Lp-multipliers associated to the Harmonic oscillator”, in Analysis in Developing Countries, Springer Proceedings in Mathematics & Statistics, Springer (2018), M. Ruzhansky and J. Delgado (Eds), to appear.
[3] Cardona D. and Barraza E.S., “Characterization of nuclear pseudo-multipliers associated to the harmonic oscillator”, to appear in, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys (2018), arXiv:1709.07961.
[4] Cardona D. and Ruzhansky M., “Hörmander condition for pseudo-multipliers associated to the harmonic oscillator”, preprint.
[5] Delgado J., A trace formula for nuclear operators on Lp, in Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations, Operator Theory: Advances and Applications 205, Schulze, B.W., Wong, M.W. (eds.), Birkhäuser, Basel (2010), 181–193.
[6] Delgado J., “The trace of nuclear operators on Lp(μ) for -finite Borel measures on second countable spaces”, Integr. Equ. Oper. Theory 68 (2010), No- 1, 61–74.
[7] Delgado J., “On the r-nuclearity of some integral operators on Lebesgue spaces”, Tohoku Math. J. (2) 67 (2015), No. 1, 125–135.
[8] Delgado J., RuzhanskyM. andWang B., “Approximation property and nuclearity on mixednorm Lp, modulation and Wiener amalgam spaces”, J. Lond. Math. Soc.(2) 94 (2016), 391–408.
[9] Delgado J., Ruzhansky M. and Wang B., “Grothendieck-Lidskii trace formula for mixednorm Lp and variable Lebesgue spaces”, to appear in J. Spectr. Theory, arXiv:1604.00198.
[10] Delgado J. and Ruzhansky M., “Schatten-von Neumann classes of integral operators”, arXiv:1709.06446.
[11] Epperson J., “Hermite multipliers and pseudo-multipliers”, Proc. Amer. Math. Soc. 124 (1996), No. 7, 2061–2068.
[12] Grothendieck A., “Produits tensoriels topologiques et espaces nucléaires”, in: Mem. Amer. Math. Soc. 16, Providence, 1955.
[13] Pietsch A., Operator ideals, Mathematische Monographien 16, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.
[14] Pietsch A., History of Banach spaces and linear operators, Birkhäuser Boston Inc., Boston, 2007.
[15] Prugove˘cki E., Quantum mechanics in Hilbert space, Pure and Applied Mathematics 92, Academic Press Inc., New York-London, 1981.
[16] Reinov O.I. and Latif Q., “Grothendieck-Lidskii theorem for subspaces of Lp-spaces”, Math. Nachr. 286 (2013), No. 2-3, 279–282.
[17] Ruzhansky M. and Tokmagambetov N., “Nonharmonic analysis of boundary value problems”, Int. Math. Res. Notices 12 (2016), 3548–3615.
[18] RuzhanskyM. and Tokmagambetov N., “Nonharmonic analysis of boundary value problems without WZ condition”, Math. Model. Nat. Phenom. 12 (2017), No. 1, 115–140.
[19] Simon B., “Distributions and their Hermite expansions”, J. Math. Phys. 12 (1971), No. 1, 140–148.
[20] Stempak K., “Multipliers for eigenfunction expansions of some Schrödinger operators”, Proc. Amer. Math. Soc. 93 (1985), No. 3, 477–482.
[21] Stempak, K. and Torrea J.L., “On g-functions for Hermite function expansions”, Acta Math. Hung. 109 (2005), No. 1-2, 99–125.
[22] Stempak K. and Torrea J.L., “BMO results for operators associated to Hermite expansions”, Illinois J. Math. 49 (2005), No. 4, 1111–1132.
[23] Thangavelu S., Lectures on Hermite and Laguerre Expansions, Math. Notes 42, Princeton University Press, Princeton, 1993.
[24] Thangavelu S., “Hermite and special Hermite expansions revisited”, Duke Math. J. 94 (1998), No. 2, 257–278.
[25] Thangavelu S., “Multipliers for Hermite expansions”, Rev. Mat. Iberoam. 3 (1987), 1–24.
[26] Thangavelu S., “Summability of Hermite expansions I”, Trans. Amer. Math. Soc. 314 (1989), No. 1, 119–142.
[27] Thangavelu S., “Summability of Hermite expansions II”, Trans. Amer. Math. Soc. 314 (1989), No. 1, 143–170.
[28] Thangavelu S., “Hermite expansions on R2n for radial functions”, Rev. Mat. Iberoam. 6 (1990), No. 2, 61–73.
[2] Barraza E.S. and Cardona D., “On nuclear Lp-multipliers associated to the Harmonic oscillator”, in Analysis in Developing Countries, Springer Proceedings in Mathematics & Statistics, Springer (2018), M. Ruzhansky and J. Delgado (Eds), to appear.
[3] Cardona D. and Barraza E.S., “Characterization of nuclear pseudo-multipliers associated to the harmonic oscillator”, to appear in, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys (2018), arXiv:1709.07961.
[4] Cardona D. and Ruzhansky M., “Hörmander condition for pseudo-multipliers associated to the harmonic oscillator”, preprint.
[5] Delgado J., A trace formula for nuclear operators on Lp, in Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations, Operator Theory: Advances and Applications 205, Schulze, B.W., Wong, M.W. (eds.), Birkhäuser, Basel (2010), 181–193.
[6] Delgado J., “The trace of nuclear operators on Lp(μ) for -finite Borel measures on second countable spaces”, Integr. Equ. Oper. Theory 68 (2010), No- 1, 61–74.
[7] Delgado J., “On the r-nuclearity of some integral operators on Lebesgue spaces”, Tohoku Math. J. (2) 67 (2015), No. 1, 125–135.
[8] Delgado J., RuzhanskyM. andWang B., “Approximation property and nuclearity on mixednorm Lp, modulation and Wiener amalgam spaces”, J. Lond. Math. Soc.(2) 94 (2016), 391–408.
[9] Delgado J., Ruzhansky M. and Wang B., “Grothendieck-Lidskii trace formula for mixednorm Lp and variable Lebesgue spaces”, to appear in J. Spectr. Theory, arXiv:1604.00198.
[10] Delgado J. and Ruzhansky M., “Schatten-von Neumann classes of integral operators”, arXiv:1709.06446.
[11] Epperson J., “Hermite multipliers and pseudo-multipliers”, Proc. Amer. Math. Soc. 124 (1996), No. 7, 2061–2068.
[12] Grothendieck A., “Produits tensoriels topologiques et espaces nucléaires”, in: Mem. Amer. Math. Soc. 16, Providence, 1955.
[13] Pietsch A., Operator ideals, Mathematische Monographien 16, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.
[14] Pietsch A., History of Banach spaces and linear operators, Birkhäuser Boston Inc., Boston, 2007.
[15] Prugove˘cki E., Quantum mechanics in Hilbert space, Pure and Applied Mathematics 92, Academic Press Inc., New York-London, 1981.
[16] Reinov O.I. and Latif Q., “Grothendieck-Lidskii theorem for subspaces of Lp-spaces”, Math. Nachr. 286 (2013), No. 2-3, 279–282.
[17] Ruzhansky M. and Tokmagambetov N., “Nonharmonic analysis of boundary value problems”, Int. Math. Res. Notices 12 (2016), 3548–3615.
[18] RuzhanskyM. and Tokmagambetov N., “Nonharmonic analysis of boundary value problems without WZ condition”, Math. Model. Nat. Phenom. 12 (2017), No. 1, 115–140.
[19] Simon B., “Distributions and their Hermite expansions”, J. Math. Phys. 12 (1971), No. 1, 140–148.
[20] Stempak K., “Multipliers for eigenfunction expansions of some Schrödinger operators”, Proc. Amer. Math. Soc. 93 (1985), No. 3, 477–482.
[21] Stempak, K. and Torrea J.L., “On g-functions for Hermite function expansions”, Acta Math. Hung. 109 (2005), No. 1-2, 99–125.
[22] Stempak K. and Torrea J.L., “BMO results for operators associated to Hermite expansions”, Illinois J. Math. 49 (2005), No. 4, 1111–1132.
[23] Thangavelu S., Lectures on Hermite and Laguerre Expansions, Math. Notes 42, Princeton University Press, Princeton, 1993.
[24] Thangavelu S., “Hermite and special Hermite expansions revisited”, Duke Math. J. 94 (1998), No. 2, 257–278.
[25] Thangavelu S., “Multipliers for Hermite expansions”, Rev. Mat. Iberoam. 3 (1987), 1–24.
[26] Thangavelu S., “Summability of Hermite expansions I”, Trans. Amer. Math. Soc. 314 (1989), No. 1, 119–142.
[27] Thangavelu S., “Summability of Hermite expansions II”, Trans. Amer. Math. Soc. 314 (1989), No. 1, 143–170.
[28] Thangavelu S., “Hermite expansions on R2n for radial functions”, Rev. Mat. Iberoam. 6 (1990), No. 2, 61–73.