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ADIABATIC EVOLUTION OF 1D SHAPE RESONANCES: AN ARTIFICIAL INTERFACE CONDITIONS APPROACH

ALI FARAJ

IRMAR, UMR — CNRS 6625, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

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ANDREA MANTILE

IRMAR, UMR — CNRS 6625, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

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, and

FRANCIS NIER

IRMAR, UMR — CNRS 6625, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

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https://doi.org/10.1142/S0218202511005143Cited by:5 (Source: Crossref)

Abstract

Artificial interface conditions parametrized by a complex number θ₀ are introduced for 1D-Schrödinger operators. When this complex parameter equals the parameter θ ∈ iℝ of the complex deformation which unveils the shape resonances, the Hamiltonian becomes dissipative. This makes possible an adiabatic theory for the time evolution of resonant states for arbitrarily large time scales. The effect of the artificial interface conditions on the important stationary quantities involved in quantum transport models is also checked to be as small as wanted, in the polynomial scale (h^N)_N∈ℕ as h → 0, according to θ₀.

Keywords:

AMSC: 35J10, 35P20, 35B34, 47A20

References

W. K. Abou Salem and J. Fröhlich, Commun. Math. Phys. 273, 651 (2007). Crossref, Web of Science, Google Scholar
S. Agmon , Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-Body Schrödinger Operators , Mathematical Notes 29 ( Princeton Univ. Press , 1982 ) . Google Scholar
J. Aguilar and J. M. Combes, Commun. Math. Phys. 22, 269 (1971), DOI: 10.1007/BF01877510. Crossref, Web of Science, Google Scholar
S. Albeverio and K. Pankrashkin, J. Phys. A 38, 4859 (2005), DOI: 10.1088/0305-4470/38/22/010. Crossref, Google Scholar
X. Antoineet al., Comm. Comput. Phys. 4, 729 (2008). Web of Science, Google Scholar
A. Arnold, Transp. Th. Statist. Phys. 30, 561 (2001), DOI: 10.1081/TT-100105939. Crossref, Web of Science, Google Scholar
J. E. Avronet al., J. Statist. Phys. 116, 425 (2004), DOI: 10.1023/B:JOSS.0000037245.45780.e1. Crossref, Web of Science, Google Scholar
J. E. Avron, R. Seiler and L. G. Yaffe, Commun. Math. Phys. 110, 33 (1987), DOI: 10.1007/BF01209015. Crossref, Web of Science, Google Scholar
J. E. Avronet al., Comm. Pure Appl. Math. 57, 528 (2004), DOI: 10.1002/cpa.3051. Crossref, Web of Science, Google Scholar
E. Balslev and J. M. Combes, Commun. Math. Phys. 22, 280 (1971), DOI: 10.1007/BF01877511. Crossref, Web of Science, Google Scholar
M. Baroet al., J. Math. Phys. 45, 21 (2004), DOI: 10.1063/1.1628385. Crossref, Web of Science, Google Scholar
M. Baroet al., Rev. Math. Phys. 16, 281 (2004). Link, Web of Science, Google Scholar
M. Baro, H. Neidhardt and J. Rehberg, SIAM J. Math. Anal. 37, 941 (2005), DOI: 10.1137/040611690. Crossref, Web of Science, Google Scholar
J. Behrndt, M. M. Malamud and H. Neidhardt, Math. Phys. Anal. Geom. 10, 313 (2007), DOI: 10.1007/s11040-008-9035-x. Crossref, Web of Science, Google Scholar
N. Ben Abdallah, J. Math. Phys. 41, 4241 (2000). Crossref, Web of Science, Google Scholar
N. Ben Abdallah, P. Degond and P. A. Markowich, Z. Angew. Math. Phys. 48, 135 (1997), DOI: 10.1007/PL00001463. Crossref, Web of Science, Google Scholar
N. Ben Abdallah and O. Pinaud, J. Comput. Phys. 213, 288 (2006). Crossref, Web of Science, Google Scholar
V. Bonnaillie-Noël, A. Faraj and F. Nier, J. Comput. Elec. 8, 11 (2009). Crossref, Web of Science, Google Scholar
V. Bonnaillie-Noël, F. Nier and Y. Patel, J. Comput. Phys. 219, 644 (2006). Crossref, Web of Science, Google Scholar
V. Bonnaillie-Noël, F. Nier and Y. Patel, Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 937 (2008). Crossref, Web of Science, Google Scholar
V. Bonnaillie-Noël, F. Nier and Y. Patel, J. Math. Soc. Jpn. 61, 65 (2009). Crossref, Web of Science, Google Scholar
H. D. Corneanet al., J. Math. Phys. 49, 102106 (2008), DOI: 10.1063/1.2992839. Crossref, Web of Science, Google Scholar
H. L. Cycon et al. , Schrödinger Operators with Application to Quantum Mechanics and Global Geometry ( Springer-Verlag , 1987 ) . Crossref, Google Scholar
M. Dimassi and J. Sjöstrand , Spectral Asymptotics in the Semi-Classical Limit , London Mathematical Society Lecture Note Series 268 ( Cambridge Univ. Press , 1999 ) . Crossref, Google Scholar
M. Ehrhardt and A. Arnold, Riv. Mat. Univ. Parma 4, 57 (2001). Web of Science, Google Scholar
H. O. Fattorini , The Cauchy Problem , Encyclopedia of Mathematics and its Applications 18 ( Addison-Wesley , 1983 ) . Google Scholar
C. Gérard and I. M. Sigal, Commun. Math. Phys. 145, 281 (1992). Crossref, Web of Science, Google Scholar
B. Helffer , Semi-Classical Analysis for the Schrödinger Operator and Applications , Lecture Notes in Mathematics 1336 ( Springer-Verlag , 1988 ) . Crossref, Google Scholar
B. Helffer and A. Martinez, Helv. Phys. Acta 60, 992 (1987). Google Scholar
B. Helffer and J. Sjöstrand, Résonances en Limite Semi-Classique, Mém. Soc. Math. Nos (1986) pp. 24–25. Google Scholar
P. D. Hislop and I. M. Sigal , Semiclassical Theory of Shape Resonances in Quantum Mechanics 78 ( Amer. Math. Soc. , 1989 ) . Crossref, Google Scholar
P. D. Hislop and I. M. Sigal , Introduction to Spectral Theory, Applied Mathematical Sciences , With applications to Schrödinger operators 113 ( Springer-Verlag , 1996 ) . Crossref, Google Scholar
G. Jona-Lasinio, C. Presilla and J. Sjöstrand, Ann. Phys. 240, 1 (1995). Crossref, Web of Science, Google Scholar
A. Joye and C. E. Pfister, Exponential estimates in adiabatic quantum evolution, XII Int. Congr. Mathematical Physics (ICMP '97) (Int. Press, Brisbane, 1999) pp. 309–315. Google Scholar
A. Joye, Commun. Math. Phys. 275, 139 (2007), DOI: 10.1007/s00220-007-0299-y. Crossref, Web of Science, Google Scholar
T. Kato, J. Math. Soc. Jpn. 5, 208 (1953). Crossref, Web of Science, Google Scholar
T. Kato, J. Fac. Sci. Univ. Tokyo Sect. I 17, 241 (1970). Web of Science, Google Scholar
M. Klein and J. Rama, Almost exponential decay of quantum resonance states and Paley–Wiener type estimates in Gevrey spaces, preprint, mp-arc 09-64, 2009 . Google Scholar
M. Klein, J. Rama and R. Wüst, Asymptot. Anal. 51, 1 (2007). Web of Science, Google Scholar
A. Lahmar-Benbernou and A. Martinez, Int. Math. Res. Not. 13, 697 (2002). Google Scholar
S. E. Laux, A. Kumar and M. V. Fischetti, J. Appl. Phys. 95, 5545 (2004), DOI: 10.1063/1.1695597. Crossref, Web of Science, Google Scholar
P. Lochak, Ann. Inst. H. Poincaré Sect. A (N.S.) 39, 119 (1983). Web of Science, Google Scholar
G. Nenciu, Commun. Math. Phys. 152, 479 (1993), DOI: 10.1007/BF02096616. Crossref, Web of Science, Google Scholar
G. Nenciu and G. Rasche, J. Phys. A 25, 5741 (1992), DOI: 10.1088/0305-4470/25/21/027. Crossref, Google Scholar
F. Nier, Nonlinearity 11, 1127 (1998), DOI: 10.1088/0951-7715/11/4/022. Crossref, Web of Science, Google Scholar
F. Nier, Serdica Math. J. 34, 113 (2008). Web of Science, Google Scholar
K. Pankrashkin, Rep. Math. Phys. 58, 207 (2006). Crossref, Web of Science, Google Scholar
G. Perelman, Asymptot. Anal. 22, 177 (2000). Web of Science, Google Scholar
O. Pinaud, J. Appl. Phys. 92, 1987 (2002), DOI: 10.1063/1.1494127. Crossref, Web of Science, Google Scholar
C. Presilla and J. Sjöstrand, J. Math. Phys. 37, 4816 (1996), DOI: 10.1063/1.531671. Crossref, Web of Science, Google Scholar
M. Reed and B. Simon , Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness ( Academic Press , 1975 ) . Google Scholar
M. Reed and B. Simon , Methods of Modern Mathematical Physics. IV. Analysis of Operators ( Academic Press , 1978 ) . Google Scholar
B. Simon, Int. J. Quantum Chem. 14, 529 (1978), DOI: 10.1002/qua.560140415. Crossref, Web of Science, Google Scholar
B. Simon, Phys. Lett. 71A, 211 (1979). Google Scholar
J. Sjöstrand, C. R. Acad. Sci. Paris Sér. I Math. 317, 217 (1993). Google Scholar
J. Sjöstrand and M. Zworski, J. Amer. Math. Soc. 4, 729 (1991). Crossref, Google Scholar
E. Skibsted, Commun. Math. Phys. 104, 591 (1986), DOI: 10.1007/BF01211067. Crossref, Web of Science, Google Scholar
E. Skibsted, Ann. Inst. H. Poincaré Phys. Théor. 46, 131 (1987). Web of Science, Google Scholar
E. Skibsted, J. Math. Anal. Appl. 141, 27 (1989). Crossref, Web of Science, Google Scholar
A. Soffer and M. I. Weinstein, Geom. Funct. Anal. 8, 1086 (1998), DOI: 10.1007/s000390050124. Crossref, Web of Science, Google Scholar
K. Yosida, Functional Analysis (Springer-Verlag, 1995), Reprint of 6th (1980) edition . Google Scholar