The mathematical theory of resonances whose widths are exponentially small

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The mathematical theory of resonances whose widths are exponentially small

E. Harrell and B. Simon

Source: Duke Math. J. Volume 47, Number 4 (1980), 845-902.

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Primary Subjects: 81C05

Secondary Subjects: 34B25

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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077314340
Mathematical Reviews number (MathSciNet): MR596118
Zentralblatt MATH identifier: 0455.35091
Digital Object Identifier: doi:10.1215/S0012-7094-80-04750-X

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References

[1] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964.

Mathematical Reviews (MathSciNet): MR29:4914

Zentralblatt MATH: 0171.38503

[2] J. E. Avron and I. W. Herbst, Spectral and scattering theory of Schrödinger operators related to the Stark effect, Comm. Math. Phys. 52 (1977), no. 3, 239–254.

Mathematical Reviews (MathSciNet): MR57:8666

Zentralblatt MATH: 0351.47007

Digital Object Identifier: doi:10.1007/BF01609485

Project Euclid: euclid.cmp/1103900538

[3] E. Balslev and J. M. Combes, Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions, Comm. Math. Phys. 22 (1971), 280–294.

Mathematical Reviews (MathSciNet): MR49:10288

Zentralblatt MATH: 0219.47005

Digital Object Identifier: doi:10.1007/BF01877511

Project Euclid: euclid.cmp/1103857513

[4] L. Benassi and V. Grecchi, Resonances in the Stark effect and strongly asymptotic approximations, J. Phys. B, to appear, 1980.

[5] Carl M. Bender and Tai Tsun Wu, Anharmonic oscillator, Phys. Rev. (2) 184 (1969), 1231–1260.

Mathematical Reviews (MathSciNet): MR41:4951

Digital Object Identifier: doi:10.1103/PhysRev.184.1231

[6] C. Bender and T. T. Wu, Large order behavior of perturbation theory, Phys. Rev. Lett. 16 (1971), 461–465, Anharmonic oscillator, II. A study of perturbation theory in large order, Phys. Rev. D7 (1973), 1620–1636.

[7] T. Banks, C. M. Bender, and T. T. Wu, Coupled anharmonic oscillators. I. Equal-mass case, Phys. Rev. D (3) 8 (1973), 3346–3366.

Mathematical Reviews (MathSciNet): MR50:15673

Digital Object Identifier: doi:10.1103/PhysRevD.8.3346

[8] E. Brézin, J. C. LeGuillou, and J. Zinn-Justin, Perturbation theory at large order, I. The $\phi^2N$ interaction, Phys. Rev. D15 (1971), 1544–1557.

[9] H. Buchholz, The confluent hypergeometric function with special emphasis on its applications, Translated from the German by H. Lichtblau and K. Wetzel. Springer Tracts in Natural Philosophy, Vol. 15, Springer-Verlag New York Inc., New York, 1969.

Mathematical Reviews (MathSciNet): MR39:1692

Zentralblatt MATH: 0169.08501

[10] J. M. Combes, An algebraic approach to quantum scattering, unpublished manuscript, 1971.

[11] W. Y. Crutchfield, Method for Borel-summing instanton singularities: introduction, Phys. Rev. D (3) 19 (1979), no. 8, 2370–2384.

Mathematical Reviews (MathSciNet): MR80c:81041

Digital Object Identifier: doi:10.1103/PhysRevD.19.2370

[12] R. J. Damburg and V. V. Kolosov, An asymptotic approach to the Stark effect for the hydrogen atom, J. Phys. B11 (1978), 1921–1930.

[13] P. S. Epstein, Zur Theorie des Starkeffektes, Ann. der Physik 50 (1916), 489–520.

[14] P. S. Epstein, The Stark effect from the point of view of Schroedinger's quantum theory, Phys. Rev. 28 (1926), 695–710.

Zentralblatt MATH: 52.0980.06

[15] N. Fröman and P.-O. Fröman, JWKB approximation. Contributions to the theory, North-Holland Publishing Co., Amsterdam, 1965.

Mathematical Reviews (MathSciNet): MR30:3694

Zentralblatt MATH: 0129.41907

[16] G. Gamow, Constitution of Atomic Nuclei and Radioactivity, Oxford University Press, Oxford, 1931.

[17] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Fourth edition prepared by Ju. V. Geronimus and M. Ju. Ceĭtlin. Translated from the Russian by Scripta Technica, Inc. Translation edited by Alan Jeffrey, Academic Press, New York, 1965.

Mathematical Reviews (MathSciNet): MR33:5952

[18] S. Graffi and V. Grecchi, Resonances in Stark effect and perturbation theory, Commun. Math. Phys. 62 (1979), 83–96.

Mathematical Reviews (MathSciNet): MR506369

Digital Object Identifier: doi:10.1007/BF01940333

Project Euclid: euclid.cmp/1103904305

[19] S. Graffi and V. Grecchi, Confinement of the resonances in hydrogen Stark effect, J. Phys. B12 (1979), L 265–267.

[20] S. Graffi, V. Grecchi, S. Levoni, and M. Maioli, Resonances in one-dimensional Stark effect and continued fractions, J. Math. Phys. 20 (1979), no. 4, 685–690.

Mathematical Reviews (MathSciNet): MR80h:81010

Zentralblatt MATH: 0443.34022

Digital Object Identifier: doi:10.1063/1.524111

[21] S. Graffi, V. Grecchi, and B. Simon, Complete separability of the Stark problem in hydrogen, J. Phys. A 12 (1979), no. 7, L193–L197.

Mathematical Reviews (MathSciNet): MR83d:81116

Zentralblatt MATH: 0441.35065

Digital Object Identifier: doi:10.1088/0305-4470/12/7/009

[22]¹ E. Harrell, On the rate of asymptotic eigenvalue degeneracy, Comm. Math. Phys. 60 (1978), no. 1, 73–95.

Mathematical Reviews (MathSciNet): MR58:6464

Zentralblatt MATH: 0395.34023

Digital Object Identifier: doi:10.1007/BF01609474

Project Euclid: euclid.cmp/1103904022

[22]² E. Harrell, The band-structure of a one-dimensional, periodic system in a scaling limit, Ann. Physics 119 (1979), no. 2, 351–369.

Mathematical Reviews (MathSciNet): MR80i:34031

Zentralblatt MATH: 0412.34013

Digital Object Identifier: doi:10.1016/0003-4916(79)90191-X

[23] I. Herbst, Unitary equivalence of stark Hamiltonians, Math. Z. 155 (1977), no. 1, 55–70.

Mathematical Reviews (MathSciNet): MR56:7623

Zentralblatt MATH: 0338.47009

Digital Object Identifier: doi:10.1007/BF01322607

[24] I. Herbst, Dilation analyticity in constant electric field. I. The two body problem, Comm. Math. Phys. 64 (1979), no. 3, 279–298.

Mathematical Reviews (MathSciNet): MR81a:81020

Zentralblatt MATH: 0447.47028

Digital Object Identifier: doi:10.1007/BF01221735

Project Euclid: euclid.cmp/1103904724

[25] I. W. Herbst and B. Simon, Stark effect revisited, Phys. Rev. Lett. 41 (1978), no. 2, 67–69.

Mathematical Reviews (MathSciNet): MR58:14542

Digital Object Identifier: doi:10.1103/PhysRevLett.41.67

[26] I. Herbst and B. Simon, Dilation analyticity in constant electric field, II: $N$-body problem, Borel summability, submitted to Commun. Math. Phys.

Mathematical Reviews (MathSciNet): MR623157

Digital Object Identifier: doi:10.1007/BF01213010

Project Euclid: euclid.cmp/1103919875

[27] J. Howland, Spectral concentration and virtual poles, Amer. J. Math. 91 (1969), 1106–1126.

Mathematical Reviews (MathSciNet): MR40:7863

Zentralblatt MATH: 0208.38801

Digital Object Identifier: doi:10.2307/2373318

JSTOR: links.jstor.org

[28] C. G. J. Jacobi, Vorlesungen über Dynamik, G. Reiner, Berlin, 1984.

[29] T. Kato, On the convergence of the perturbation method, J. Fac. Sci. Univ. Tokyo 6 (1951), 145–226.

Zentralblatt MATH: 0045.21602

[30] C. Lanczos, Zur Theorie des Starkeffektes in hohen Feldern., Z. für Physik 62 (1930), 518–544, Zur Verschiebung der Wasserstoffterme in hohen elektrischen Feldern, Z. für Physik 65, (1930), 431–455; Zur Intensitätsschwächung der Spektrallinien in hohen Feldern, Z. für Physik 68, (1931), 204–232.

Zentralblatt MATH: 56.0749.03

[31] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Pergamon Press, New York, 1977.

[32] R. E. Langer, The asymptotic solutions of ordinary linear differential equations of the second order, with special reference to a turning point, Trans. Amer. Math. Soc. 67 (1949), 461–490.

Mathematical Reviews (MathSciNet): MR11,438b

Zentralblatt MATH: 0041.05901

Digital Object Identifier: doi:10.2307/1990486

JSTOR: links.jstor.org

[33] R. Lavine, Spectral density and sojourn times, Atomic Scattering Theory ed. J. Nuttall, Univ. of Western Ontario Press, London, Ontario, 1978.

[34] L. N. Lipatov, Divergence of the perturbation theory series and pseudo-particles, Soviet Phys. JETP Lett. 25 (1977), 104–107.

Mathematical Reviews (MathSciNet): MR456084

[35] J. J. Loeffel and A. Martin, Propriétés Analytiques des Niveaux de l'Oscillateur Anharmonique et Convergence des Approximants de Padé, vol. 25, Proc. R.C.P., May 1970.

[36] J. J. Loeffel, A. Martin, B. Simon, and A. Wightman, Padé approximants and the anharmonic oscillator, Phys. Lett. 30B (1969), 656–658.

[37] C. Lovelace, Three-particle systems and unstable particles, Strong Interactions and High Energy Physics (Scottish Universities' Summer School, 1963), Plenum Press, New York, 1964, pp. 437–475.

Mathematical Reviews (MathSciNet): MR30:5732

[38] J. R. Oppenheimer, Three notes on the quantum theory of aperiodic effects, Phys. Rev. 31 (1928), 66–81.

[39] M. Reed and B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978.

Mathematical Reviews (MathSciNet): MR58:12429c

Zentralblatt MATH: 0401.47001

[40] V. de Alfaro and T. Regge, Potential scattering, North-Holland Publishing Co., Amsterdam, 1965.

Mathematical Reviews (MathSciNet): MR32:8724

Zentralblatt MATH: 0141.23202

[41] E. Schrödinger, Quantisierung als Eigenwertproblem, III, Ann. der Physik 80 (1926), 457–490.

Zentralblatt MATH: 52.0966.02

[42] K. Schwarzschild, Zur Quantenhypothese, Sitzungsber. der kön. preuss. Akad. der Wiss. 25 (1916), 548–568.

[43] J. Schwinger, Field theory of unstable particles, Ann. Physics 9 (1960), 169–193.

Mathematical Reviews (MathSciNet): MR22:2362

Zentralblatt MATH: 0089.44902

Digital Object Identifier: doi:10.1016/0003-4916(60)90027-0

[44] A. J. F. Siegert, On the Derivation of the Dispersion Formula for Nuclear Reactions, Phys. Rev. 56 (1939), 750–752.

[45] H. J. Silverstone, Perturbation theory of the Stark effect in hydrogen to arbitrarily high order, Phys. Rev. A18 (1978), 1853–1864.

[46] H. J. Silverstone, Asymptotic relations between the energy shift and ionization rate in the Stark effect in hydrogen, Johns Hopkins Univ., preprint.

[47] H. J. Silverstone, B. G. Adams, J. Čižek, and P. Otto, Asymptotic formula for the perturbed energy coefficients and calculation of the ionization rate by means of high-order perturbation theory for hydrogen in the Stark effect, Johns Hopkins Univ. and Univ. of Waterloo preprint.

[48] B. Simon, Coupling constant analyticity for the anharmonic oscillator. (With appendix), Ann. Physics 58 (1970), 76–136.

Mathematical Reviews (MathSciNet): MR54:4397

Digital Object Identifier: doi:10.1016/0003-4916(70)90240-X

[49] B. Simon, Resonances in $n$-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory, Ann. of Math. (2) 97 (1973), 247–274.

Mathematical Reviews (MathSciNet): MR50:6378

Zentralblatt MATH: 0252.47009

Digital Object Identifier: doi:10.2307/1970847

JSTOR: links.jstor.org

[50] B. Simon, Phase space analysis of simple scattering systems: extensions of some work of Enss, Duke Math. J. 46 (1979), no. 1, 119–168.

Mathematical Reviews (MathSciNet): MR80j:35081

Zentralblatt MATH: 0402.35076

Digital Object Identifier: doi:10.1215/S0012-7094-79-04607-6

Project Euclid: euclid.dmj/1077313257

[51] B. Simon, Functional integration and quantum physics, Pure and Applied Mathematics, vol. 86, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1979.

Mathematical Reviews (MathSciNet): MR84m:81066

Zentralblatt MATH: 0434.28013

[52] T. Spencer, in prep.

[53] J. Stark, Beobachtungen über den Effekt des electrischen Feldes auf Spektrallinien, Sitzungsber. der kön, preuss. Akad. der Wiss. 47 (1913), 932–946.

[54] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Vol. 2, Oxford, at the Clarendon Press, 1958.

Mathematical Reviews (MathSciNet): MR20:1065

Zentralblatt MATH: 0097.27601

[55]¹ E. C. Titchmarsh, Some theorems on perturbation theory, Proc. Roy. Soc. London. Ser. A. 200 (1949), 34–46.

Mathematical Reviews (MathSciNet): MR11,596c

Zentralblatt MATH: 0035.18001

Digital Object Identifier: doi:10.1098/rspa.1949.0157

[55]² E. C. Titchmarsh, Some theorems on perturbation theory. II, Proc. Roy. Soc. London. Ser. A. 201 (1950), 473–479.

Mathematical Reviews (MathSciNet): MR12,337f

Zentralblatt MATH: 0041.22102

Digital Object Identifier: doi:10.1098/rspa.1950.0072

[55]³ E. C. Titchmarsh, Some theorems on perturbation theory. III, Proc. Roy. Soc. London. Ser. A. 207 (1951), 321–328.

Mathematical Reviews (MathSciNet): MR13,844f

Zentralblatt MATH: 0045.19503

Digital Object Identifier: doi:10.1098/rspa.1951.0120

[55]⁴ E. C. Titchmarsh, Some theorems on perturbation theory. IV, Proc. Roy. Soc. London. Ser. A. 210 (1951), 30–47.

Mathematical Reviews (MathSciNet): MR13,844g

Zentralblatt MATH: 0045.19503

Digital Object Identifier: doi:10.1098/rspa.1951.0229

[55]⁵ E. C. Titchmarsh, Some theorems on perturbation theory. V, J. Analyse Math. 4 (1954/55), 187–208.

Mathematical Reviews (MathSciNet): MR17,853c

Zentralblatt MATH: 0067.23103

Digital Object Identifier: doi:10.1007/BF02787720

[55]⁶ E. C. Titchmarsh, Eigenfunction expansions associated with partial differential equations. V, Proc. London Math. Soc. (3) 5 (1955), 1–21.

Mathematical Reviews (MathSciNet): MR16,827a

Zentralblatt MATH: 0053.39301

Digital Object Identifier: doi:10.1112/plms/s3-5.1.1

[56] K. Veselić and J. Weidmann, Potential scattering in a homogeneous electrostatic field, Math. Z. 156 (1977), no. 1, 93–104.

Mathematical Reviews (MathSciNet): MR58:23173

Zentralblatt MATH: 0364.35042

Digital Object Identifier: doi:10.1007/BF01215131

[57] K. Yajima, Spectral and scattering theory for Schrödinger operators with Stark field. I, J. Fac. Sci. Univ. Tokyo, to appear. II. Univ. of Virginia preprint.

[58] T. Yamabe, A. Tachibana, and H. J. Silverstone, Theory of the ionization of the hydrogen atom by an electrostatic field, Phys. Rev. A 16 (1977), 877–890.

[59] T. Yamabe, A. Tachibana, and H. J. Silverstone, Perturbation theory of resonant states induced by an electrostatic field: One dimensional model, J. Phys. B 10 (1977), 2083–2100.

[60] B. Simon, Complex scaling; A rigorous overview, Int. J. Quant. Chem. 14 (1978), 529–542.

[61] J. Howland, Puiseux series for resonances at an embedded eigenvalue, Pacific J. Math. 55 (1974), 157–176.

Mathematical Reviews (MathSciNet): MR54:5871

Zentralblatt MATH: 0312.47010

Project Euclid: euclid.pjm/1102911145

[62] I. Herbst, Exponential decay in the Stark effect, Commun. Math. Phys., to appear.

Mathematical Reviews (MathSciNet): MR581945

Digital Object Identifier: doi:10.1007/BF01212708

Project Euclid: euclid.cmp/1103908145

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