Delabaere, Eric; Dillinger, Hervé; Pham, Frédéric Exact semiclassical expansions for one-dimensional quantum oscillators. (English) Zbl 0896.34051 J. Math. Phys. 38, No. 12, 6126-6184 (1997). The time independent one-dimensional Schrödinger equation is considered. WKB expansions encoding exact (not only approximate) solutions of the Schrödinger equation are studied. Rules for computing the Stokes automorphism and the connection isomorphisms are given. Generic values of energy for which all turning points are simple are dealt with. The connection isomorphisms are given by combinations of analytic continuations along suitable paths of the complex \(q\) plane and they are described explicitly by simple pictograms. The pictograms are extended to the case, when there are double turning points. The corresponding WKB expansions then involve special prefactors. The problem of solving the quantization condition for bound states or resonances with respect to the energy parameter is examined. Expressions for the energy levels yielding rigorous justifications of such results as the Zinn-Justin expansions are obtained. Reviewer: V.Burjan (Praha) Cited in 3 ReviewsCited in 62 Documents MSC: 34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory Keywords:turning point theory; WKB methods; one-dimensional Schrödinger equation; Stokes automorphism; resonances; Zinn-Justin expansions PDFBibTeX XMLCite \textit{E. Delabaere} et al., J. Math. Phys. 38, No. 12, 6126--6184 (1997; Zbl 0896.34051) Full Text: DOI Link References: [1] DOI: 10.1103/PhysRev.184.1231 [2] DOI: 10.1016/0003-4916(70)90497-5 · Zbl 0207.40202 [3] DOI: 10.1016/0003-4916(70)90497-5 · Zbl 0207.40202 [4] DOI: 10.1016/0003-4916(70)90497-5 · Zbl 0207.40202 [5] DOI: 10.1016/0003-4916(71)90032-7 [6] DOI: 10.1016/0003-4916(74)90421-7 · Zbl 0281.35029 [7] Voros A., Ann. Inst. H. Poincaré Phys. Theor. 39 pp 211– (1983) [8] DOI: 10.5802/aif.1381 · Zbl 0807.35105 [9] Delabaere E., C. R. Acad. Sci. Paris Ser. I 310 pp 141– (1990) [10] DOI: 10.1103/PhysRevD.7.1620 [11] DOI: 10.1016/0003-4916(88)90003-6 · Zbl 0646.58038 [12] DOI: 10.1016/0003-4916(88)90004-8 · Zbl 0646.58039 [13] DOI: 10.5802/aif.1327 · Zbl 0785.30017 [14] DOI: 10.1098/rspa.1990.0111 · Zbl 0745.34052 [15] DOI: 10.1098/rspa.1991.0119 · Zbl 0764.30031 [16] DOI: 10.1098/rspa.1993.0134 · Zbl 0803.58008 [17] Aoki T., Sugaku Expo. 8 pp 217– (1995) [18] DOI: 10.1070/RM1966v021n01ABEH004145 [19] Jidoumou A. O., J. Math. Pures Appl. 73 pp 111– (1994) [20] DOI: 10.1103/PhysRevA.40.6814 [21] Harrel E., Duke Math. J. 47 pp 845– (1980) · Zbl 0455.35091 [22] DOI: 10.1002/qua.560210103 [23] DOI: 10.1016/0003-4916(70)90240-X [24] DOI: 10.1007/BF01385638 · Zbl 0701.30002 [25] DOI: 10.1016/0550-3213(83)90369-3 [26] DOI: 10.1063/1.526205 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.