参考文献:1.Alexandersson, P., Gabrielov, A.: On eigenvalues of the Schr?dinger operator with a complex-valued polynomial potential. CMFT 12(1), 119-44 (2012)MATH MathSciNet 2.Baryshnikov, Yu.: On Stokes sets. In: New developments in singularity theory (Cambridge, 2000), pp. 65-6. NATO Sci. Ser. II Math. Phys. Chem., vol. 21. Kluwer, Dordrecht (2001) 3.Birkhoff, G.D.: Quantum mechanics and asymptotic series. Bull. Am. Math. Soc. 32, 681-00 (1933)View Article MathSciNet 4.Delabaere, E., Pham, F.: Unfolding the quartic oscillator. Ann. Phys. 261, 180-18 (1997)View Article MATH MathSciNet 5.Eremenko, A., Gabrielov, A.: Analytic continuation of eigenvalues of a quartic oscillator. Commun. Math. Phys. 287(2), 431-57 (2009)View Article MATH MathSciNet 6.Eremenko, A., Gabrielov, A.: Singular perturbation of polynomial potentials in the complex domain with applications to PT-symmetric families. Mosc. Math. J. 11(3), 473-03 (2011)MATH MathSciNet 7.Evgrafov, M., Fedoryuk, M.: Asymptotic behavior of solutions of the equation \(w^{\prime \prime }(z)-p(z,\,\lambda )w(z)=0\) as \(\lambda \rightarrow \infty \) in the complex \(z\) -plane (Russian). Usp. Math. Nauk 21(1), 3-0 (1966)MATH 8.Fedoruyk, M.: Asymptotic analysis. Linear Ordinary Differential Equations. Springer, Berlin (1993) 9.Fedoryuk, M.: Asymptotics of the discrete spectrum of the operator \(w^{\prime \prime }(x)-\lambda ^2w(x)\) . Math. Sb. 68(1), 81-09 (1965) 10.Fedoryuk, M.: One-dimensional scattering in quasiclassical approximation. Diff. Uravn. 1(5), 483-95 (1965) 11.Fedoryuk, M.: Topology of Stokes lines for the 2nd order equations. Izv. AS USSR 29, 645-56 (1965) 12.Fedoryuk, M.: Analytic spectral problems. Diff. Uravn. 26(2), 258-67 (1990)MathSciNet 13.Hsieh, P.S., Sibuya, Y.: On the asymptotic integration of second order linear ordinary differential equation with polynomial coefficient. J. Math. Anal. Appl. 16, 84-03 (1966)View Article MATH MathSciNet 14.Jenkins, J.A.: Univalent functions and conformal mapping. Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Heft 18. Reihe: Moderne Funktionentheorie. Springer, Berlin (1958) 15.Mumford, D., Fogarty, J., Kirwan, F.: Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 3rd edn. Springer, Berlin (1994) 16.Nevanlinna, R.: über Riemannsche Fl?chen mit endlich vielen Windungspunkten. Acta Math. 58, 295-73 (1932)View Article MathSciNet 17.Shapere, A.D., Vafa, C.: BPS structure of Argyres–Douglas superconformal theories. hep-th/-910172 18.Shin, K. C.: Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT -Symmetry, Symmetry, Integrability and Geometry: Methods and Applications, SIGMA 6, 015 (2010) 19.Sibuya, Y.: Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient, Noth-Holland Math. Studies, vol. 18. North-Holland, Amsterdam (1974) 20.Sibuya, Y.: Subdominant solutions of the differential equation\(y^{\prime \prime }-\lambda ^2(x-a_1)(x-a_2)..(x-a_m)y=0\) . Acta Math. 119(1), 235-72 (1967)View Article MATH MathSciNet 21.Strebel, K.: Quadratic differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer, Berlin (1984) 22.Wasow, W.: Linear turning point theory. Applied Mathematical Sciences. Springer, New York (1985)
作者单位:Boris Shapiro (1)
1. Department of Mathematics, Stockholm University, 106 91, Stockholm, Sweden
刊物类别:Mathematics and Statistics
刊物主题:Analysis Mathematical Methods in Physics
出版者:Birkh盲user Basel
ISSN:1664-235X
文摘
The purpose of this note is to recall the theory of the (homogenized) spectral problem for Schr?dinger equation with a polynomial potential and its relation with quadratic differentials. We derive from results of this theory that the accumulation rays of the eigenvalues of the latter problem are in \(1-1\)-correspondence with the short geodesics of the singular planar metrics induced by the corresponding quadratic differential. We prove that for a polynomial potential of degree \(d,\) the number of such accumulation rays can be any positive integer between \((d-1)\) and \(d \atopwithdelims ()2\).