On the asymptotic integration of second order linear ordinary differential equations with polynomial coefficients

Hsieh, Po-Fang; Sibuya, Yasutaka

doi:10.1016/0022-247x(66)90188-0

Cited by 65 publications

(48 citation statements)

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“…(6) shows, among other things, that \pN -* 0 along the imaginary axis (by (2)). Curiously, \¡/'N "•*> 0 along the imaginary axis.…”

Section: A Subclass Of Anharmonic Oscillators Whose Eigenfunctions Hamentioning

confidence: 90%

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A Subclass of Anharmonic Oscillators Whose Eigenfunctions Have No Recurrence Relations

Gundersen¹

1976

Proceedings of the American Mathematical Society

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Abstract.The equation w"(z) + (X -p(z))w(z) = 0 (z e C,A e R) with a fixed p(z) = a2mz2m + a2m_2z2m-2 + ---+ a2z2(a,>0Vi,a2m>0), possesses a set of solutions {i//"(z)}^L0 (with associated {a"}^°=0) which form a complete orthonormal set for L2(-co, oo) (as a real space). Here it is shown that any g(z) 1when q = 0) cannot be expressed as a finite linear combination of {ipn(z))^L0, when deg(p) is a multiple of 4. It is well known that g(z)\¡fj¡j\z) can always be expressed as such when p(z) = z2 (the Hermite case).

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“…(6) shows, among other things, that \pN -* 0 along the imaginary axis (by (2)). Curiously, \¡/'N "•*> 0 along the imaginary axis.…”

Section: A Subclass Of Anharmonic Oscillators Whose Eigenfunctions Hamentioning

confidence: 90%

“…Curiously, \¡/'N "•*> 0 along the imaginary axis. Although this fact comes out in the course of obtaining (6) [5, p. 27], below we give an elementary proof which uses none of the machinery developed to get to (6).…”

Section: A Subclass Of Anharmonic Oscillators Whose Eigenfunctions Hamentioning

confidence: 99%

A Subclass of Anharmonic Oscillators Whose Eigenfunctions Have No Recurrence Relations

Gundersen¹

1976

Proceedings of the American Mathematical Society

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“…For p > 2 the solutions of (1.2) are not adequately known, so our results for such cases will not be so useful until those solutions become better known. (See P. F. Hsieh and Y. Sibuya [5] and Y. Sibuya [6].)…”

Section: Of the Comparison Equationmentioning

confidence: 99%

Asymptotic solution of eigenvalue problems for second‐order ordinary differential equations

Anyanwu

Keller

1975

Comm Pure Appl Math

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“…The normalized groundstate £2(x) is strictly positive and can be written as Í2(;c) = exp(-0), for <j> satisfying aU the requirements of Theorem 5 with s =p [7], [11], [12].…”

Section: Jay Rosenmentioning

confidence: 99%

Sobolev Inequalities for Weight Spaces and Supercontractivity

Rosen¹

1976

Transactions of the American Mathematical Society

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On the asymptotic integration of second order linear ordinary differential equations with polynomial coefficients

Cited by 65 publications

References 0 publications

A Subclass of Anharmonic Oscillators Whose Eigenfunctions Have No Recurrence Relations

A Subclass of Anharmonic Oscillators Whose Eigenfunctions Have No Recurrence Relations

Asymptotic solution of eigenvalue problems for second‐order ordinary differential equations

Sobolev Inequalities for Weight Spaces and Supercontractivity

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