On Sturm–Liouville equations with several spectral parameters

Porter, R. Michael

doi:10.1007/s40590-015-0078-2

Cited by 2 publications

(7 citation statements)

References 37 publications

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“…Proof. First we show the convergence of (12). By continuity, there is a uniform bound were also given in [9].…”

Section: Derivatives Of Iterated Integrals and The Pólya Systemmentioning

confidence: 94%

“…For n =2, the powers

P_{1}^{false(m false)}

and

P_{2}^{false(m false)}

have been denoted X (2 m +1) and

{\overset{X}{true}}^{false(2 m false)}

in most of the SPPS literature for equations of second order, as in the literature. () Here for 1≤ k ≤ n we define the k ‐th sequence of secondary formal powers for L and r by setting

X_{k}^{false(j false)} \equiv 0

for all j <0,

X_{k}^{false(0 false)} \equiv 1

and then recursively for j ≥1:

X_{k}^{false(j false)} = ({, \begin{matrix} center \\ array j {boldI}_{k - j, k - j} X_{k}^{(j - 1)}, & array 1 \leq j \leq k - 1, \\ array j {boldI}_{n, n} b_{0} r X_{k}^{(j - 1)}, & array j \equiv k mod n, \\ array j {boldI}_{n - j^{'}, n - j^{'}} X_{k}^{(j - 1)}, & array j \equiv k + j^{'} mod n, 1 \leq j^{'} \leq n - 1 . \end{matrix})

…”

Section: Solution In Terms Of Formal Powersmentioning

confidence: 99%

“…For n = 2, the powers P (m) 1 and P (m) 2 have been denoted X (2m+1) andX (2m) in most of the SPPS literature for equations of second order, as in the literature. 2,3,11,12 Here for 1 ≤ k ≤ n we define the k-th sequence of secondary formal powers for L and r by setting X ( ) k ≡ 0 for all j < 0, X (0) k ≡ 1 and then recursively for j ≥ 1:…”

Section: Spps Solutions Of N-th Ordermentioning

confidence: 99%

“…In the third case of (13), clearly X ( ) k = b n− ′ X ( −1) k . Further, by comparing with (10) we see that the subsequence of main formal powers is included as P (m) k = X (mn+k−1) Further, for 1 ≤ ≤ n − 1, the -th derivative of u k is given by substituting (12), where (x) n is the Pochhammer symbol and the coefficients A , are the functions defined in (10), the P (m) k are obtained by integrations of the form I 1,j · · · where i = 1.…”

Section: Spps Solutions Of N-th Ordermentioning

confidence: 99%

“…Further, for 1 ≤ ℓ ≤ n − 1, the ℓ-th derivative of u k is given by substituting 12), where (x) n is the Pochhammer symbol and the coefficients A ℓ,α are the functions defined in Lemma 2 for the particular case i…”

Section: Derivatives Of Iterated Integrals and The Pólya Systemmentioning

confidence: 99%

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Spectral parameter power series for arbitrary order linear differential equations

Kravchenko

Porter

Torba

2018

Math Methods in App Sciences

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Let L be the n‐th order linear differential operator Ly=ϕ0y(n)+ϕ1y(n−1)+⋯+ϕny with variable coefficients. A representation is given for n linearly independent solutions of Ly=λry as power series in λ, generalizing the SPPS (spectral parameter power series) solution that has been previously developed for n=2. The coefficient functions in these series are obtained by recursively iterating a simple integration process, beginning with a solution system for λ=0. It is shown how to obtain such an initializing system working upwards from equations of lower order. The values of the successive derivatives of the power series solutions at the basepoint of integration are given, which provides a technique for numerical solution of n‐th order initial value problems and spectral problems.

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“…Proof. First we show the convergence of (12). By continuity, there is a uniform bound were also given in [9].…”

Section: Derivatives Of Iterated Integrals and The Pólya Systemmentioning

confidence: 94%

“…For n =2, the powers

P_{1}^{false(m false)}

and

P_{2}^{false(m false)}

have been denoted X (2 m +1) and

{\overset{X}{true}}^{false(2 m false)}

in most of the SPPS literature for equations of second order, as in the literature. () Here for 1≤ k ≤ n we define the k ‐th sequence of secondary formal powers for L and r by setting

X_{k}^{false(j false)} \equiv 0

for all j <0,

X_{k}^{false(0 false)} \equiv 1

and then recursively for j ≥1:

X_{k}^{false(j false)} = ({, \begin{matrix} center \\ array j {boldI}_{k - j, k - j} X_{k}^{(j - 1)}, & array 1 \leq j \leq k - 1, \\ array j {boldI}_{n, n} b_{0} r X_{k}^{(j - 1)}, & array j \equiv k mod n, \\ array j {boldI}_{n - j^{'}, n - j^{'}} X_{k}^{(j - 1)}, & array j \equiv k + j^{'} mod n, 1 \leq j^{'} \leq n - 1 . \end{matrix})

…”

Section: Solution In Terms Of Formal Powersmentioning

confidence: 99%

Section: Spps Solutions Of N-th Ordermentioning

confidence: 99%

Section: Spps Solutions Of N-th Ordermentioning

confidence: 99%

Section: Derivatives Of Iterated Integrals and The Pólya Systemmentioning

confidence: 99%

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Spectral parameter power series for arbitrary order linear differential equations

Kravchenko

Porter

Torba

2018

Math Methods in App Sciences

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Supersymmetric generalized power functions

Ouellet

Tremblay

2020

Journal of Mathematical Physics

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Complex-valued functions defined on a finite interval [a, b] generalizing power functions of the type (x−x0)n for n ≥ 0 are studied. These functions called Φ-generalized powers, Φ being a given nonzero complex-valued function on the interval, were considered to construct a general solution representation of the Sturm–Liouville equation in terms of the spectral parameter [V. V. Kravchenko and R. M. Porter, Math. Methods Appl. Sci. 33(4), 459–468 (2010)]. The Φ-generalized powers can be considered as natural basis functions for the one-dimensional supersymmetric quantum mechanics systems taking Φ=ψ02, where the function ψ0(x) is the ground state wave function of one of the supersymmetric scalar Hamiltonians. Several properties are obtained such as Φ-symmetric conjugate and antisymmetry of the Φ-generalized powers, a supersymmetric binomial identity for these functions, a supersymmetric Pythagorean elliptic (hyperbolic) identity involving four Φ-trigonometric (Φ-hyperbolic) functions, as well as a supersymmetric Taylor series expressed in terms of the Φ-derivatives. We show that the first n Φ-generalized powers are a fundamental set of solutions associated with nonconstant coefficient homogeneous linear ordinary differential equations of order n + 1. Finally, we present a general solution representation of the stationary Schrödinger equation in terms of geometric series where the Volterra compositions of the first type are considered.

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On Sturm–Liouville equations with several spectral parameters

Cited by 2 publications

References 37 publications

Spectral parameter power series for arbitrary order linear differential equations

Spectral parameter power series for arbitrary order linear differential equations

Supersymmetric generalized power functions

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