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On Sturm-Liouville Equations with Several Spectral Parameters

R. Michael Porter

Abstract

We give explicit formulas for a pair of linearly independent solutions of $(py')'(x)+q(x)=(λ_1r_1(x)+\cdots+λ_dr_d(x))y(x)$, thus generalizing to arbitrary $d$ previously known formulas for $d=1$. These are power series in the spectral parameters $λ_1,\dots,λ_d$ (real or complex), with coefficients which are functions on the interval of definition of the differential equation. The coefficients are obtained recursively using indefinite integrals involving the coefficients of lower degree. Examples are provided in which these formulas are used to solve numerically some boundary value problems for $d=2$, as well as an application to transmission and reflectance in optics.

On Sturm-Liouville Equations with Several Spectral Parameters

Abstract

We give explicit formulas for a pair of linearly independent solutions of , thus generalizing to arbitrary previously known formulas for . These are power series in the spectral parameters (real or complex), with coefficients which are functions on the interval of definition of the differential equation. The coefficients are obtained recursively using indefinite integrals involving the coefficients of lower degree. Examples are provided in which these formulas are used to solve numerically some boundary value problems for , as well as an application to transmission and reflectance in optics.

Paper Structure

This paper contains 15 sections, 9 theorems, 73 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Formal powers
  3. Construction of $\widetilde{X}^{(\vec{\jmath}\space)}$
  4. Construction of $X^{(\vec{\jmath}\space)}$
  5. SPPS series and characteristic function
  6. General solution
  7. Generalized Sturm-Liouville equation
  8. Spectral problems
  9. Remarks
  10. Reduction to simple cases
  11. Computational aspects
  12. Numerical examples
  13. Boundary value problems
  14. Application to electromagnetic transmission
  15. Closing remarks

Key Result

Lemma 1

Let $u_0$ be a nonvanishing function on $[x_1,x_2]$ and suppose that $Lu_0=0$. Then for any nonnegative even multiindex $2\vec{n}$,

Figures (7)

  • Figure 1: Construction of $\widetilde{X}^{(\vec{\jmath}\space)}$.
  • Figure 2: Formal powers $X^{(\vec{\jmath}\space)}$ for $d=2$.
  • Figure 3: Characteristic function and zero-level curves (Example 1).
  • Figure 4: Characteristic function and zero-level curves (Example 2).
  • Figure 5: $\chi(\lambda_1,\lambda_2)$ for fixed value of $\lambda_2=1.0$.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 5
  • Corollary 6
  • Lemma 7
  • Lemma 8
  • Theorem 9