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A method for obtaining bounds on eigenvalues and eigenfunctions by solving non-homogeneous integral equations

Eine Methode zur Berechnung der Grenzen von Eigenwerten und Eigenfunktionen durch Lösen inhomogener Integralgleichungen

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Abstract

A new technique is presented for obtaining upper and lower bounds on eigenvalues and eigenfunctions for linear integral equations. The method is unique in that the bounds are obtained by solving non-homogeneous equations. In order to solve the non-homogeneous equations, non-linear sequence-to-sequence transformations are used to accelerate convergence of the Neumann series inside the radius of convergence and are used to “sum” the Neumann series outside the radius of convergence. Since the reciprocals of the eigenvalues appear as poles in the solution of the non-homogeneous equation, a very sensitive bounding criterion can be given. The method applies to quite general kernels, and has been successfully applied to symmetric and non-symmetric kernels. In addition, thek th eigenfunction may be obtained without a knowledge of the first (k−1) eigenvalues or eigenfunctions.

Zusammenfassung

Es wird eine neue Methode zur Berechnung oberer und unterer Grenzen der Eigenwerte und Eigenfunktionen linearer Integralgleichungen dargestellt. Dadurch, daß die Lösungen inhomogener Gleichungen diese Grenzwerte bestimmen, ist die Methode neuartig. Zur Lösung der inhomogenen Gleichungen werden nichtlineare Reihentransformationen verwendet um die Konvergenz der Neumann-Reihen innerhalb des Konvergenzradius zu beschleunigen und werden ferner zur Summierung der Neumann-Reihen außerhalb des Konvergenzradius benützt. Da die reziproken Eigenwerte Wurzelsingularitäten der inhomogenen Gleichung sind, kann eine sehr empfindliche Begrenzungsbedingung angegeben werden. Die Methode bezieht sich auf ganz allgemeine Kerne und konnte im Falle symmetrischer sowohl wie unsymmetrischer Kerne angewandt werden. Ferner kann diek-te Eigenfunktion ohne Kenntnis der ersten (k−1) Eigenwerte oder Eigenfunktionen berechnet werden.

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Burgmeier, J.W., Scott, M.R. A method for obtaining bounds on eigenvalues and eigenfunctions by solving non-homogeneous integral equations. Computing 10, 9–22 (1972). https://doi.org/10.1007/BF02242379

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Keywords

  • Integral Equation
  • Lower Bound
  • Computational Mathematic
  • Neumann Series
  • Linear Integral Equation