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Energy Eigenvalue Level Motion with Two Parameters

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Energy Eigenvalue Level Motion with Two Parameters

Willi-Hans Steeb, Yorick Hardy, and Ruedi Stoop

International School for Scientific Computing, Rand Afrikaans University, P.O. Box 524, Auckland Park 2006, South Africa

Reprint requests to Prof. W.-H. Steeb; E-mail: whs@na.rau.ac.za Z. Naturforsch. 56 a, 565–567 (2001); received April 17, 2001

From the eigenvalue equation ˆH

j

n

()i=E

n

()j

n

()iwhere ˆH

= ˆH0+Vˆ one can derive an autonomous system of first order ordinary differential equations for the eigenvaluesE

n

() and

the matrix elementsV

mn

() :=h

m

()jVˆj

n

()iwhereis the independent variable. We derive the partial differential equations for the extended case ˆH

1

;

2 = ˆH0+1Vˆ1+2Vˆ2, where1and

2are the independent variables. Some applications of this system of partial differential equations are discussed.

Key words: Eigenvalue Level Motion; Partial Differential Equations.

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