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Heun’s equation with nearby singularities. (English) Zbl 0956.34077

In this paper, the following Fuchsian equation with four singularities is studied:

L z [y(z)]=p 0 (z)y '' (z)+p 1 (z)y ' (z)+p 2 (z)y(z)=λy(z),(1)

with p 0 (z)=z(z-1)(z+s), p 1 (z)=c(z-1)(z+s)+dz(z+s)+ez(z-1), p 2 (z)=abz and the parameters a, b, c, d, e satisfy the Fuchs identity a+b+1-c-d-e=0 and the additional conditions c1, d1, e1, ab. The other parameter s is a small positive one.

A boundary problem on the interval [0,1] is posed for equation (1) with the boundary conditions |y(0)|<, |y(1)|<. The eigenvalues λ n of this problem are functions of the parameter s.

The authors study the behaviour of the eigenvalue curves λ n =λ n (s) in the vicinity of s=0, i.e. for s0+. The qualitative behaviour of the eigenfunctions y n (z) is also studied.

A nice application of the obtained results is demonstrated on an important physical model – the plastic deformation of crystalline materials under stress.

MSC:
34M35Singularities, monodromy, local behavior of solutions, normal forms
34M30Asymptotics, summation methods (ODE in the complex domain)
34M40Stokes phenomena and connection problems (ODE in the complex domain)
74E15Crystalline structure