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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: $$L_z[y(z)]= p_0(z) y''(z)+ p_1(z) y'(z)+ p_2(z) y(z)= \lambda y(z),\tag 1$$ 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 $c\ge 1$, $d\ge 1$, $e\ge 1$, $a\ge b$. 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 $|y(0)|< \infty$, $|y(1)|< \infty$. The eigenvalues $\lambda_n$ of this problem are functions of the parameter $s$. The authors study the behaviour of the eigenvalue curves $\lambda_n= \lambda_n(s)$ in the vicinity of $s= 0$, i.e. for $s\to 0+$. 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.

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
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