In this article, we recursively obtain the general solution of the Schrödinger equation $$y_{\nu }''(x;\lambda )+[\lambda -\nu (\nu +1)v(x)]y_{\nu }(x;\lambda )=0$$ y ν ′ ′ ( x ; λ ) + [ λ - ν ( ν + 1 ) v ( x ) ] y ν ( x ; λ ) = 0 for non negative integer values of $$\nu$$ ν and an arbitrary values of the eigenvalue parameter $$\lambda$$ λ where v ( x ) is certain trigonometric potentials. The recursions are obtained from the contour integral solutions of Tanriverdi. By using these contour integral solutions the author have obtained the first few solutions when $$\nu =n$$ ν = n , a non negative integer, by means of residue calculations which becomes considerably troublesome or almost impossible for larger values of n . Therefore, the recursive procedure of the present article can be seen as a superior alternative to the method of residue calculation for deriving the general solution for arbitrary values of $$\lambda$$ λ and non negative integer n . Moreover, the eigenpairs with the homogeneous Drichlet and Neumann boundary conditions are also derived from the general solution.

中文翻译：

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某些三角势的薛定谔方程的通解

在本文中，我们递归地得到薛定谔方程 $$y_{\nu }''(x;\lambda )+[\lambda -\nu (\nu +1)v(x)]y_{\ nu }(x;\lambda )=0$$ y ν ′ ′ ( x ; λ ) + [ λ - ν ( ν + 1 ) v ( x ) ] y ν ( x ; λ ) = 0 对于非负整数值$$\nu$$ ν 和特征值参数 $$\lambda$$ λ 的任意值，其中 v ( x ) 是某些三角势。递归是从 Tanriverdi 的轮廓积分解中获得的。通过使用这些轮廓积分解决方案，作者通过残差计算获得了前几个解决方案，当 $$\nu =n$$ ν = n ，一个非负整数时，残差计算对于较大的 n 值变得相当麻烦或几乎不可能。所以，本文的递归过程可以看作是残差计算方法的一种更好的替代方法，用于推导 $$\lambda$$ λ 和非负整数 n 的任意值的一般解决方案。此外，具有齐次 Drichlet 和 Neumann 边界条件的特征对也由通解导出。