It is the purpose of this paper to make a study of the solutions of the following k-formally self-adjoint differential equations
where ar, br), r = 1, 2, …, k, denote k semi-open intervals in which ar is finite and br is arbitrary and the λs, s = 1, 2, …, k, are spectral parameters.
The main theme of the paper is that of extending the Hermann Weyl limit-point, limit-circle theory to the multi-parameter case. That is we consider under which circumstances there exist, for each r, one or two solutions yr(xr) of (*) which are square integrable in a suitably defined Hilbert space Hr. This is then generalised to consider the problem of investigating the possibility of the product
of solutions of (*) being square integrable in H, the tensor product of the separate spaces Hr. The analyticky of the corresponding generalised Hermann Weyl coefficients mr(λ1, λ2,…, λk), r = 1,…, k, is also investigated. Some examples illustrating the theory are given and an alternative formulation of the problem is suggested.