Peierls ( 7 ) has gone into details , but his treatment , he admits , is non-rigorous . as Dolph ( 8 ) points out , the promised justification of this has never appeared . Schwartz ( 9 ) , in a very important and powerful paper , treats the Sturm-Liouville case ( and also certain singular cases ) , but only as a special case of a long and complicated function-theoretic argument . Keldysh ( 10 ) has also given a linear-operator approach to the problem . altogether , there does seem a case for a direct justification of Peierls &apos;s work that does not depend on function-theoretic arguments , and this is particularly so when it appears that , without any great complication , it is possible at the same time to make a contribution to the singular case in which the range of x remains finite but q(x) becomes discontinuous at one or other or both of the end-points . this contribution does not seem to be covered by the existing function-theoretic arguments . the problem we shall consider is the following . we take the equation &amp;formula; where q(r) may be complex but is continuous except at r = 0 , and where &amp;formula; exists . we suppose that l is a positive integer or zero . the reader will readily verify that the analysis is not restricted to those values of l , but this is the case of practical importance . ( the equation is the well-known equation that arises when a three-dimensional equation with spherical symmetry is solved by the method of separation of variables . ) the boundary conditions we impose are ( 1.3 ) , for some b &gt; 0 , together with the requirement that y(x) be L2(0,b) . this , as the analysis shows , is sufficient to define an eigenvalue problem , except in the case l = 0 , when we have to impose a further condition of the type ( 1.2 ) at a = 0 . despite this , the case l = 0 is similar enough to the case l &gt; 0 , so that we can safely restrict ourselves to l &gt; 0 . the case l = 0 , with q(r) continuous , is just the Sturm-Liouville case , which therefore comes out as a particular case of the argument . we shall examine the eigenfunctions associated with this eigenvalue problem . as usual , an eigenfunction is a non-trivial solution of the equation ( 1.4 ) which satisfies the boundary conditions . in the self-adjoint case , the set of eigenfunctions would be complete , i.e any reasonable function could be expanded in a series of them . in the non-self-adjoint case , we shall see that in general this no longer holds , but that the set of eigenfunctions can be made complete by adding to it certain other functions which , though not eigenfunctions , are related to them . ( their precise form will be found in &amp;symbol;5 . ) I shall refer to these additional functions as adjoint functions . the problem can be extended to the case in which r = b is also a discontinuity of q(r) , of the same type as at r = 0 . it will not be necessary to discuss in detail this extension , but it will be clear that the same general conclusions hold on the completeness of the set of eigenfunctions and adjoint functions . I have limited myself to proving completeness , but , at least in certain cases , much more can be proved . for example , in the Sturm-Liouville case , a very straightforward adaptation of ( 1 ) ( Ch 1 ) shows that not only is the set of eigenfunctions and adjoint functions complete , but also that , if f(r) is any function of L(0,b) , then the eigenfunction expansion of f(r) ( an expansion which , of course , includes adjoint functions ) converges under Fourier conditions to f(r) . this analysis does not seem to extend to the singular cases considered in this paper . 2 . if &amp;formula; , then ( 1.4 ) has solutions &amp;formula; , of which &amp;formula; is L2(0,b) . if we then write ( 1.4 ) in the form &amp;formula; we see that it is formally equivalent to the integral equation &amp;formula; . our first objective is to prove that , for &amp;formula; , and all &amp;formula; sufficiently large , the solution of ( 1.4 ) that is L2(0,b) is , apart from a multiplicative constant , &amp;formula; , where o(1) denotes a term small where &amp;formula; is large , uniformly for r in [0,b] , and where &amp;formula; . we do this by investigating ( 2.1 ) . let &amp;formula; . then &amp;formula; for all r , l , where A denotes various positive constants . let &amp;formula; . then , if &amp;formula; , ( 2.1 ) gives &amp;formula; since &amp;formula; exists , the o(1) term denoting a quantity which tends to zero as &amp;formula; . also , if &amp;formula; , &amp;formula; where &amp;formula; . but , for &amp;formula; , we have &amp;formula; . for , for all z , &amp;formula; , so that &amp;formula; . the required estimate for G(r,t,l) follows from this by using the asymptotic expressions for &amp;formula; . substituting this estimate in ( 2.2 ) , we obtain &amp;formula; . the first of the two integrals in the last line is &amp;formula; since &amp;formula; in the range of integration and &amp;formula; . the second integral is &amp;formula; , by a similar type of argument . ( the second integral will not , of course , appear if &amp;formula; . ) it thus follows from ( 2.1 b ) and ( 2.4 ) that , for &amp;formula; , if &amp;formula; is large enough , i.e that &amp;formula; . if we substitute this result back in the integral in ( 2.1 ) and re-estimate this integral on the same lines as has just been done , we emerge with ( 2.1 a ) . thus any solution of ( 2.1 ) satisfies ( 2.1 a ) . that there is one ( and just one ) solution of ( 2.1 ) can be proved by the usual iteration process , of which the work above is effectively the first step . then ( 2.1 ) can be differentiated back to show that the solution is a solution of ( 1.4 ) . we have thus found a solution of ( 1.4 ) that is L2(0,b) . if we denote this solution by f(r,l) , then any other solution apart from a constant multiple of f(r,l) is given by a constant multiple of &amp;formula; , and knowing now the behaviour of f(r,l) near r = 0 , we can readily verify that ps(r,l) is not L2(0,b) . the L2(0,b) solution is therefore ( apart from a multiplicative constant ) unique . we remark finally that , since &amp;formula; is an integral function of l , the process of solving ( 2.1 ) by iteration shows that &amp;formula; is also an integral function of l . 3 . we now consider the solution ch(r,l) which satisfies ( 1.4 ) and the boundary conditions &amp;formula; . as in ( 1 ) ( Ch 1 ) ch(r,l) is an integral function of l . the Wronskian of f , ch is independent of r and so may be written as o(l) , and &amp;formula; will be an integral function of l . further , the vanishing of o(l) is a necessary and sufficient condition for f , ch to be multiples the one of the other , i.e for l to be an eigenvalue . for large values of &amp;formula; , &amp;formula; . ( the asymptotic behaviour of &amp;formula; is obtained by differentiating ( 2.1 ) with respect to r and proceeding as before . ) hence , for large values of &amp;formula; , the zeros of o(l) must be near the zeros of &amp;formula; , which are , of course , independent of q(r) . further , for large &amp;formula; , the zeros of o(l) are simple . this is best seen by writing &amp;formula; where C is a circle with centre l , and by using the asymptotic expression ( 3.1 ) for o(l) to give an asymptotic expression for o&amp;prime;(l) . it is then clear that values of l near the zeros of &amp;formula; do not satisfy o&amp;prime;(l) = 0 . we now construct the function F(r,l) , where &amp;formula; , and f(t) is any function which is L2(0,b) . this is a meromorphic function of l , having poles at the zeros of o(l) . it will be our object in the next section to show that , if f(t) is such that all the residues of F(r,l) at its poles vanish , then f(t) = 0 almost everywhere . 4 . if all the residues vanish , F(r,l) becomes an integral function of l . let us suppose that we can prove ( as we shall do ) that we can find a sequence of circles &amp;formula; , with &amp;formula; , such that F(r,l) is bounded on the circles , with the bound possibly dependent on r , but independent of n . then , by Liouville &apos;s theorem , F(r,l) is a constant , independent of l . suppose then that F(r,l)=g(r) . it follows by differentiation that &amp;formula; , with the result holding at least almost everywhere . by varying l , we have g(r) = 0 , and hence f(r) = 0 almost everywhere . it remains to prove the boundedness of F(r,l) , with r fixed , but &amp;formula; , on the circles &amp;formula; . since we are concerned only with results almost everywhere , we may exclude r = 0 . the differential equation is thus non-singular in the interval ( r , b ) , and we can appeal to ( 1 ) ( equation ( 1.7.8 ) ) to get an asymptotic form of ch(r,l) for sufficiently large &amp;formula; . in fact , we have &amp;formula; , where A denotes various positive constants independent of l . from &amp;symbol;2 we have , again for fixed r and sufficiently large &amp;formula; , &amp;formula; . finally , if we choose the sequence &amp;formula; to be such that &amp;formula; , we see that &amp;formula; on each of the circles &amp;formula; , and so , on those circles , for n sufficiently large , we have from ( 3.1 ) that &amp;formula; . if we now substitute ( 4.1 ) , ( 4.2 ) , ( 4.3 ) in the definition of F(r,l) , and use Schwarz &apos;s inequality to estimate the integrals , we see readily that , on the circles &amp;formula; , F(r,l) is bounded with bound independent of n . 5 . from this , we can deduce the completeness of the eigenfunctions and adjoint functions . before we do this , however , we must examine the nature of these eigenfunctions and adjoint functions . in the real self-adjoint case , it is well known that the zeros of o(l) are real and simple , and , if ln is such a zero , ch(r,ln) is a multiple of f(r,ln) , so that we may write &amp;formula; . then , near l = ln , the singular part of F(r,l) is &amp;formula; . hence the residue at l = ln is &amp;formula; , and this vanishes for all r if and only if the Fourier coefficient of f(t) with respect to the eigenfunction f(t,ln) vanishes . the argument remains valid even in the non-self-adjoint case provided that ln is a simple zero of o(l) . however , there is no longer any guarantee that the eigenvalues of o(l) will be simple , and counterexamples are easily provided . suppose now that ln is a zero of order p of o(l) . then , at l = ln , F(r,l) has a residue of the form &amp;formula; where the As(ln) are constants depending on the derivatives of o(l) at l = ln and whose precise value will not concern us . now o(l) can be written in the form &amp;formula; , and we know that o(ln) = o&amp;prime;(ln) = 0 . hence &amp;formula; , and interchange of the order of differentiation gives that &amp;formula; is independent of r . if we repeat this process with higher differentiations with respect to l , we obtain finally that &amp;formula; is independent of r for s = 0 , 1 , &amp;hellip; , p-1 . this implies that , for these values of s , &amp;formula; so that ( 5.1 ) can be expressed as a linear combination of the p functions &amp;formula; , the coefficients being homogeneous linear combinations of the p expressions &amp;formula; , or , what is the same thing , homogeneous linear combinations of the p expressions &amp;formula; . for the residue to vanish it is therefore sufficient that all the Fourier coefficients of f(t) with respect to the p functions &amp;formula; should vanish . hence , if all the Fourier coefficients of f(t) vanish at all zeros of o(l) , then all the residues of F(r,l) vanish , and so , as already proved , f(t) = 0 almost everywhere . this shows , by application of a standard theorem , that the system of eigenfunctions and adjoint functions , where the adjoint functions are &amp;formula; , is complete . the question does arise whether the adjoint functions are indeed necessary for completeness , or whether on the contrary they themselves can be expressed as linear combinations of the eigenfunctions , and so be eliminated from the expansion of an arbitrary function . it is a standard theorem in the theory of orthogonal functions that all the eigenfunctions and adjoint functions are necessary if they form an orthonormal set , and we shall prove that they are substantially orthonormal in &amp;symbol; 6 . what we shall actually prove ( and it is clear that this will be sufficient ) is ( 1 ) that all the eigenfunctions and adjoint functions associated with an eigenvalue ln are orthogonal to all the eigenfunctions and adjoint functions associated with an eigenvalue lm , where &amp;formula; ; ( 2 ) that the eigenfunctions and adjoint functions associated with an eigenvalue ln of multiplicity p can be expressed by a non-singular transformation as linear combinations of p orthonormal functions . it should be remarked that the number of multiple eigenvalues is at most finite , and so the number of adjoint functions is at most finite . 