Finite spectrum of 2n2nth order boundary value problems

For any even positive integer 2n2n and any positive integer mm we construct a class of regular self-adjoint and non-self-adjoint boundary value problems whose spectrum consists of at most (2n−1)m+1(2n−1)m+1 eigenvalues. Our main result reduces to previously known results for the cases n=1n=1 and n=2n=2. In the self-adjoint case with separated boundary conditions this upper bound can be improved to n(m+1)n(m+1).