Self-adjoint eigenvalue problems and non-Hermitian quantum mechanics

Based on the location of quotients to estimate eigenvalues, self-adjointness for (generalized) eigenvalue problems in finite dimensions is investigated. It is shown that eigenvalue problems with real spectrum are, under very mild assumptions, self-adjoint in an adjustment of the inner-product. In this connection, the geometrical treatment of Cn involves two inner-products. In non-Hermitian quantum mechanics and in using the finite element method, there is a strong incentive to recover these inner-products. The task consists of finding a certain Gram matrix of eigenvectors. This, however, can be converted into a computational geometric problem of positive definiteness on the kernel of an elementary operator, requiring solving Hermitian extreme eigenvalue problems only. Well compressible inner-products are sought in terms of a preconditioner up-dated by small rank corrections. Classical properties of self-adjoitness are retained, such as orthogonality of eigenvectors, variational principles for eigenvalues and iterative methods attaining cubic convergence.