Upper and lower bounds for an eigenvalue associated with a positive eigenvector. (English) Zbl 1111.47065
Summary: When an eigenvector of a semibounded operator is positive, we show that a remarkably simple argument allows to obtain upper and lower bounds for its associated eigenvalue. This theorem is a substantial generalization of Barta-type inequalities and can be applied to non-necessarily purely quadratic Hamiltonians. An application for a magnetic Hamiltonian is given and the case of a discrete Schrödinger operator is also discussed. It is shown how this approach leads to some explicit bounds on the ground-state energy of a system made of an arbitrary number of attractive Coulombian particles.
MSC:
47N50 | Applications of operator theory in the physical sciences |
35P15 | Estimates of eigenvalues in context of PDEs |
47B25 | Linear symmetric and selfadjoint operators (unbounded) |
47B80 | Random linear operators |
81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
Keywords:
semiboundedness; positivity; upper and lower bounds for eigenvalues; Barta-type inequality; magnetic Hamiltonian; discrete Schrödinger operator; ground-state energyReferences:
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