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Name:
Miloslav Znojil
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
DE 017 218 37X
Author(s):
Kong, Q.; Wu, H.; Zettl, A.:
Shorttitle:
Left-definite Sturm-Liouville problems.
Source:
J. Differ. Equations 177, No. 1, 1 - 26, doi:10.1006/jdeq.2001.3997 (2001).
Classification:
34L15Estimation of eigenvalues, upper and lower bounds
Primary Classification:
34B24Sturm-Liouville theory
Secondary Classification:
46C20Spaces with indefinite inner product Krein spaces, Pontryagin spaces,...
65L15Eigenvalue problems
Keywords:
Sturm-Liouville problems; indefinite weight function; left-definiteness; existence of eigenvalues; inequalities for eigenvalues; dependence on parameters;
Review:

For a linear and regular differential equation of the second order
the self-adjoint Sturm-Liouville problem is usually studied in the
so called right-definite case (meaning that the right-hand side
``weight" w of the eigenvalues does not change sign). This means
that we may work in a Hilbert space with the standard inner
product. By analogy, whenever the ``weight" w is permitted to
change sign, the resulting ``right-indefinite" problems have to be
studied within a Krein space (with indefinite metric). Still,
there exists a subset of the latter cases (called left-definite)
characterized by the possibility of a return to the Hilbert space
theory after a suitable change of the inner product.

Such a trick proves extremely productive in the applied functional
analysis - cf., e.g., F. G. Scholz et al in Annals of Physics 213,
74 - 101 (1992) in the context of the fermion-boson mappings, or
A. Mostafazadeh in J. Math. Phys. 43, 2814 - 2816 (2002) in the
pseudo-Hermitian, so called PT symmetric quantum mechanics. Its
strict mathematical study in question paying detailed attention to
``the most elementary" Sturm-Liouville problem is extremely
important, therefore.

Its introductory parts clarify in which sense the
left-definiteness means just a guarantee that all the eigenvalues
remain real. The subsequent detailed study of eigenvalues reveals,
when and how do they depend smoothly on the coefficient functions
and on the boundary condition parameters, and when this dependence
is monotonic and/or leads to some useful inequalities. The study,
definitely, fills several unpleasant gaps in the available
literature.
Remarks to the editors:

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