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Name:  
Miloslav Znojil  
Reviewer number:  
9689  
Email:  
znojil@ujf.cas.cz  
Item's zblNumber:  
DE 017 218 37X  
Author(s):  
Kong, Q.; Wu, H.; Zettl, A.:  
Shorttitle:  
Leftdefinite SturmLiouville problems.  
Source:  
J. Differ. Equations 177, No. 1, 1  26, doi:10.1006/jdeq.2001.3997 (2001).  
Classification:  
 
Primary Classification:  
 
Secondary Classification:  
 
Keywords:  
SturmLiouville problems; indefinite weight function; leftdefiniteness; existence of eigenvalues; inequalities for eigenvalues; dependence on parameters;  
Review:  
For a linear and regular differential equation of the second order the selfadjoint SturmLiouville problem is usually studied in the so called rightdefinite case (meaning that the righthand 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 ``rightindefinite" problems have to be studied within a Krein space (with indefinite metric). Still, there exists a subset of the latter cases (called leftdefinite) 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 fermionboson mappings, or A. Mostafazadeh in J. Math. Phys. 43, 2814  2816 (2002) in the pseudoHermitian, so called PT symmetric quantum mechanics. Its strict mathematical study in question paying detailed attention to ``the most elementary" SturmLiouville problem is extremely important, therefore. Its introductory parts clarify in which sense the leftdefiniteness 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:  