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Name:  
Miloslav Znojil  
Reviewer number:  
9689  
Email:  
znojil@ujf.cas.cz  
Item's zblNumber:  
DE 017 580 944  
Author(s):  
Exner, P.; Yoshitomi, K.  
Shorttitle:  
Asymptotics of eigenvalues of the Schroedinger operator with a strong deltainteraction on a loop.  
Source:  
J. Geom. Phys. 41, No. 4, 344  358 (2002). http://dx.doi.org/10.1016/S03930440(01)000717  
Classification:  
 
Primary Classification:  
 
Secondary Classification:  
 
Keywords:  
Schroedinger operator in two dimensions; delta interaction on a smooth curve; eigenvalues; strong coupling estimates  
Review:  
Historically, the routine studies of quantum systems in one dimension found an interesting sophistication in the two different directions. In one of them, superpositions of the Dirac deltafunction potentials have been found to provide a sufficiently flexible class of the solvable models (cf., e.g., the book cited in this paper as [1]). In another, mathematically equally challenging context, people imagined that many phenomenologically relevant quasionedimensional, wireshaped systems (i.e., domains on which the quantized particle resides) may be smooth but thick (so that one has to return to the old books on classical electrodynamics and study the so called quantum waveguides) and, in principle, curved (so that the curvature itself starts playing the role of another type of an effective potential  cf. e.g., [6] for more references). In a combination of these two tendencies, the work with an infinitely thin quantum wires may profit from the efficiency of the weakcoupling techniques as reviewed, e.g., in ref. [3]. In such a setting, the paper in question offers another approach (basically, the Dirichlet  Neumann bracketing) which becomes efficient in the strongcoupling regime. In particular, the authors get (and prove, rigorously) an asymptotic formula for the nth eigenvalue (which exists and decreases, quadratically, with the strength of the ``attraction" at any n) and for the number of the (negative) eigenvalues (with a linear growth and with a logarithmic error estimate). It is worth noting that the project itself is far from being in its final stage: Very recently there appeared new results (by the partially overlapping groups of authors) on the wires with points of an infinite curvature (cf. arXiv: condmat/0206397) and on the nonplanar surface motion in three dimensions (cf. the paper on the curvedsurfacesupported delta interaction in arXiv: mathph/0207025 which offers a generalization of some of the present techniques and results).  
Remarks to the editors:  