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Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 017 580 944
Exner, P.; Yoshitomi, K.
Asymptotics of eigenvalues of the Schroedinger operator with a strong delta-interaction on a loop.
J. Geom. Phys. 41, No. 4, 344 - 358 (2002). http://dx.doi.org/10.1016/S0393-0440(01)00071-7
35J10Schroedinger operator
Primary Classification:
35P15Estimation of eigenvalues, upper and lower bounds
Secondary Classification:
81Q20Semiclassical techniques including WKB and Maslov methods
Schroedinger operator in two dimensions; delta interaction on a smooth curve; eigenvalues; strong coupling estimates

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
delta-function 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 quasi-one-dimensional, wire-shaped
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
wave-guides) 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 weak-coupling
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
strong-coupling regime. In particular, the authors get (and prove,
rigorously) an asymptotic formula for the n-th 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:
cond-mat/0206397) and on the non-planar surface motion in three
dimensions (cf. the paper on the curved-surface-supported delta
interaction in arXiv: math-ph/0207025 which offers a
generalization of some of the present techniques and results).
Remarks to the editors:

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