Spectral analysis of the Regge problem with parameters

Abstract

done if we show that 02N/cOT 2 # 0 for T = T~k and E = vk(T~). If this were not the case, then for the same T and E we would obtain the equation V'(T)(Ul A u~ --u~ A u2) = 0 by a straightforward computation, which contradicts (I).Now it is easy to complete the proof of the theorem. First, let 0 < T < T~k+l. Then, by virtue of (III), V(T) < V(T~+I ) = vk+l(T~k+l ) < vk+l(T), and hence, the lemma implies #k(T) < Vk+l(T). By passing to the limit, we obtain #k(T~k+l ) < I/k+l(T~k+l ) . Now let T > T~+I. Then, by virtue of (II) and (III), Vk+l(T) > vk+,(T~k+l ) > pk(T~+l) > p~(T). It remains to check that pk(T~k+l ) ~ Vk+l(T~+l ) . If this is not the case, then it readily follows from the equations pk(T~k+l ) ----vk+l(T~k+l ) and v[+l(T~+l ) = 0 and from the inequalities proved above that p~(T~k+l ) ----0, but this contradicts (II). The theorem is thereby proved. O Remark. Since the potential U is even, we actually have pk -)~k+l and vk+l = ~k+l. It was proved in [4] that in the subspace E0 C L2(-T, T) of 2T-periodic even potentials U with zero mean on the interval [-T, T], the correspondence U ~-+ (vk+l -Pk)k>l is a real-analytic isomorphism of E0 onto l 2. Our results provide examples of potentials whose image under this isomorphism is contained in the positive cone l~.. We must also point out that, unlike [3, 4], our results do not allow us to obtain a potential from a prescribed set of spectral data.Let us finally remark that the rigorous justification of results concerning the absence of the edge state backscattering in the quantum Hall effect [6] is reduced to a proof of the inequality pj,(T) < vj,+~(T).These inequalities were obtained by numerical methods in a series of papers.In conclusion, the authors are grateful to the referee for his suggestions, which made for an improvement of our manuscript.South-Ukrainian Pedagogical University; Odessa State Academy of Civil Engineering and Architecture.

Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications

“…Since then many results concerning eigenvalue asymptotics for concrete boundary value problems generated by the Sturm-Liouville equation and similar equations have been published, see, e. g., [17], [84] and the references therein. Asymptotics for the eigenvalues of the generalized Regge problem (7.2.5)-(7.2.7) with α = 1 were obtained in [101]. However, asymptotics of the classical Regge problem, that is, the case when α = 1 and β = 0, being investigated for a long time are not completely known.…”

Spectral asymptotics for Sturm-Liouville equations with indefinite weight

“…We note that applications of s-dependent boundary conditions can be found in [6] and the references therein.…”

On small vibrations of a damped Stieltjes string

“…The corresponding boundary value problem with the spectral parameter in the boundary conditions was considered in many publications (see [7,21,22,27] and 144 Olga Boyko and Vyacheslav Pivovarchik references therein). In [10] and [23] conditions on a sequence of complex numbers were given necessary and sufficient to be the spectrum of a smooth inhomogeneous string damped at one end for any value of the damping parameter except for one crucial value.…”