Citation Context

...reduced to a simple Wronski determinant of C-valued distributional solutions of (6.5). This fact goes back to Jost and Pais [23] (see also [14], [37], [39], [41, Sect. 12.1.2], [52, Proposition 5.7], =-=[53]-=-, and the extensive literature 16 F. GESZTESY, Y. LATUSHKIN, M. MITREA, AND M. ZINCHENKO cited in these references). The principal aim of this section is to explore possibilities to extend this fact t...

Citation Context

...ported Jacobi parameters is the study of the sets of allowed resonance positions for half-line Schrödinger operators with compactly supported potentials. There is a large literature on this question =-=[6, 7, 17, 18, 21, 29, 30]-=-. In particular in [17, 18], Korotyaev makes some progress in classifying all Jost functions in this case. We announced the results in [4] and some of them have been presented in [23], but we note an ...

Citation Context

...ter this book was translated into English, see [M]. In fact he really reproved correctly the well known Faddeev's result [F], where there were some mistakes (see [DT]). Remark that in the papers [H], =-=[-=-S], [Z], [K] there are results about the resonances for the Schrodinger operator in the 1D case. In the paper [K] we solved the characterization problem for the Schrodinger operator on the half line w...

Citation Context

... (cf. (3.100)). We are not aware of any earlier references mentioning these solutions ψ±. In addition, we obtain in Section 3 in passing an elementary proof of an interesting formula derived by Simon =-=[37]-=- for the Jost solutions Ψ±(z, · ) in terms of Fredholm determinants. Of course, the approach of Section 3 applies equally well to the general onedimensional case, yielding in principle a similarly com...

Citation Context

... −1 has no poles; for d = 2 we give examples with no poles except, perhaps, at the origin. These results are surprising, as there are no such nontrivial potentials with this property in one dimension =-=[4, 10, 14, 20]-=-. Moreover, it is known that for nontrivial real-valued, smooth, compactly supported potentials in all dimensions greater than two the resolvent must have infinitely many poles [9, 12, 13]. We also sh...

Citation Context

... tjc@math.missouri.edu. 12 T. CHRISTIANSEN Let RV be the set of poles of RV (λ), repeated with multiplicity. Let Then, if d = 1, NV (r) = #{zj ∈ RV : |zj| < r}. NV (r) lim = r→∞ r 2 diam(supp(V )) π =-=[2, 19, 15]-=-. This is true for complex-valued potentials as well as for real-valued ones. Much less is known about the higher-dimensional case, and there is evidence that the question of distribution of resonance...

Citation Context

...V : |λj| < r}. The large r properties of NV (r) have been extensively studied, and we refer the reader to the review article of Zworski [17]. The leading asymptotic behavior is known in one dimension =-=[3, 12, 20]-=-, and for certain spherically symmetric potentials for odd d ≥ 3 [18]. Moreover, the following upper bound on NV (r) for compactly supported potentials is well-known (3) NV (r) ≤ CV,d(1 + r d ), TC pa...

Citation Context

...h the case when H1 is a scalar Schrödinger operator. It must be noted that none of the results given in Section 2 is new. Instead, they can, in various and, in fact, much stronger forms, be found in =-=[3, 5, 6, 8, 9, 18, 19, 20, 21, 23, 24, 25, 26]-=- to name a few examples. In these references the eigenvalue problem is studied either via classical scattering theory or via the Fredholm determinant. As it will be seen, the Evans function approach i...

Citation Context

... a quasi-periodic, finite-gap background. In the case of zero background it is well-known that the transmission coefficient is the perturbation determinant in the sense of Krein [22] (see e.g., [19], =-=[32]-=-, [38] see also [15], [16] and the references therein for generalizations to non trace class situations) and our first aim is to establish this result for the case considered here; thereby establishin...