TY  - JOUR
A1  - Klein, Markus
A1  - Korotyaev, Evgeni
A1  - Pokrovski, A.
T1  - Spectral asymptotics of the harmonic oscillator perturbed by bounded potentials
N2  - Consider the operator T = -d(2)/dx(2) + x(2) + q(x) in L-2 (R), where q is a real function with q' and integral(0)(x) q(s) ds bounded. The spectrum of T is purely discrete and consists of simple eigenvalues. We determine their asymptotics mu(n) = (2n + 1) + (2 pi)(-1) integral(-pi)(pi) q(root 2n+1 sin theta)d theta + O(n(-1/3)) and we extend these results for complex q.
Y1  - 2005
SN  - 1424-0637
ER  - 
TY  - BOOK
A1  - Klein, Markus
A1  - Badanin, Andrei
A1  - Korotyaev, Evgeni
T1  - The Marchenko-Ostrovski mapping and the trace formula for the Camassa-Holm equation
T3  - Preprint / SFB 288, Differentialgeometrie und Quantenphysik
Y1  - 2001
UR  - 1960 = preprint559.ps.gz
VL  - 559
CY  - Berlin
ER  - 
TY  - JOUR
A1  - Chelkak, D.
A1  - Kargaev, P.
A1  - Korotyaev, Evgeni
T1  - Inverse problem for harmonic oscillator perturbed by potential, characterization
N2  - Consider the perturbed harmonic oscillator Ty=-y''+x(2)y+q(x)y in L-2(R), where the real potential q belongs to the Hilbert space H={q', xq is an element of L-2(R)}. The spectrum of T is an increasing sequence of simple eigenvalues lambda(n)(q)=1+2n+mu(n), ngreater than or equal to0, such that mu(n)-->0 as n-->infinity. Let psi(n)(x,q) be the corresponding eigenfunctions. Define the norming constants nu(n)(q)=lim(xup arrowinfinity)log |psi(n) (x,q)/psi(n) (-x,q)|. We show that {mu(n)}(0)(infinity) is an element of H {nu(n)}(0)(infinity) is an element of H-0 for some real Hilbert space and some subspace H-0 subset of H. Furthermore, the mapping Psi:q-- >Psi(q)=({lambda(n)(q)}(0)(infinity), {nu(n)(q)}(0)(infinity)) is a real analytic isomorphism between H and S x H-0, where S is the set of all strictly increasing sequences s={s(n)}(0)(infinity) such that s(n)=1+2n+h(n), {h(n)}(0)(infinity) is an element of H. The proof is based on nonlinear functional analysis combined with sharp asymptotics of spectral data in the high energy limit for complex potentials. We use ideas from the analysis of the inverse problem for the operator -y"py, p is an element of L-2(0,1), with Dirichlet boundary conditions on the unit interval. There is no literature about the spaces H,H-0. We obtain their basic properties, using their representation as spaces of analytic functions in the disk
Y1  - 2004
SN  - 0010-3616
ER  -