Spectral properties of Schrödinger operators and time-decay of the wave functions

Infinite non homogeneous chain of harmonic oscillators: Large-time behavior of solutions

“…This theorem is proved in Section 4 using the technique of Jensen and Kato [7] which was developed in the works of Komech at al. [8].…”

The flux-across-surfaces theorem for short range potentials and wave functions without energy cutoffs

“…Therefore to prove (3) we need only show that the last three terms in (9) For some xed T > 0 this will follow from the estimates (which w e shall prove b e l o w) sup x2SR j (x t)j t ; 3 2 f 1 (R t) 8t T (11) sup x2SR jr (x t)j t ; 3 2 f 2 (R t) 8t T (12) where there exists a c < 1 such that f i (R t) satisfy lim R!1 f i (R t) = 08 t T (13) and sup R2 0 1) t T f i (R t) < c (14) for i = 1 2, and there is R 0 0 s u c h that…”

“…We turn now to the proof of the estimates (11)(12)(13)(14)(15)(16). Recalling that (x t) = ( e ;iH0t out )(x) and, since r commutes with the free time evolution, r (x t) = ( e ;iH0t r out )(x), we can write …”

“…1 . If H has a zero-energy resonance or eigenvalue, according to Jensen and Kato, 14 the spectral density is singular at E = 0. Since the spectral density and the generalized eigenfunctions are closely related, 20 we expect that in this case also the generalized eigenfunctions become singular at k = 0 .…”

One-dimensional Schrödinger operators with singular potentials: A Schwartz distributional formulation

“…These objects yield exactly solvable models [2,4,5,6,8,11,12,26,38,45] and have been widely used in applications in quantum mechanics (e.g. in models of low-energy scattering [3,13,14,35] and quantum systems with boundaries [22,23,24,27,32]), condensed matter physics [10,17,25] and, more recently, on the approximation of thin quantum waveguides by quantum graphs [1,15,16,20].…”