Integrals of the motion and exact solutions of the problem of two dispersing delta-wells

Bound States of a Particle in the Field of Receding Point Potentials

“…The validity of expression (11) can be verified by direct substitution in the equation. This solution was also found in [6].…”

Exact Propagators for Soliton Potentials

“…In contrast, our formulas ( 12) and ( 14) give an explicit expression for the propagator. To use formula (14), we first calculate necessary Wronskians with the transformation functions (5), where for simplicity we choose b 1 = b 2 = 0, i.e. u 1 (x) = cosh(a 1 x), u 2 (x) = sinh(a 2 x) and then after some algebra we get the propagator for the two-soliton potential…”

“…One can find the value of the integral at the right hand side of this equation in [14] ∞ −∞ dp e −2p 2 +4px p + iα = −iπe 2α(α−2ix) erfc √ 2(α − ix)…”

“…Let us consider how expressions (12) and (14) lead to the propagators for the one-and two-soliton potentials.…”

Excitation of a moving oscillator

“…The moving delta-potential in one dimension was discussed in [2,[10][11][12][13]. Two moving delta-potentials in one dimension were investigated in [14][15][16][17][18][19][20][21][22][23][24]. Generalizations to three dimensions were made in [18,23,24], while a constant harmonic oscillator potential was added in [24].…”