Scattering for 1D Schrödinger equation with energy-dependent potentials and the recovery of the potential from the reflection coefficient.(English)Zbl 1050.81706

Summary: We consider the 1D Schrödinger equation with a potential proportional to energy. When the spatial part of the potential is twice continuously differentiable, is less than 1 everywhere, and satisfies a certain integrability condition, we compute the scattering matrix. For the same class of potentials, under the further assumption that the potential is non-negative, we obtain the potential from one of the reflection coefficients.

MSC:

 81U05 $$2$$-body potential quantum scattering theory 34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) 35J10 Schrödinger operator, Schrödinger equation 35P25 Scattering theory for PDEs 35R30 Inverse problems for PDEs
Full Text:

References:

 [1] , J. Acoust. Soc. Am. 45 pp 911– (1969) [2] , Am. Math. Soc. Transl. 2 pp 139– (1964) [3] , J. Acoust. Soc. Am. 58 pp 956– (1975) [4] , Phys. Rev. 87 pp 977– (1952) [5] , in: The Inverse Problem of Scattering Theory (1963) [6] , J. Math. Phys. 21 pp 493– (1980) [7] , Am. Math. Soc. Transl. 1 pp 253– (1955) · Zbl 0066.33603 [8] , Trudy Tbilissk. Mat. Inst. 12 pp 1– (1943) [9] , in: Singular Integral Equations (1953) [10] , in: Asymptotic Expansions (1956) [11] , Commun. Pure Appl. Math. 32 pp 121– (1979) [12] , in: Boundary Value Problems (1966)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.