TY - JOUR
T1 - Exact solutions of the Schrödinger equation with inverse-power potential
AU - Dong, Shi Hai
AU - Ma, Zhong Qi
AU - Esposito, Giampiero
N1 - Funding Information:
Acknowledgments. This work was supported by the National Natural Science Foundation of China and Grant No. LWTZ-1298 from the Chinese Academy of Sciences. The work of GE has been partially supported by PRJN97 "Sintesi."
PY - 1999/10
Y1 - 1999/10
N2 - The Schrödinger equation for stationary states is studied in a central potential V(r) proportional to r-β in an arbitrary number of spatial dimensions. The presence of a single term in the potential makes it impossible to use previous algorithms, which only work for quasi-exactly-solvable problems. Nevertheless, the analysis of the stationary Schrödinger equation in the neighbourhood of the origin and of the point at infinity is found to provide relevant information about the desired solutions for all values of the radial coordinate. The original eigenvalue equation is mapped into a differential equation with milder singularities, and the role played by the particular case β = 4 is elucidated. In general, whenever the parameter β is even and larger than 4, a recursive algorithm for the evaluation of eigenfunctions is obtained. Eventually, in the particular case of two spatial dimensions, the exact form of the ground-state wave function is obtained for a potential containing a finite number of inverse powers of r, with the associated energy eigenvalue.
AB - The Schrödinger equation for stationary states is studied in a central potential V(r) proportional to r-β in an arbitrary number of spatial dimensions. The presence of a single term in the potential makes it impossible to use previous algorithms, which only work for quasi-exactly-solvable problems. Nevertheless, the analysis of the stationary Schrödinger equation in the neighbourhood of the origin and of the point at infinity is found to provide relevant information about the desired solutions for all values of the radial coordinate. The original eigenvalue equation is mapped into a differential equation with milder singularities, and the role played by the particular case β = 4 is elucidated. In general, whenever the parameter β is even and larger than 4, a recursive algorithm for the evaluation of eigenfunctions is obtained. Eventually, in the particular case of two spatial dimensions, the exact form of the ground-state wave function is obtained for a potential containing a finite number of inverse powers of r, with the associated energy eigenvalue.
KW - Bound states
KW - Quantum mechanics
KW - Scattering states
UR - http://www.scopus.com/inward/record.url?scp=0033261979&partnerID=8YFLogxK
U2 - 10.1023/A:1021633411616
DO - 10.1023/A:1021633411616
M3 - Artículo
SN - 0894-9875
VL - 12
SP - 465
EP - 474
JO - Foundations of Physics Letters
JF - Foundations of Physics Letters
IS - 5
ER -