TY - JOUR
T1 - Exact solutions of the Schrödinger equation with a complex periodic potential
AU - Dong, Shi Hai
AU - Sun, Guo Hua
N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2023/9
Y1 - 2023/9
N2 - The exact solutions of 1D Schrödinger equation subject to a complex periodic potential V(x)=-[iasin(bx)+c]2 (a, b, c∈ R) are found as a confluent Heun function (CHF) HC(α, β, γ, δ, η; z) . The energy spectra which are solved exactly by calculating the Wronskian determinant are found as real except for complex values. It is found that the eigenvalues obtained by two constraints on the CHF are not reliable or complete any more since they are only one small part of those evaluated by the Wronskian determinant. The wave functions are illustrated when eigenvalues are substituted into the eigenfunctions. We also notice that the energy spectra remain invariant when one substitutes a→ - a or b→ - b or c→ - c due to the PT symmetry with the property V(x) = V(- x) ∗ .
AB - The exact solutions of 1D Schrödinger equation subject to a complex periodic potential V(x)=-[iasin(bx)+c]2 (a, b, c∈ R) are found as a confluent Heun function (CHF) HC(α, β, γ, δ, η; z) . The energy spectra which are solved exactly by calculating the Wronskian determinant are found as real except for complex values. It is found that the eigenvalues obtained by two constraints on the CHF are not reliable or complete any more since they are only one small part of those evaluated by the Wronskian determinant. The wave functions are illustrated when eigenvalues are substituted into the eigenfunctions. We also notice that the energy spectra remain invariant when one substitutes a→ - a or b→ - b or c→ - c due to the PT symmetry with the property V(x) = V(- x) ∗ .
KW - 1D Schrödinger equation
KW - Complex periodic potential
KW - Confluent Heun differential equation (CHDE)
KW - The Wronskian determinant
UR - http://www.scopus.com/inward/record.url?scp=85160305229&partnerID=8YFLogxK
U2 - 10.1007/s10910-023-01483-7
DO - 10.1007/s10910-023-01483-7
M3 - Artículo
AN - SCOPUS:85160305229
SN - 0259-9791
VL - 61
SP - 1684
EP - 1695
JO - Journal of Mathematical Chemistry
JF - Journal of Mathematical Chemistry
IS - 8
ER -