Exact analytical solutions of a two-dimensional hydrogen atom in a constant magnetic field

Exact analytical solutions of the Schrödinger equation for a two dimensional purely sextic double-well potential

“…19,23 For these cases, the wave function (15) is definitely normalizable because the polynomial of ρ goes to infinity slower than the term exp { ρ 4 /8} that goes to zero as ρ → +∞. In order that the function f (ρ) becomes a polynomial at a certain order N, …”

“…The inspiration to consider the two-dimensional case is the connection between the two-dimensional hydrogen atom in the magnetic field and the two-dimensional purely sextic double-well problem by the Levi-Civita transformation [20][21][22] since the former has exact solutions investigated in Ref. 23. This connection, which is often used to relate exact solutions of a family of Schrödinger equations together (for example, see Ref.…”

A family of analytically solvable Schrödinger equations related by Levi-Civita transformation

“…Particularly, Levi-Civita transformation (Levi-Civita, 1906) has been widely used as an efficient tool to connect the Schrödinger equations which describe two-dimensional motions under the influence of some central potentials, e.g Coulomb and harmonic oscillator potentials (Le & Nguyen, 1993) or Coulomb potential plus harmonic oscillator term (C+HO) and sextic double-well anharmonic oscillator (sextic DWAO) (Hoang-Do, Pham & Le, 2013). The former problem, which arises from two-dimensional hydrogen atom model under presence of a homogeneous magnetic field, has been analytically solved both in recurrence form (Taut, 1995) and in compact form (Le, Hoang & Le, 2017) while the exact analytical solution for later one is also obtained in compact form via the inspiration of one-dimension case (Le, Hoang & Le, 2018). Hence, repeatedly applying Levi-Civita transformation into C+HO and sextic DWAO problems can leads us to other problems whose analytical solution can be exactly obtained in compact form as the same as known ones shown by Le, Hoang & Le (2017, 2018.…”

“…The former problem, which arises from two-dimensional hydrogen atom model under presence of a homogeneous magnetic field, has been analytically solved both in recurrence form (Taut, 1995) and in compact form (Le, Hoang & Le, 2017) while the exact analytical solution for later one is also obtained in compact form via the inspiration of one-dimension case (Le, Hoang & Le, 2018). Hence, repeatedly applying Levi-Civita transformation into C+HO and sextic DWAO problems can leads us to other problems whose analytical solution can be exactly obtained in compact form as the same as known ones shown by Le, Hoang & Le (2017, 2018. As an expected results, these problems combining with C+HO and sextic DWAO problems can form one of families whose members are interrelated by Levi-Civita problems; also, this family is exactly analytically solved in compact form.…”

“…On the other hand, Feranchuk-Komarov operator method whose idea is based on harmonic oscillator (Feranchuk et al, 2015) has been applied to achieve exact numerical solution for C+HO problem (Hoang-Do, Pham & Le, 2013) and sextic DWAO problem (Hoang-Do, 2016). After that, exact analytical solutions for these problems have also been used as good references to test how efficient Feranchuk-Komarov operator method is (Le, Hoang & Le, 2017, 2018. Similarly, other problems which belong to the expected family and their corresponding exact analytical solutions can also be used as other significant references for any numerical method including Feranchuk-Komarov operator method.…”