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The Schrödinger equation ψ ′′ (x) + κ 2 ψ(x) = 0 where κ 2 = k 2 − V (x) is rewritten as a more popular form of a second order differential equation through taking a similarity transformation ψ(z) = φ(z)u(z) with z = z(x). The Schrödinger invariant I S (x) can be calculated directly by the Schwarzian derivative {z, x} and the invariant I(z) of the differential equation u zz + f (z)u z + g(z)u = 0. We find an important relation for moving particle as ∇ 2 = −I S (x) and thus explain the reason why the Schrödinger invariant I S (x) keeps constant. As an illustration, we take the typical Heun differential equation as an object to construct a class of soluble potentials and generalize the previous results through choosing different ρ = z ′ (x) as before. We get a more general solution z(x) * E-mail address: dongsh2@yahoo. com through integrating (z ′ ) 2 = α 1 z 2 + β 1 z + γ 1 directly and it includes all possibilities for those parameters. Some particular cases are discussed in detail.

A few important integrals involving the product of two universal associated Legendre polynomials , and x2a(1 − x2)−p−1, xb(1 ± x)−p−1 and xc(1 −x2)−p−1 (1 ± x) are evaluated using the operator form of Taylor’s theorem and an integral over a single universal associated Legendre polynomial. These integrals are more general since the quantum numbers are unequal, i.e. l′ ≠ k′ and m′ ≠ n′. Their selection rules are also given. We also verify the correctness of those integral formulas numerically.

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