content_image/355920.jpg content_image/355921.jpg content_image/355929.jpg content_image/355928.jpg content_image/355927.jpg
eigenfunctions H δ $_{n}$( x ) new generalized Hermite functions . 

A plot of the first generalized Hermite functions is shown in Fig.(1). Notice the change in the functions behaviour as δ goes from zero to infinity, showing the continuos passage from the quantum mechanics eigenfunctions to the Hermite polynomials. 

## IV. RAISING AND LOWERING OPERATORS 

The corresponding raising and lowering operators, c ∗ , c , are easily deduced from the action of a ∗ and a over ψ$_{n}$ ( x ). The non-mutually adjoint operators 

become the creation/anihilation operators of the quantum mechanics problem as δ → 0, and the raising and lowering operators for the Hermite polynomials as δ → ∞ . They satisfy the relations 

<img src='content_image/355920.jpg'>

<img src='content_image/355921.jpg'>

<img src='content_image/355929.jpg'>

which are the quantum eigenfunctions creation/anihilation operations when δ = 1, and become the raising/lowering operations for the Hermite polynomials in the limit δ → ∞ . 

They also satisfy that 

and, therefore, 

<img src='content_image/355928.jpg'>

<img src='content_image/355927.jpg'>

Finally, due to the relation a ∗ = (1 /α ) B ∗ and eq.(19), c ∗ and c correspond to the cre- ation/anihilation operators introduced by Mielnik[2] as triple operator products. 

