Hankel-type Segal-Bargmann transform and its connection with certain special operators

In this paper, we discuss the Hankel-type Segal-Bargmann space \(\mathscr {F}_{\alpha ,*}(\mathbb {C}^d)\) $\mathcal{F}_{\alpha ,\ast }^{}({\mathbb{C}}^d)$ and the Hankel-type Segal-Bargmann transform \(\mathscr {B}_\alpha\) ${\mathcal{B}}_{\alpha }$ , and we give some intertwining relations on the space \(\mathscr {F}_{\alpha ,*}(\mathbb {C}^d)\) $\mathcal{F}_{\alpha ,\ast }^{}({\mathbb{C}}^d)$ . By means of these intertwining relations, we define and give the properties of the Hankel harmonic oscillator \(\mathscr {L}\) $\mathcal{L}$ . As an application, we prove one result involving complex Euler operator and also state alternate forms of complex operators. One result based on uncertainty principle is given. Finally, we state and solve the time-dependent Hankel-Schrödinger equation. The results of this paper are illustrated by some examples, numerical tables, and figures.