Solving the heat equation for a perturbed magnetic Laplacian on the complex plane

A Bargmann-like transform associated with the polyanalytic Intissar–Hermite polynomials is efficiently employed to solve the heat-like equations associated with Dirac type operators by giving their explicit solutions. We also provide the integral representation of the heat equation associated with a magnetic-like Laplacian. The constructed Bargmann transform is shown to define a unitary transform from the configuration space on the real line onto a Segal–Bargmann type space. The latter is shown to be realizable as the null space of a special first order partial differential operator involving both the holomorphic and anti-holomorphic derivatives for which the polyanalytic Intissar–Hermite polynomials constitute an orthogonal basis. A closed expression of its reproducing kernel function is also given.