Unnormalized tomograms and quasidistributions of quantum states

We consider tomograms and quasidistributions, such as the Wigner functions, the Glauber–Sudarshan $P$-functions, and the Husimi $Q$-functions, that violate the standard normalization condition for probability distribution fucntions. We introduce special conditions for the Wigner function to determine the tomogram with the Radon transform and study three different examples of states like the de Broglie plane wave, the Moschinsky shutter problem, and the stationary state of a charged particle in a uniform constant electric field. We show that their tomograms and quasidistribution functions expressed in terms of the Dirac delta function, the Airy function, and Fresnel integrals violate the standard normalization condition and the density matrix of the state therefore cannot always be reconstructed. We propose a method that allows circumventing this problem using a special tomogram in the limit form.