Quantum mechanics of singular inverse square potentials under usual boundary conditions Kolawol´ e Kˆ egnid´ e Damien Adja¨ ı a , Jean Akande a , Lucas Herv´ e Koudahoun a , Biswanath Rath b* , Pravanjan Mallick b , Rati Ranjan Sahoo c , Fernando Y´ elom` e Judica¨ el Kpomahou d and Marc Delphin Monsia a+1 a. Department of Physics, University of Abomey- Calavi, Abomey-Calavi, 01.B.P.526, Cotonou, BENIN (Corresponding author † E.mail: monsiadelphin@yahoo.fr). b. Department of Physics, North Orissa University, Takatpur, Baripada -757003, Odisha, INDIA(* E.mail: biswanathrath10@gmail.com). c. Department of Computer Science and Engineering, High-Tech Institute of Technology, Bhubaneswar-752057, Odisha, INDIA. d. Department of Industrial and Technical Sciences, ENSET-Lokossa, University of Abomey, Abomey, BENIN. The quantum mechanics of inverse square potentials in one dimension is usually studied through renormalization, self-adjoint extension and WKB approximation. This paper shows that such potentials may be investigated within the framework of the position-dependent mass quantum mechanics formalism under the usual boundary conditions. As a result, exact discrete bound state solutions are expressed in terms of associated Laguerre polynomials with negative energy spectrum using the Nikiforov-Uvarov method for the repulsive inverse square potential. Keywords: Singular repulsive potentials, Schr¨ odinger equation, position-dependent mass, bound state solutions, quantum mechanics, Nikiforov-Uvarov method. 1 Introduction The problem of exact discrete bound states for strongly singular potentials has been a subject of in- tensive analytic studies in the research field of mathematical physics. In particular, the one-dimensional Schr¨ odinger equation for the singular inverse square potential is well known to require a special mathemati- cal treatment. In [1] for example, the renormalization technique has been used to study the one-dimensional attractive inverse square potential. Recently, this potential has been analyzed in [2] in the context of renor- malization and self-adjoint extension of the Hamiltonian operator. In [3] WKB approximation with special mathematical treatments are used to solve the Schr¨ odinger wave problem for the strongly repulsive poten- tials. Such inverse square potentials are known to be used for modeling many practical problems in modern engineering studies. The question of discrete bound state solutions for repulsive potentials is not yet com- pletely resolved [2,4,5]. In such a situation, there appears logic to investigate the quantum mechanics of strongly repulsive singular inverse square potentials under usual boundary conditions. In this work, the position-dependent mass formalism is shown to have the ability to remedy some inherent difficultes related to the naturel domain of this potential. Due to its applications for the quantum control [6] the Schr¨ odinger equation with position-dependent mass has fast become an important research field from mathematical as well as physical point of view. Many problems have been solved in various areas of science on the basis of 1 corresponding author. E.mail: monsiadelphin@yahoo.fr 1