from the exact result ($1/4(\sigma+1)$ [@Sneddon1960]) may open a door to the exact evaluation of $\sum_{\sigma,n}\alpha^{2}_{\sigma,n}$ and to reproduce the Dirichlet b.c. part. This method is to be generalized for the Neumann b.c. part to predict the actual scenario from the roots of the first derivative of the Bessel functions of the first kind.

Casimir effect could have been further studied by considering a small gap between the condensate and the cylindrical surface generalizing the effect for two concentric cylinders [@Teo]. Instead of keeping the BEC inside the cylinder, we could keep a Fermi liquid (say $^3$He). How to calculate the Casimir force, for a Fermi liquid in the confined geometry, is an open question.

S. Biswas and A.A. gratefully acknowledge funding from the INSPIRE Faculty Award by the DST (Govt. of India). S. Biswas further acknowledges the hospitality of the Department of Physics, IIT-Kanpur during the initial phase of this work. A.A. also acknowledges funding from the Faculty Initiation Grant of IIT-Kanpur, India. Several useful discussions with J.K. Bhattacharjee (HRI, India) and S. Dutta Gupta (University of Hyderabad, India) are gratefully acknowledged. Useful comments of K.A. Milton (University of Oklahoma, USA) prior to the preparation of the present form 