14 ε$_{n}$ ( φ ) = ∞ ∑ s =0 a$_{sN}$ cos [ k$_{n}$ ( φ ) sNa ] + M ∑ j =1 ∞ ∑ s =0 { a$_{sN}$$_{+}$$_{j}$ cos [ k$_{n}$ ( φ )( sN + j ) a ] + a$_{(}$$_{sN}$$_{+}$$_{N}$$_{−}$$_{j}$$_{)}$ cos [ k$_{n}$ ( φ ) ( sN + N − j ) a ] } . ε$_{n}$ ( φ ) = Ω$_{0}$+ M ∑ j =1 { Ω R j cos ( k$_{n}$ ( φ ) ja )+Ω I j sin ( k$_{n}$ ( φ ) ja ) } , (101) Ω$_{0}$( φ ) = ∞ ∑ s =0 a$_{sN}$ cos(2 π φ φ$_{0}$ s ) , Ω R j ( φ ) = a$_{j}$ + ∞ ∑ s =1 cos(2 π φ φ$_{0}$ s ) [ a$_{sN}$$_{+}$$_{j}$ + a$_{sN}$$_{−}$$_{j}$ ] , Ω I $_{j}$( φ ) = ∞ ∑ s =1 sin(2 π φ φ$_{0}$ s ) [ a$_{sN}$$_{−}$$_{j}$ − a$_{sN}$$_{+}$$_{j}$ ] . (102) Ω R N/ $_{2}$( φ ) = ∞ ∑ s =0 a$_{sN}$$_{+}$$_{N/}$$_{2}$ cos(2 π φ φ$_{0}$ s ) , Ω I N/ $_{2}$( φ ) = − ∞ ∑ s =1 a$_{sN}$$_{+}$$_{N/}$$_{2}$ sin(2 π φ φ$_{0}$ s ) . (103) ε$_{n}$ ( φ ) = ∞ ∑ j ′ =0 a$_{j}$ ′ cos ( k$_{n}$ ( φ ) j $^{′}$a ) , (98) ε$_{n}$ ( φ ) = N − 1 ∑ j =0 ∞ ∑ s =0 a$_{sN}$$_{+}$$_{j}$ cos [ k$_{n}$ ( φ )( sN + j ) a ] . (99) ε$_{n}$ ( φ ) = T$_{0}$ + M ∑ j =1 e − ik$_{n}$ ( φ ) $^{ja}$T$_{j}$ + N − 1 ∑ j = M +1 e − ik$_{n}$ ( φ ) $^{ja}$T$_{j}$ , (104) ε$_{n}$ ( φ ) = ∞ ∑ s =0 a$_{sN}$ cos [ k$_{n}$ ( φ ) sNa ] + M ∑ j =1 ∞ ∑ s =0 a$_{sN}$$_{+}$$_{j}$ cos [ k$_{n}$ ( φ )( sN + j ) a ] + N − 1 ∑ j = M +1 ∞ ∑ s =0 a$_{sN}$$_{+}$$_{j}$ cos [ k$_{n}$ ( φ )( sN + j ) a ] , (100) ε$_{n}$ ( φ ) = T$_{0}$ + M ∑ j =1 e − ik$_{n}$ ( φ ) $^{ja}$T$_{j}$ + M ∑ j =1 e − ik$_{n}$ ( φ )( N − j ) $^{a}$T$_{N}$$_{−}$$_{j}$ = T$_{0}$ + M ∑ j =1 e − ik$_{n}$ ( φ ) $^{ja}$T$_{j}$ + e ik$_{n}$ ( φ ) $^{ja}$T ∗ j , (105) Utilizing definition (29) we find from Eq. (101) the result would only renormalize the electron transmission through the potential barrier separating two potential wells. That means that the electron-electron interaction would not essentially change our results. where M = Int( N/ 2) . Equation (100) can be rewritten as Finally, we relate our results to the paper by Kohn [17], which defines the insulating state as a property of the many- body ground state (see also the review by Resta [32]). Kohn’s paper contains a discussion implicitly admitting the existence of a negligibly small persistent current in the insulating ring [see the discussion of equations (2.23) - (2.31) in that work]. Kohn points out the exponential localization of the electron states in the ring. However, he clearly states by writing his equation (2.30), that the current is zero only approximately. Just this negligibly small current is calculated in our paper. Surprisingly, it can be of measurable size. where This work was supported by grants APVV-51-003505 and VVCE-0058-07 from the Grant Agency APVV and by grants VEGA-2/0633/09 and VEGA-2/0206/11 from the Grant Agency VEGA. We thank Pavol Quittner for important help with the mathematical derivation in Appendix D. ACKNOWLEDGEMENTS APPENDIX A: DERIVATION OF EQUATION (30) FROM EQUATION (28) APPENDIX B: VARIOUS FORMS OF EQUATION (60) We have sofar assumed odd N . For even N , derivation is similar and we find again the results (101) and (102), except that for j = N/ 2 we have instead of Eqs. (102) the equations We start by writing Eq. (28) as Using substitution j ′ = Ns + j , where s = 0 , 1 , 2 , . . . and j = 0 , 1 , . . . ( N − 1) , we obtain Equation (101) coincides with Eq. (30). We first assume odd N . We can rewrite Eq. (60) as Assuming that N is odd we rewrite Eq. (99) as where M = Int(N / 2) . Further,