CORRECTIONS TO “ON QUANTUM UNIQUE ERGODICITY FOR Γ\H × H”

ELON LINDENSTRAUSS

We note that (∗∗) on page 920 of [1], as well as its derivation on page 921, has a sign error. In fact, and this will actually be useful for us later, one can in addition improve the error estimate: as we shall show for −N ≤ l, m, k, t ≤ N with N < r_{1}/2

⟨fϕ_{2l,2m}, ϕ_{2k,2t}⟩ = ⟨fϕ_{2l+2,2m}, ϕ_{2k+2,2t}⟩ + O_{f}(r_{1}^{−1}).  (* * ’)

The original estimate O_{f}(Nr_{1}^{−1}) for the error term can be obtained by simply correcting signs in the calculation on the top of page 921 of [1]; the same idea, but with slightly more careful analysis gives (∗∗′). Without loss of generality |l − k| ≤ N_{0} (otherwise both sides of (∗∗′) are 0), and

⟨fϕ_{2l,2m}, ϕ_{2k,2t}⟩ = \frac{1}{(ir_{1} − 2l − 1)(−ir_{1} − 2k − 1)}⟨fE_{1}^{−}ϕ_{2l+2,2m}, E_{1}^{−}ϕ_{2k+2,2t}⟩

= \frac{1}{(ir_{1} − 2l − 1)(−ir_{1} − 2k − 1)}(⟨E_{1}^{−}(fϕ_{2l+2,2m}), E_{1}^{−}ϕ_{2k+2,2t}⟩

− ⟨E_{1}^{−}(f)ϕ_{2l+2,2m}, E_{1}^{−}ϕ_{2k+2,2t}⟩)

= −\frac{1}{(ir_{1} − 2l − 1)(−ir_{1} − 2k − 1)}(⟨fϕ_{2l+2,2m}, E_{1}^{+}E_{1}^{−}ϕ_{2k+2,2t}⟩ − O_{f}(r_{1}))

= −\frac{(−ir_{1} + 2k + 1)(−ir_{1} − 2k − 1)}{(ir_{1} − 2l − 1)(−ir_{1} − 2k − 1)}⟨fϕ_{2l+2,2m}, ϕ_{2k+2,2t}⟩ + O_{f}(r_{1}^{−1})

= ⟨fϕ_{2l+2,2m}, ϕ_{2k+2,2t}⟩ + O_{f}(|k − l| r_{1}^{−1}) + O_{f}(r_{1}^{−1})

= ⟨fϕ_{2l+2,2m}, ϕ_{2k+2,2t}⟩ + O_{f}(r_{1}^{−1}).

This calculation needs to be iterated N times, and not N_{0} times as claimed on page 921, which does indeed give

⟨fϕ_{2l,2m}, ϕ_{2k,2t}⟩ = ⟨fϕ_{2l−2k,2m−2t}, ϕ_{0,0}⟩ + O_{f}(N max(r_{1}^{−1}, r_{2}^{−1})),

as stated in [1], page 921.

The proof now proceeds as written, with the obvious typo corrected in the formula for Fejer summation: the coefficients of the sum should be \frac{(2N−2|l|+1)(2N−2|m|+1)}{(2N+1)^{2}}. Notice that in the same formula the summation is effectively over |l| , |m| ≤ N_{0}, and so since the sum is of fixed length the error term remains unchanged.

Date: August 7, 2002.

References

[1] E. Lindenstrauss. On quantum unique ergodicity for Γ\H × H. IMRN, 17:913– 933, 2001.

Department of Mathematics, Stanford University, Stanford, CA 94305

  E-mail address: elon@math.stanford.edu