D. Jakobson, Quantum unique ergodicity for Eisenstein series on PSL2 (Z)\PSL2(R), Ann. Inst. Fourier (Grenoble), 44 (1994), 1477--1504.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....explicit description in terms of either a Maass waveform of weight zero or a holomorphic cusp form, and the coe#cients in the Fourier expansions of the two forms are proportional. We will state this fact in a precise form in the next two lemmas, thus slightly generalizing formulas given earlier in [J1, J2]. For the proofs of the main results in the present paper it will not be essential to know the exact formulas for the proportionality factors involved. However, these formulas, as well as the uniformity in v obtained in Lemma 2.1, might be of importance in future applications, e.g. to obtain ....

....that all square roots appearing in the lemma are well defined, since clearly (s) and for all s , 1) and (1 , 1) We also remark that the above formulas are consistent with [J1, 1. 8) J2, 6) 7) in the case Re s = # = PSL(2, Z) since the # j,k expansions in [J1, J2] agree with our formulas above after multiplication by a certain constant c = c(s, v) C with = 1) Proof. The first assertion follows from [H2, p. 382(d) e) f) h) used repeatedly. It remains to prove the formulas for the c n s. The case v = 0 is trivial since K s (y) # 2y) W ....

D. Jakobson, Quantum unique ergodicity for Eisenstein series on PSL2 (Z)\PSL2(R), Ann. Inst. Fourier (Grenoble), 44 (1994), 1477--1504.

....that the degeneracies are coupled to the existence of quantum symmetries. There is a commutative group of unitary operators on HN which commute with UN (A) and therefore act on each eigenspace of UN (A) We will call these Hecke operators , in analogy with the setting of the modular surface 4 [20, 15, 10]. We may thus consider eigenfunctions of the desymmetrized quantum map, that is eigenstates of both UN (A) and of all the Hecke operators. We call these Hecke eigenfunctions. Our second goal is to show that these become equidistributed with respect to Liouville measure, that is the expectation ....

D. Jakobson, Quantum unique ergodicity for Eisenstein series on PSL 2 (Z)nPSL 2 (R), Ann. Inst. Fourier (Grenoble) 44 (1994), 1477-1504.

....for all but possibly a zero density subsequence of eigenfunctions. This phenomenon is commonly referred to as quantum ergodicity 2 . There are no examples where it is known if there are any exceptional subsequences. The case where there are none is referred to as quantum unique ergodicity (QUE) [15, 13, 10]. In this paper, we consider a compact model of the above situation, where the dynamics, instead of taking place in the co tangent bundle, occurs in a compact symplectic manifold, namely the 2 torus T 2 . The (classical) evolution is then given by iterating a symplectic map of the torus. In ....

D. Jakobson, Quantum unique ergodicity for Eisenstein series on PSL 2 (Z)n PSL 2 (R), Ann. Inst. Fourier (Grenoble) 44 (1994) 1477-1504.

.... For more comprehensive surveys of the results on high energy spectra and eigenfunctions of Laplacians for arithmetic hyperbolic manifolds and related results in Number Theory, we refer the reader to [Sar2, Sar3, I S2] The Quantum Unique Ergodicity conjecture has been established in [L S2] and [Jak2] for (generalized) eigenfunctions corresponding to the continuous spectrum of on PSL 2 (Z)nH. Analogous results for arithmetic hyperbolic 3 manifolds were reduced in [Ko2] to certain L function estimates; those estimates were established in [P S] Rudnick and Sarnak showed in [R S] that on ....

D. Jakobson. Quantum Unique Ergodicity for Eisenstein Series on PSL 2 (Z)nPSL2 (R). Annales de l'Institut Fourier 44 (1994), 1477-1504.

....measure d on GammanG is given by (dxdy d ) 2 y 2 ) where j;k (z) are shifted Maass cusp forms of weight 2k. j;k e Gamma2ik is an eigenfunctions of Casimir operator with the same eigenvalue j = 1=4 r 2 j for every k. The Fourier expansion of j and j;k s was computed in [Ja94]. The Fourier expansion of j in (x; y; coordinates is given by j (z) X n6=0 c j (jnj) q jnj W 0;ir j (4 jnjy)e(nx) 5) where j = 1=4 r 2 j , W 0;ir j is a Whittaker function, e(nx) denotes e 2 inx and c j (n) c j (1) j (n) where j (n) s are Hecke eigenvalues of j . ....

....1 2 k ir j ) X n 0 c j (jnj) q jnj W k;ir j (4 jnjy)e(nx) 6) Gamma1) k Gamma(1=2 ir j ) Gamma( 1 2 Gamma k ir j ) X n 0 c j (jnj) q jnj W Gammak;ir j (4 jnjy)e(nx) Here W k;ir j is a Whittaker function and c j (n) s are as before. 1 We use the same notation as in [Ja94], so that x iy = z 2 H, the hyperbolic metric is j dz j=y, Delta = y 2 ( 2 = x 2 2 = y 2 ) and dv = dxdy =y 2 . EQUIDISTRIBUTION OF CUSP FORMS 3 The Fourier expansion of j; Gammak ; k 0 (weight Gamma2k) is given by j; Gammak (z) Gamma1) k Gamma(1=2 ir j ) ....

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D. Jakobson. Quantum Unique Ergodicity for Eisenstein Series on PSL 2 (Z)n PSL 2 (R). Annales de l'Institut Fourier, 44(5):1477--1504, 1994.

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