General Toeplitz Matrices Subject to Gaussian Perturbations
- PDF / 771,815 Bytes
- 33 Pages / 439.37 x 666.142 pts Page_size
- 38 Downloads / 206 Views
Annales Henri Poincar´ e
General Toeplitz Matrices Subject to Gaussian Perturbations Johannes Sj¨ostrand and Martin Vogel Abstract. We study the spectra of general N × N Toeplitz matrices given by symbols in the Wiener Algebra perturbed by small complex Gaussian random matrices, in the regime N 1. We prove an asymptotic formula for the number of eigenvalues of the perturbed matrix in smooth domains. We show that these eigenvalues follow a Weyl law with probability subexponentially close to 1, as N 1, in particular that most eigenvalues of the perturbed Toeplitz matrix are close to the curve in the complex plane given by the symbol of the unperturbed Toeplitz matrix.
Contents 1. Introduction and main result 2. The unperturbed operator 3. A Grushin problem for PN − z 4. Second Grushin problem 5. Determinants 6. Lower bounds with probability close to 1 7. Counting eigenvalues in smooth domains 8. Convergence of the empirical measure Acknowledgements References
1. Introduction and main result Let aν ∈ C, for ν ∈ Z and assume that |aν | ≤ O(1)m(ν),
(1.1)
J. Sj¨ ostrand, M. Vogel
Ann. Henri Poincar´e
where m : Z →]0, +∞[ satisfies (1 + |ν|)m(ν) ∈ 1 ,
(1.2)
m(−ν) = m(ν), ∀ν ∈ Z.
(1.3)
and Let p(τ ) =
+∞
aν τ ν ,
(1.4)
−∞
act on complex valued functions on Z. Here τ denotes translation by 1 unit to the right: τ u(j) = u(j − 1), j ∈ Z. By (1.2) we know that p(τ ) = O(1) : 2 (Z) → 2 (Z). Indeed, for the corresponding operator norm, we have (1.5) p(τ ) ≤ |aj |τ j = a1 ≤ O(1)m1 . From the identity, τ (eikξ ) = e−iξ eikξ , we define the symbol of p(τ ) by p(e−iξ ) =
∞
aν e−iνξ .
(1.6)
−∞
It is an element of the Wiener algebra [4] and by (1.2) in C 1 (S 1 ). We are interested in the Toeplitz matrix def
PN = 1[0,N [ p(τ )1[0,N [ ,
(1.7)
acting on C ([0, N [), for 1 N < ∞. Furthermore, we frequently identify 2 ([0, N [) with the space 2[0,N [ (Z) of functions u ∈ 2 (Z) with support in [0, N [. The spectra of such Toeplitz matrices have been studied thoroughly, see [4] for an overview. Let P∞ denote p(τ ) as an operator 2 (Z) → 2 (Z). It is a normal operator and by Fourier series expansions, we see that the spectrum of P∞ is given by σ(P∞ ) = p(S 1 ). (1.8) 2 The restriction PN = P∞ |2 (N) of P∞ to (N) is in general no longer normal, except for specific choices of the coefficients aν . The essential spectrum of the Toeplitz operator PN is given by p(S 1 ) and we have pointspectrum in all loops of p(S 1 ) with nonzero winding number, i.e., N
2
σ(PN ) = p(S 1 ) ∪ {z ∈ C; indp(S 1 ) (z) = 0}.
(1.9)
By a result of Krein [4, Theorem 1.15], the winding number of p(S 1 ) around the point z ∈ p(S 1 ) is related to the Fredholm index of PN − z: Ind(PN − z) = −indp(S 1 ) (z). The spectrum of the Toeplitz matrix PN is contained in a small neighborhood of the spectrum of PN . More precisely, for every > 0, σ(PN ) ⊂ σ(PN ) + D(0, )
(1.10)
for N > 0 sufficiently large, where D(z, r) denotes the open disc of radius r, centered at z. Moreover, the limit of σ(PN ) as N → ∞ is contained
Data Loading...
