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**Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic Schrödinger operators.**
*(English)*
Zbl 1260.35027

The authors prove the complete aymptotic expansion (1) \(N(\lambda) \sim \lambda^{d/2}(C_d+ \sum_{j=1}^{\infty} e_j \lambda^{-j}) \) of the integrated density of states \(N(\lambda)\) of a self-adjoint operator \(H=-\Delta+h\) acting in \(\mathbb R^d\) with \(h\) a smooth periodic function or belonging to a wide class of almost-periodic functions. The conditions imposed on the almost-periodic functions are satisfied by a generic class of quasi-periodic functions (finite linear combinations of exponential functions). The constant \(C_d\) in formula (1) is equal to \(2\pi^{-d} \omega_d\), where \(\omega_d\) is the volume of the unit ball in \(\mathbb R^d\) and \(e_j\) are real numbers which can be computed using the heat kernel invariants.

Formula (1) was proved for periodic potentials and \(d=1\) in [D. Schenk and M. A. Shubin, Math. USSR, Sb. 56, 473–490 (1987); translation from Mat. Sb., Nov. Ser. 128(170), No. 4(12), 474–491 (1985; Zbl 0604.34015)] and recently for periodic potentials and \(d=2\) in the authors’ paper [Invent. Math. 176, No. 2, 275–323 (2009; Zbl 1171.35092)].

One of the main difficulties that the authors have to overcome is the study of the resonant eigenvalues. They treat unstable eigenvalues by the gauge transform method, which consists in this case in constructing two pseudodifferential operators \(H_1\) and \(H_2\), \(H_1= e^{i\Psi} H e^{-i\Psi}\), \(\Psi\) a bounded periodic self-adjoint operator of order \(0\) and \(H_2\) close to \(H_1\) in norm. Another idea that helped in extending the result to almost periodic potentials is to consider pseudodifferential operators acting in Besicovitch space instead of \(L^2 (\mathbb R^d)\) (as a substitute for the Floquet-Bloch decomposition).

Finally, besides the importance of the result which is proved, we have also to mention the accurate and clear writing.

Formula (1) was proved for periodic potentials and \(d=1\) in [D. Schenk and M. A. Shubin, Math. USSR, Sb. 56, 473–490 (1987); translation from Mat. Sb., Nov. Ser. 128(170), No. 4(12), 474–491 (1985; Zbl 0604.34015)] and recently for periodic potentials and \(d=2\) in the authors’ paper [Invent. Math. 176, No. 2, 275–323 (2009; Zbl 1171.35092)].

One of the main difficulties that the authors have to overcome is the study of the resonant eigenvalues. They treat unstable eigenvalues by the gauge transform method, which consists in this case in constructing two pseudodifferential operators \(H_1\) and \(H_2\), \(H_1= e^{i\Psi} H e^{-i\Psi}\), \(\Psi\) a bounded periodic self-adjoint operator of order \(0\) and \(H_2\) close to \(H_1\) in norm. Another idea that helped in extending the result to almost periodic potentials is to consider pseudodifferential operators acting in Besicovitch space instead of \(L^2 (\mathbb R^d)\) (as a substitute for the Floquet-Bloch decomposition).

Finally, besides the importance of the result which is proved, we have also to mention the accurate and clear writing.

Reviewer: Mihai Pascu (Bucureşti)

### MSC:

35J10 | Schrödinger operator, Schrödinger equation |

47A10 | Spectrum, resolvent |

35P99 | Spectral theory and eigenvalue problems for partial differential equations |

47F05 | General theory of partial differential operators |

81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |

### Keywords:

integrated density of states; Schrödinger operator; periodic functions; almost periodic functions; complete asymptotic expansion; pseudodifferential operator; gauge transform method
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\textit{L. Parnovski} and \textit{R. Shterenberg}, Ann. Math. (2) 176, No. 2, 1039--1096 (2012; Zbl 1260.35027)

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