The periodic Schrödinger operators with potentials in the Morrey class. (English) Zbl 1119.35316

Shen, Zhongwei

J. Funct. Anal. 193, No. 2, 314-345 (2002).

Summary: We consider the periodic Schrödinger operator \(H = - \Delta + V(x)\) in \(\mathbb R^d\), \(d\geq 3\) with potential \(V\) in the Morrey class. Let \(\Omega\) be a periodic cell for \(V\). We show that, for \(p\in ((d-1)/2,d/2]\), there exists a positive constant \(\varepsilon\) depending only on the shape of \(\Omega\), \(p\) and \(d\) such that, if \[ \limsup_{r\to 0} \sup_{x \in \Omega} r^2 \left\{{1 \over r^d} \int_{Q(x,r)} | V(y)|^p dy \right\}^{1/p} < \varepsilon_0,\tag{\({*}\)} \] then the spectrum of \(- \Delta + V(x)\) is purely absolutely continuous. We obtain this result as a consequence of certain weighted \(L^2\) Sobolev inequalities on the \(d\)-torus. It improves an early result by the author for potentials in \(L^{d/2}\) or weak-\(L^{d/2}\) space.

MSC:

35J10	Schrödinger operator, Schrödinger equation
47F05	General theory of partial differential operators
35B10	Periodic solutions to PDEs
35P05	General topics in linear spectral theory for PDEs
81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis

References:

[1]	Sh. Birman, M.; Suslina, T. A., Two-dimensional periodic magnetic Hamiltonian is absolutely continuous, Algebra Anal., 9, 32-48 (1997) · Zbl 0890.35096
[2]	Sh. Birman, M.; Suslina, T. A., Absolute continuity of the two-dimensional periodic magnetic Hamiltonian with discontinuous vector-valued potential, Algebra Anal., 10, 1-36 (1998) · Zbl 0922.35101
[3]	Sh. Birman, M.; Suslina, T. A., Periodic magnetic Hamiltonian with a variable metric. The problem of absolute continuity, Algebra. Ana., 11, 1-40 (1999) · Zbl 0941.35015
[4]	Chanillo, S.; Sawyer, E., Unique continuation for \(Δ + v\) and the C. Feffermann-Phong class, Trans. Amer. Math. Soc., 318, 275-300 (1990) · Zbl 0702.35034
[5]	Chanillo, S.; Wheeden, R., \(L^p\) estimates for fractional integrals and Sobolev inequalities with applications to Schrödinger operators, Comm. Partial Differential Equations, 10, 1077-1116 (1985) · Zbl 0578.46024
[6]	Chiarenza, F.; Ruiz, A., Uniform \(L^2\)-weighted Sobolev inequalities, Proc. Amer. Math. Soc., 112, 53-64 (1991) · Zbl 0745.35007
[7]	Danilov, L., On the spectrum of the Dirac operator with periodic potential in \(R^n\), Theor. Math. Phys., 85, 1039-1048 (1990) · Zbl 0723.47043
[8]	Fefferman, C., The uncertainty principle, Bull. Amer. Math. Soc. (N.S), 9, 129-206 (1983) · Zbl 0526.35080
[9]	Gilbarg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0562.35001
[10]	Hempel, R.; Herbst, I., Strong magnetic fields, Dirichlet boundaries, and spectral gaps, Common. Math. Phys., 164, 237-259 (1995) · Zbl 0827.35031
[11]	Hempel, R.; Herbst, I., Bands and gaps for periodic magnetic Hamiltonians, Operator Theory: Adv. Appl., 78, 175-184 (1995) · Zbl 0849.35091
[12]	Kato, T., “Perturbation Theory for Linear Operators” (1979), Springer-Verlag: Springer-Verlag Berlin
[13]	Kenig, C. E., “Restriction Theorems, Carleman Estimates, Uniform Sobolev Inequalities and Unique Continuation”. “Restriction Theorems, Carleman Estimates, Uniform Sobolev Inequalities and Unique Continuation”, Lecture Notes in Mathematics (1979), Springer-Verlag: Springer-Verlag Berlin · Zbl 0685.35003
[14]	Kenig, C. E.; Ruiz, A.; Sogge, C. D., Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J., 55, 329-347 (1987) · Zbl 0644.35012
[15]	Kerman, R.; Sawyer, E., The trace inequality and eigenvalue estimates for Schrödinger operators, Ann. Inst. Fourier (Grenoble)
[16]	Kuchment, P., Floquet Theory for Partial Differential Equations (1993), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0789.35002
[17]	Kurtz, D., Littlewood-Paley and multiplier theorems on weighted \(L^2\) spaces, Trans. Amer. Math. Soc., 259, 235-254 (1980) · Zbl 0436.42012
[18]	Lebedev, N. N., Special Functions and Their Applications (1972), Dover Publications: Dover Publications New York · Zbl 0271.33001
[19]	Morame, A., Absence of singular spectrum for a perturbation of a two-dimensional Laplace-Beltrami operator with periodic electro-magnetic potential, J. Phys., 31, 7593-7601 (1998) · Zbl 0931.35143
[20]	Reed, M.; Simon, B., Methods of Modern Mathematical Physics (1978), Academic Press: Academic Press New York
[21]	Shen, Z., On absolute continuity of the periodic Schrödinger operators, Internat. Math. Res. Notices, 2001, 1-31 (2001) · Zbl 0998.35007
[22]	Sobolev, A. V., Absolute continuity of the periodic magnetic Schrödinger operator, Invent. Math., 137, 85-112 (1999) · Zbl 0932.35049
[23]	Sogge, C. D., Concerning the \(L^p\) norm of spectral clusters of second order elliptic operators on compact manifolds, J. Funct. Anal., 77, 123-138 (1988) · Zbl 0641.46011
[24]	Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0207.13501
[25]	Stein, E. M., Harmonic Analysis, Real-variable Methods, Orthogonality, and Oscillatory Integrals (1993), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0821.42001
[26]	Stein, E. M.; Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces (1971), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0232.42007
[27]	Thomas, L. E., Time dependent approach to scattering from impurities in a crystal, Comm. Math. Phys., 33, 335-343 (1973)
[28]	Zygmund, A., Trigonometric Series (1968), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · JFM 58.0280.01