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35
The integrated density of states for random Schrödinger operators
 in “Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday
, 2007
"... Abstract. We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is di ..."
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Cited by 20 (1 self)
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Abstract. We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed in the second section. One central topic of this survey is the asymptotic behavior of the integrated density of states at the boundary of the spectrum. In particular, we are interested in Lifshitz tails and the occurrence of a classical and a quantum regime. In the last section we discuss regularity properties of the IDS. Our emphasis is on the discussion of fundamental problems and central ideas to handle them. Finally, we discuss further developments and problems of current
Eigenvalue Bounds for Perturbations of Schrödinger Operators and Jacobi Matrices With Regular Ground States
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2008
"... We prove general comparison theorems for eigenvalues of perturbed Schrödinger operators that allow proof of Lieb–Thirring bounds for suitable nonfree Schrödinger operators and Jacobi matrices. ..."
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Cited by 11 (11 self)
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We prove general comparison theorems for eigenvalues of perturbed Schrödinger operators that allow proof of Lieb–Thirring bounds for suitable nonfree Schrödinger operators and Jacobi matrices.
Eigenvalue bounds in the gaps of Schrödinger operators and Jacobi matrices
, 2007
"... We consider C = A + B where A is selfadjoint with a gap (a, b) in its spectrum and B is (relatively) compact. We prove a general result allowing B of indefinite sign and apply it to obtain a (δV) d/2 bound for perturbations of suitable periodic Schrödinger operators and a (not quite) Lieb–Thirring ..."
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Cited by 11 (7 self)
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We consider C = A + B where A is selfadjoint with a gap (a, b) in its spectrum and B is (relatively) compact. We prove a general result allowing B of indefinite sign and apply it to obtain a (δV) d/2 bound for perturbations of suitable periodic Schrödinger operators and a (not quite) Lieb–Thirring bound for perturbations of algebrogeometric almost periodic Jacobi matrices.
Integral representations and Liouville theorems for solutions of periodic elliptic equations
 J. Funct. Anal
"... The paper contains integral representations for certain classes of exponentially growing solutions of second order periodic elliptic equations. These representations are the analogs of those previously obtained by S. Agmon, S. Helgason, and other authors for solutions of the Helmholtz equation. When ..."
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Cited by 9 (6 self)
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The paper contains integral representations for certain classes of exponentially growing solutions of second order periodic elliptic equations. These representations are the analogs of those previously obtained by S. Agmon, S. Helgason, and other authors for solutions of the Helmholtz equation. When one restricts the class of solutions further, requiring their growth to be polynomial, one arrives to Liouville type theorems, which describe the structure and dimension of the spaces of such solutions. The Liouville type theorems previously proved by M. Avellaneda and F.H. Lin, and J. Moser and M. Struwe for periodic second order elliptic equations in divergence form are significantly extended. Relations of these theorems with the analytic structure of the Fermi and Bloch surfaces are explained.
An estimate of the gap of spectrum of Schrödinger operators which generate hyperbounded semigroup
, 2001
"... ..."
TwoDimensional Periodic Pauli OPERATOR. THE EFFECTIVE MASSES AT THE LOWER EDGE OF THE SPECTRUM
, 1998
"... We calculate the tensor of effective masses for the twodimensional periodic Pauli operator. The explicit representation for this tensor is given in terms of the magnetic field. It is proved that the tensor of effective masses is circular symmetric and always proportional to the unit matrix. We als ..."
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Cited by 7 (3 self)
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We calculate the tensor of effective masses for the twodimensional periodic Pauli operator. The explicit representation for this tensor is given in terms of the magnetic field. It is proved that the tensor of effective masses is circular symmetric and always proportional to the unit matrix. We also consider the generalized Pauli operator with variable metric. In Appendix we study the periodic elliptic operators of the second order and discuss the behavior of the first band function near its minimum point.
Wegner estimate and the density of states of some indefinite alloy type Schrödinger operators
 Lett. Math. Phys
, 2002
"... We study Schrödinger operators with a random potential of alloy type. The single site potentials are allowed to change sign. For a certain class of them we prove a Wegner estimate. This is a key ingredient in an existence proof of pure point spectrum of the considered random Schrödinger operators. O ..."
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Cited by 7 (1 self)
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We study Schrödinger operators with a random potential of alloy type. The single site potentials are allowed to change sign. For a certain class of them we prove a Wegner estimate. This is a key ingredient in an existence proof of pure point spectrum of the considered random Schrödinger operators. Our estimate is valid for all bounded energy intervals and all space dimensions and implies the existence of the density of states.
The AllegrettoPiepenbrink Theorem for Strongly Local Dirichlet Forms
 DOCUMENTA MATH.
, 2009
"... The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. ..."
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Cited by 7 (4 self)
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The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
Wegner Estimate for some Indefinite Andersontype Schrödinger Operators
, 2000
"... We study Schrödinger operators with a random potential of Andersontype. The single site potentials are allowed to change sign. For a certain class of them we prove a Wegner estimate (Theorem 5). This is a key ingredient in an existence proof of pure point spectrum of the considered random Schröd ..."
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Cited by 6 (2 self)
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We study Schrödinger operators with a random potential of Andersontype. The single site potentials are allowed to change sign. For a certain class of them we prove a Wegner estimate (Theorem 5). This is a key ingredient in an existence proof of pure point spectrum of the considered random Schrödinger operators. Our estimate is valid for all bounded energy intervals and all space dimensions.
Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds
"... Abstract. The paper describes relations between Liouville type theorems for solutions of a periodic elliptic equation (or a system) on an abelian cover of a compact Riemannian manifold and the structure of the dispersion relation for this equation at the edges of the spectrum. Here one says that the ..."
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Cited by 5 (3 self)
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Abstract. The paper describes relations between Liouville type theorems for solutions of a periodic elliptic equation (or a system) on an abelian cover of a compact Riemannian manifold and the structure of the dispersion relation for this equation at the edges of the spectrum. Here one says that the Liouville theorem holds if the space of solutions of any given polynomial growth is finite dimensional. The necessary and sufficient condition for a Liouville type theorem to hold is that the real Fermi surface of the elliptic operator consists of finitely many points (modulo the reciprocal lattice). Thus, such a theorem generically is expected to hold at the edges of the spectrum. The precise description of the spaces of polynomially growing solutions depends upon a ‘homogenized ’ constant coefficient operator determined by the analytic structure of the dispersion relation. In most cases, simple explicit formulas are found for the dimensions of the spaces of polynomially growing solutions in terms of the dispersion curves. The role of the base of the covering (in particular its dimension) is rather limited, while the deck group is of the most importance. The results are also established for overdetermined elliptic systems, which in particular leads to Liouville theorems for polynomially growing holomorphic functions on abelian coverings of compact analytic manifolds. Analogous theorems hold for abelian coverings of compact combinatorial or quantum graphs. 1.