244 research outputs found
The a priori Tan Theta Theorem for spectral subspaces
Let A be a self-adjoint operator on a separable Hilbert space H. Assume that
the spectrum of A consists of two disjoint components s_0 and s_1 such that the
set s_0 lies in a finite gap of the set s_1. Let V be a bounded self-adjoint
operator on H off-diagonal with respect to the partition spec(A)=s_0 \cup s_1.
It is known that if ||V||<\sqrt{2}d, where d=\dist(s_0,s_1), then the
perturbation V does not close the gaps between s_0 and s_1 and the spectrum of
the perturbed operator L=A+V consists of two isolated components s'_0 and s'_1
grown from s_0 and s_1, respectively. Furthermore, it is known that if V
satisfies the stronger bound ||V||< d then the following sharp norm estimate
holds: ||E_L(s'_0)-E_A(s_0)|| \leq sin(arctan(||V||/d)), where E_A(s_0) and
E_L(s'_0) are the spectral projections of A and L associated with the spectral
sets s_0 and s'_0, respectively. In the present work we prove that this
estimate remains valid and sharp also for d \leq ||V||< \sqrt{2}d, which
completely settles the issue.Comment: v3: some typos fixed; Examples adde
On the Lieb-Thirring constants L_gamma,1 for gamma geq 1/2
Let denote the negative eigenvalues of the one-dimensional
Schr\"odinger operator on . We prove the inequality \sum_i|E_i(H)|^\gamma\leq L_{\gamma,1}\int_{\Bbb
R} V^{\gamma+1/2}(x)dx, (1) for the "limit" case This will imply
improved estimates for the best constants in (1), as
$1/2<\gamma<3/2.Comment: AMS-LATEX, 15 page
Spectral estimates for two-dimensional Schroedinger operators with application to quantum layers
A logarithmic type Lieb-Thirring inequality for two-dimensional Schroedinger
operators is established. The result is applied to prove spectral estimates on
trapped modes in quantum layers
Scattering solutions in a network of thin fibers: small diameter asymptotics
Small diameter asymptotics is obtained for scattering solutions in a network
of thin fibers. The asymptotics is expressed in terms of solutions of related
problems on the limiting quantum graph. We calculate the Lagrangian gluing
conditions at vertices for the problems on the limiting graph. If the frequency
of the incident wave is above the bottom of the absolutely continuous spectrum,
the gluing conditions are formulated in terms of the scattering data for each
individual junction of the network
A new numerical approach to Anderson (de)localization
We develop a new approach for the Anderson localization problem. The
implementation of this method yields strong numerical evidence leading to a
(surprising to many) conjecture: The two dimensional discrete random
Schroedinger operator with small disorder allows states that are dynamically
delocalized with positive probability. This approach is based on a recent
result by Abakumov-Liaw-Poltoratski which is rooted in the study of spectral
behavior under rank-one perturbations, and states that every non-zero vector is
almost surely cyclic for the singular part of the operator.
The numerical work presented is rather simplistic compared to other numerical
approaches in the field. Further, this method eliminates effects due to
boundary conditions.
While we carried out the numerical experiment almost exclusively in the case
of the two dimensional discrete random Schroedinger operator, we include the
setup for the general class of Anderson models called Anderson-type
Hamiltonians.
We track the location of the energy when a wave packet initially located at
the origin is evolved according to the discrete random Schroedinger operator.
This method does not provide new insight on the energy regimes for which
diffusion occurs.Comment: 15 pages, 8 figure
Monodromy of Cyclic Coverings of the Projective Line
We show that the image of the pure braid group under the monodromy action on
the homology of a cyclic covering of degree d of the projective line is an
arithmetic group provided the number of branch points is sufficiently large
compared to the degree.Comment: 47 pages (to appear in Inventiones Mathematicae
Schrödinger operators with δ and δ′-potentials supported on hypersurfaces
Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity
Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems
Gapped ground states of quantum spin systems have been referred to in the
physics literature as being `in the same phase' if there exists a family of
Hamiltonians H(s), with finite range interactions depending continuously on , such that for each , H(s) has a non-vanishing gap above its
ground state and with the two initial states being the ground states of H(0)
and H(1), respectively. In this work, we give precise conditions under which
any two gapped ground states of a given quantum spin system that 'belong to the
same phase' are automorphically equivalent and show that this equivalence can
be implemented as a flow generated by an -dependent interaction which decays
faster than any power law (in fact, almost exponentially). The flow is
constructed using Hastings' 'quasi-adiabatic evolution' technique, of which we
give a proof extended to infinite-dimensional Hilbert spaces. In addition, we
derive a general result about the locality properties of the effect of
perturbations of the dynamics for quantum systems with a quasi-local structure
and prove that the flow, which we call the {\em spectral flow}, connecting the
gapped ground states in the same phase, satisfies a Lieb-Robinson bound. As a
result, we obtain that, in the thermodynamic limit, the spectral flow converges
to a co-cycle of automorphisms of the algebra of quasi-local observables of the
infinite spin system. This proves that the ground state phase structure is
preserved along the curve of models .Comment: Updated acknowledgments and new email address of S
On non-local variational problems with lack of compactness related to non-linear optics
We give a simple proof of existence of solutions of the dispersion manage-
ment and diffraction management equations for zero average dispersion,
respectively diffraction. These solutions are found as maximizers of non-linear
and non-local vari- ational problems which are invariant under a large
non-compact group. Our proof of existence of maximizer is rather direct and
avoids the use of Lions' concentration compactness argument or Ekeland's
variational principle.Comment: 30 page
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