ICMP 2003 > Sessions > Quantum mechanics and spectral theory | |
Session organized by C. Gerard (Orsay), R. Weder (Mexico).
T. Chen | New York |
P. Exner | Prague |
T. Ichinose | Kanazawa |
V. Kostrykin | Aachen |
E. Skibsted | Aarhus |
G. Stolz | Birmingham, USA |
D. Yafaev | Rennes |
V.A. Zagrebnov | Marseille |
Localization lenghts and Boltzmann limit for the Anderson model at small disorder in d=3
We prove lower bounds on the localization lengths of eigenvectors in the 3-dimensional Anderson model, using an extension of techniques developed by L. Erdos and H.-T. Yau. This result is closely related to recent work of C. Shubin, W. Schlag and T. Wolff for dimensions 1 and 2. Furthermore, we demonstrate that the macroscopic limit of the corresponding lattice random Schrodinger system is governed by the linear Boltzmann equations.
Schrödinger operators with graph-type interactions
We consider a class of Schrödinger operators given formally by the expression (-i∇-A)^{2} - αδ(x-Γ), where Γ is a generalized graph, i.e. a subset of R^{d}, d=2,3, which can be decomposed into a union of curves and surfaces. We show how the geometry of Γ is reflected in the spectral properties and discuss related physical effects. Some open problems will also be listed.
Quantum formula for quantum Zeno dynamics
The fact the decay can be slowed or even prevented by frequently repeated measurements, noticed first by Beskow and Nilsson, and called quantum Zeno effect by Misra and Sudarshan, attracted recently a lot of attention since it seems to be nowadays experimentally accessible. Mathematically the issue was addressed, in particular, long time ago by Friedman and Chernoff, however, the question about existence of the "Zeno limit" remained open.
Let H and P be a non-negative self-adjoint operator acting in a separable Hilbert space H and an orthogonal projection on H, respectively. Suppose that the operator H_{P} := (H^{1/2}P)^{*}(H^{1/2}P) is densely defined, then we claim that the said limit exists and we find its value,
lim_{n→∞} [P exp(-iεtH/n)P]^{n} = exp(-iεtH_{P}) P
for ε=±1. In fact, the proof given in [1] yields a stronger result with P on the left-hand side replaced by values of a projection-valued family such that P(t)→P as t→0 and P(t)P = P(t), as well as non-symmetric versions of such formulae. The method we use employs a modification of a Kato's result on Trotter product formula in combination with analyic continuation.
As an example consider H=-Δ in L^{2}(R^{d}) and P projecting onto an open Ω⊂R^{d} with a smooth boundary. Then the limits exists and H_{P} is equal to the Dirichlet Laplacian in L^{2}(Ω); this provided a rigorous proof of the formal claim made recently in [2] using the method of stationary phase.
[Joint work with Pavel Exner (Czech Academy of Sciences)]
On Adiabatic Theorem of Quantum Mechanics
We study the variation of spectral subspaces under the Schrödinger evolution provided that the perturbation depends "slowly" on time. As a result we obtain a new proof of the Adiabatic Theorem of Quantum Mechanics and estimate the rate at wich the adiabatic limit is approached.
This is a joint work with K. A. Makarov (Missouri-Columbia).
Zero energy asymptotics of the resolvent in the long range case
We prove absence of eigenvalue at zero for two-body Schrödinger operators with long-range potentials having a positive virial at infinity. This enables us to prove a limiting absorption principle at zero and in fact a complete asymptotic expansion of the resolvent in weighted spaces in cones; one of the two branches of boundary for the cones is given by the positive real axis. The principal tools are singular Mourre theory and microlocal estimates.
[Joint work with S. Fournais (Laboratoire de Mathématiques Université Paris-Sud)]
Fractional moment methods for Anderson localization in the continuum
The fractional moment (or Aizenman-Molchanov) method has been developed in the 90s as an alternative to multiscale analysis in the proof of Anderson localization. While yielding stronger results, the method was initially only applicable to lattice models.
We will present recent joint work with M. Aizenman, A. Elgart, S. Naboko and J. Schenker which allows to extend the fractional moment method and its consequences to continuum Anderson-type models. In the present talk we will focus on the new ideas from spectral theory and harmonic/complex analysis which enter the proofs. Other aspects of this work will be discussed in Elgart's talk in the session on Condensed Matter Physics.
A particle in a magnetic field of an infinite rectilinear current
We consider the Schrödinger operator H = (i∇+A)^{2} in the space L_{2}(R^{3}) with a magnetic potential A created by an infinite rectilinear current and perform its spectral analysis almost explicitly. In particular, we show that the operator H is absolutely continuous, its spectrum has infinite multiplicity and coincides with the positive half-axis. Then we find the large-time behavior of solutions exp(-iHt)f of the time dependent Schrödinger equation. Our main observation is that a quantum particle has always a preferable (depending on its charge) direction of propagation along the current. Similar result is true in classical mechanics.
Trotter-Kato product formula: recent results
Recently T. Ichinose, Hideo Tamura, Hiroshi Tamura, and V.A. Zagrebnov have proved the following result:
THEOREM: Let A and B be non-negative self-adjoint operators
such that the operator sum C := A + B
is self-adjoint in domain
dom(C) = dom(A) ∩ dom(B). Then:
(a) the Trotter product formulae converge in operator norm
with error bound O(n^{-1}):
||(e^{-tA/n}
e^{-tB/n})^{n}
- e^{-tC}|| = O(n^{-1}),
n→∞,
||(e^{-tB/2n}
e^{-tA/n}
e^{-tB/2n})^{n}
- e^{-tC}|| = O(n^{-1}),
n→∞,
uniformly on each compact t-interval in [0, ∞). Further,
if C is strictly positive, then it holds uniformly on
[0, ∞);
(b) the error bound O(n^{-1}) is ultimate optimal.
These conditions on the generators have been formulated for the first time be E. Nelson. The hypothesis on the optimal rate of convergence discussed since 1993.
A recent result by V.A. Zagrebnov says that the same optimal estimate is valid in the trace-norm for Gibbs semigroups.
Interaction of the superpositions of squeezed displaced Fock states with atomic systems
Localization of spectrum for certain classes of operator pencils
Upper and lower limits for the number of bound states in an attractive central potential
Spectrum of quasiperiodic discrete Schrödinger operator
Localization of length and Boltzamn limit of the Anderson model at small disorders in d=3
Spectral theory and quantum damped systems
Hypergeometric type functions in a supersymmetric approach
Complete positivity from factorized dynamics
Quasi-periodic solutions and pure point spectrum for two-level systems
The heat kernel expansion for the elctromagnetic field in a cavity and its application to the Casimir energy
Product formula for quantum Zeno dynamics
On the positivity of the Jansen-Hess operators for arbitrary mass
Quantum formula for quantum zero dynamics
How do small velocity dependent forces (de)stabilize a non-conservative system?
The nature of the essential spectrum in curved quantum waveguides
Large spin and statistics in non-relativistic quantum mechanics
Quantum waveguides with combined boundary conditions
The role of the rigged Hilbert space in quantum mechanics
Spectral transitions in a class of off-diagonal Jacobi matrices: a probabilistic approach
Energy level statistics of integrable quantum systems based on the Mehta-Berry-Robink approach
Topological quantum mechanics and Bessel functions
Quantum mechanics of Dirichlet layers with a finite number of point perturbations
Study of partition functions using supersymmetric quantum mechanics
Waveguides and layers coupled through small apertures: asymptotics of bound states, bands and resonances
Spectral gaps on manifolds with noncommutative group actions
On modeling of the Maxwell-Bloch quantum optical super-radiance proccess and its application to quantum computing
On a purely real quaternionic reformulation of Dirac equation
A new numerical method to solve the inverse Sturm-Liouville problem
On a purely real quaternionic formulation of the Dirac equation
Tunneling effects between tori in double wells
Semiclassical time evolution for large times on manifolds of negative curvature
Wigner quantum systems: non-commutative sl(1|3) oscillator
Exactly solvable time-dependent Schrödinger equations
The integrated density of states of random Pauli Hamiltonians
A particle in a magnetic field of an infinite rectilinear current
Applications of Fermi's golden rule to open quantum systems
Trotter-Kato product formula: recent results
Solution of the time-dependent Schrodinger equation for decay in terms of resonant states