1,698 research outputs found
Sturm-Liouville Estimates for the Spectrum and Cheeger Constant
Buser's inequality gives an upper bound on the first non-zero eigenvalue of
the Laplacian of a closed manifold M in terms of the Cheeger constant h(M).
Agol later gave a quantitative improvement of Buser's inequality. Agol's result
is less transparent since it is given implicitly by a set of equations, one of
which is a differential equation Agol could not solve except when M is
three-dimensional. We show that a substitution transforms Agol's differential
equation into the Riemann differential equation. Then, we give a proof of
Agol's result and also generalize it using Sturm-Liouville theory. Under the
same assumptions on M, we are able to give upper bounds on the higher
eigenvalues of M, \lambda_k(M), in terms of the eigenvalues of a
Sturm-Liouville problem which depends on h(M). We then compare the Weyl
asymptotic of \lambda_k(M) given by the works of Cheng, Gromov, and
B\'erard-Besson-Gallot to the asymptotics of our Sturm-Liouville problems given
by Atkinson-Mingarelli.Comment: 41 pages, 7 figures. Answered the question from v1 in the negative;
see Example 1.5. Some changes in numbering conventions of equations and
examples to avoid confusion that occurred in v1. Improvements to the writing
of the proof of Proposition 2.1. Lemma 3.5 now references the slicing lemma,
which was omitted in v1. Some typos were fixed and references added from v
Classical and Quantum Complexity of the Sturm-Liouville Eigenvalue Problem
We study the approximation of the smallest eigenvalue of a Sturm-Liouville
problem in the classical and quantum settings. We consider a univariate
Sturm-Liouville eigenvalue problem with a nonnegative function from the
class and study the minimal number n(\e) of function evaluations
or queries that are necessary to compute an \e-approximation of the smallest
eigenvalue. We prove that n(\e)=\Theta(\e^{-1/2}) in the (deterministic)
worst case setting, and n(\e)=\Theta(\e^{-2/5}) in the randomized setting.
The quantum setting offers a polynomial speedup with {\it bit} queries and an
exponential speedup with {\it power} queries. Bit queries are similar to the
oracle calls used in Grover's algorithm appropriately extended to real valued
functions. Power queries are used for a number of problems including phase
estimation. They are obtained by considering the propagator of the discretized
system at a number of different time moments. They allow us to use powers of
the unitary matrix , where is an
matrix obtained from the standard discretization of the Sturm-Liouville
differential operator. The quantum implementation of power queries by a number
of elementary quantum gates that is polylog in is an open issue.Comment: 33 page
Spectral bounds for singular indefinite Sturm-Liouville operators with --potentials
The spectrum of the singular indefinite Sturm-Liouville operator
with a real
potential covers the whole real line and, in addition,
non-real eigenvalues may appear if the potential assumes negative values. A
quantitative analysis of the non-real eigenvalues is a challenging problem, and
so far only partial results in this direction were obtained. In this paper the
bound on the absolute values of the non-real
eigenvalues of is obtained. Furthermore, separate bounds on the
imaginary parts and absolute values of these eigenvalues are proved in terms of
the -norm of the negative part of .Comment: to appear in Proc. Amer. Math. So
Correspondence of the eigenvalues of a non-self-adjoint operator to those of a self-adjoint operator
We prove that the eigenvalues of a certain highly non-self-adjoint operator
that arises in fluid mechanics correspond, up to scaling by a positive
constant, to those of a self-adjoint operator with compact resolvent; hence
there are infinitely many real eigenvalues which accumulate only at . We use this result to determine the asymptotic distribution of the
eigenvalues and to compute some of the eigenvalues numerically. We compare
these to earlier calculations by other authors.Comment: 29 pages, corrections to section 3, added section
A Lower Bound for the Sturm-Liouville Eigenvalue Problem on a Quantum Computer
We study the complexity of approximating the smallest eigenvalue of a
univariate Sturm-Liouville problem on a quantum computer. This general problem
includes the special case of solving a one-dimensional Schroedinger equation
with a given potential for the ground state energy.
The Sturm-Liouville problem depends on a function q, which, in the case of
the Schroedinger equation, can be identified with the potential function V.
Recently Papageorgiou and Wozniakowski proved that quantum computers achieve an
exponential reduction in the number of queries over the number needed in the
classical worst-case and randomized settings for smooth functions q. Their
method uses the (discretized) unitary propagator and arbitrary powers of it as
a query ("power queries"). They showed that the Sturm-Liouville equation can be
solved with O(log(1/e)) power queries, while the number of queries in the
worst-case and randomized settings on a classical computer is polynomial in
1/e. This proves that a quantum computer with power queries achieves an
exponential reduction in the number of queries compared to a classical
computer.
In this paper we show that the number of queries in Papageorgiou's and
Wozniakowski's algorithm is asymptotically optimal. In particular we prove a
matching lower bound of log(1/e) power queries, therefore showing that log(1/e)
power queries are sufficient and necessary. Our proof is based on a frequency
analysis technique, which examines the probability distribution of the final
state of a quantum algorithm and the dependence of its Fourier transform on the
input.Comment: 23 pages, 2 figures; Major changes in Theorem 3 to previous version.
To be published in the Journal of Complexit
Eigenvalue estimates for singular left-definite Sturm-Liouville operators
The spectral properties of a singular left-definite Sturm-Liouville operator
are investigated and described via the properties of the corresponding
right-definite selfadjoint counterpart which is obtained by substituting
the indefinite weight function by its absolute value. The spectrum of the
-selfadjoint operator is real and it follows that an interval
is a gap in the essential spectrum of if and only
if both intervals and are gaps in the essential spectrum of
the -selfadjoint operator . As one of the main results it is shown that
the number of eigenvalues of in differs at most by
three of the number of eigenvalues of in the gap ; as a byproduct
results on the accumulation of eigenvalues of singular left-definite
Sturm-Liouville operators are obtained. Furthermore, left-definite problems
with symmetric and periodic coefficients are treated, and several examples are
included to illustrate the general results.Comment: to appear in J. Spectral Theor
A comparison of the eigenvalues of the Dirac and Laplace operator on the two-dimensional torus
We compare the eigenvalues of the Dirac and Laplace operator on a
two-dimensional torus with respect to the trivial spin structure. In
particular, we compute their variation up to order 4 upon deformation of the
flat metric, study the corresponding Hamiltonian and discuss several families
of examples.Comment: Latex2.09, 28 pages, with figure
On the Spectral Gap of a Quantum Graph
We consider the problem of finding universal bounds of "isoperimetric" or
"isodiametric" type on the spectral gap of the Laplacian on a metric graph with
natural boundary conditions at the vertices, in terms of various analytical and
combinatorial properties of the graph: its total length, diameter, number of
vertices and number of edges. We investigate which combinations of parameters
are necessary to obtain non-trivial upper and lower bounds and obtain a number
of sharp estimates in terms of these parameters. We also show that, in contrast
to the Laplacian matrix on a combinatorial graph, no bound depending only on
the diameter is possible. As a special case of our results on metric graphs, we
deduce estimates for the normalised Laplacian matrix on combinatorial graphs
which, surprisingly, are sometimes sharper than the ones obtained by purely
combinatorial methods in the graph theoretical literature
Variational bound on energy dissipation in plane Couette flow
We present numerical solutions to the extended Doering-Constantin variational
principle for upper bounds on the energy dissipation rate in turbulent plane
Couette flow. Using the compound matrix technique in order to reformulate this
principle's spectral constraint, we derive a system of equations that is
amenable to numerical treatment in the entire range from low to asymptotically
high Reynolds numbers. Our variational bound exhibits a minimum at intermediate
Reynolds numbers, and reproduces the Busse bound in the asymptotic regime. As a
consequence of a bifurcation of the minimizing wavenumbers, there exist two
length scales that determine the optimal upper bound: the effective width of
the variational profile's boundary segments, and the extension of their flat
interior part.Comment: 22 pages, RevTeX, 11 postscript figures are available as one
uuencoded .tar.gz file from [email protected]
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