43 research outputs found
A second eigenvalue bound for the Dirichlet Schroedinger operator
Let be the th eigenvalue of the Schr\"odinger
operator with Dirichlet boundary conditions on a bounded domain and with the positive potential . Following the spirit of the
Payne-P\'olya-Weinberger conjecture and under some convexity assumptions on the
spherically rearranged potential , we prove that . Here denotes the ball, centered at the
origin, that satisfies the condition .
Further we prove under the same convexity assumptions on a spherically
symmetric potential , that decreases
when the radius of the ball increases.
We conclude with several results about the first two eigenvalues of the
Laplace operator with respect to a measure of Gaussian or inverted Gaussian
density
Eigenvalue Ratios for Sturm-Liouville Operators
AbstractIn this paper we prove various optimal bounds for eigenvalue ratios for the Sturm-Liouville equation − [p(x) y′]′ + q(x)y = λw(x)y and certain specializations. Our results primarily concern the regular case with Dirichlet boundary conditions though various extensions and generalizations to other situations are possible. Our results here extend the result λm/λ1 ≤ m2 obtained in a previous paper for the one-dimensional Schrödinger equation, − y″ + q(x)y = λy, on a finite interval with Dirichlet boundary conditions and nonnegative potential (q ≥ 0). In particular, we obtain λm/λ1 ≤ Km2/k, where the constants k, K satisfy 0 < k ≤ p(x) w(x) ≤ K for all x. If q ≡ 0, lower bounds can also be obtained. Our methods involve a slight modification of the Prüfer variable techniques employed in the Schrödinger case. We also examine the consequences of our recent proof of the Payne-Pólya-Weinberger conjecture in the one-dimensional (Sturm-Liouville) setting. Finally, we compare our general bounds to the detailed analyses of Keller and of Mahar and Willner for the special case of the inhomogeneous stretched string
An Isoperimetric Inequality for Fundamental Tones of Free Plates
We establish an isoperimetric inequality for the fundamental tone (first
nonzero eigenvalue) of the free plate of a given area, proving the ball is
maximal. Given , the free plate eigenvalues and eigenfunctions
are determined by the equation
together with certain natural boundary conditions. The boundary conditions are
complicated but arise naturally from the plate Rayleigh quotient, which
contains a Hessian squared term . We adapt Weinberger's method from
the corresponding free membrane problem, taking the fundamental modes of the
unit ball as trial functions. These solutions are a linear combination of
Bessel and modified Bessel functions.Comment: PhD thesis. Papers are in preparatio
On the boundary of the attainable set of the Dirichlet spectrum
Denoting by the set of the pairs
for all the open sets
with unit measure, and by the union
of two disjoint balls of half measure, we give an elementary proof of the fact
that \partial\E has horizontal tangent at its lowest point
.Comment: 7 pages, 3 figure
Self Consistent Molecular Field Theory for Packing in Classical Liquids
Building on a quasi-chemical formulation of solution theory, this paper
proposes a self consistent molecular field theory for packing problems in
classical liquids, and tests the theoretical predictions for the excess
chemical potential of the hard sphere fluid. Results are given for the self
consistent molecular fields obtained, and for the probabilities of occupancy of
a molecular observation volume. For this system, the excess chemical potential
predicted is as accurate as the most accurate prior theories, particularly the
scaled particle (Percus-Yevick compressibility) theory. It is argued that the
present approach is particularly simple, and should provide a basis for a
molecular-scale description of more complex solutions.Comment: 6 pages and 5 figure
Geometric Phase, Curvature, and Extrapotentials in Constrained Quantum Systems
We derive an effective Hamiltonian for a quantum system constrained to a
submanifold (the constraint manifold) of configuration space (the ambient
space) by an infinite restoring force. We pay special attention to how this
Hamiltonian depends on quantities which are external to the constraint
manifold, such as the external curvature of the constraint manifold, the
(Riemannian) curvature of the ambient space, and the constraining potential. In
particular, we find the remarkable fact that the twisting of the constraining
potential appears as a gauge potential in the constrained Hamiltonian. This
gauge potential is an example of geometric phase, closely related to that
originally discussed by Berry. The constrained Hamiltonian also contains an
effective potential depending on the external curvature of the constraint
manifold, the curvature of the ambient space, and the twisting of the
constraining potential. The general nature of our analysis allows applications
to a wide variety of problems, such as rigid molecules, the evolution of
molecular systems along reaction paths, and quantum strip waveguides.Comment: 27 pages with 1 figure, submitted to Phys. Rev.
Hydrophobic and ionic-interactions in bulk and confined water with implications for collapse and folding of proteins
Water and water-mediated interactions determine thermodynamic and kinetics of
protein folding, protein aggregation and self-assembly in confined spaces. To
obtain insights into the role of water in the context of folding problems, we
describe computer simulations of a few related model systems. The dynamics of
collapse of eicosane shows that upon expulsion of water the linear hydrocarbon
chain adopts an ordered helical hairpin structure with 1.5 turns. The structure
of dimer of eicosane molecules has two well ordered helical hairpins that are
stacked perpendicular to each other. As a prelude to studying folding in
confined spaces we used simulations to understand changes in hydrophobic and
ionic interactions in nano droplets. Solvation of hydrophobic and charged
species change drastically in nano water droplets. Hydrophobic species are
localized at the boundary. The tendency of ions to be at the boundary where
water density is low increases as the charge density decreases. Interaction
between hydrophobic, polar, and charged residue are also profoundly altered in
confined spaces. Using the results of computer simulations and accounting for
loss of chain entropy upon confinement we argue and then demonstrate, using
simulations in explicit water, that ordered states of generic amphiphilic
peptide sequences should be stabilized in cylindrical nanopores
A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains
In the first (and abstract) part of this survey we prove the unitary
equivalence of the inverse of the Krein--von Neumann extension (on the
orthogonal complement of its kernel) of a densely defined, closed, strictly
positive operator, for some in a Hilbert space to an abstract buckling problem operator.
This establishes the Krein extension as a natural object in elasticity theory
(in analogy to the Friedrichs extension, which found natural applications in
quantum mechanics, elasticity, etc.).
In the second, and principal part of this survey, we study spectral
properties for , the Krein--von Neumann extension of the
perturbed Laplacian (in short, the perturbed Krein Laplacian)
defined on , where is measurable, bounded and
nonnegative, in a bounded open set belonging to a
class of nonsmooth domains which contains all convex domains, along with all
domains of class , .Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144
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