3,130 research outputs found
Zeros of Eigenfunctions of the Schrodinger Operator on Graphs and Their Relation to the Spectrum of the Magnetic Schrodinger Operator
In this dissertation, we analyze characteristics of eigenfunctions of the Schrödinger operator on graphs. In particular, we are interested in the zeros of the eigenfunctions and their relation to the spectrum of the magnetic Schrödinger operator.
We begin by studying the nodal count on finite quantum graphs, analyzing both the number and location of the zeros of eigenfunctions. This question was completely solved by Sturm in one dimension. In higher dimensions (including domains and graphs), we only know bounds for the nodal count. We discover more information about the nodal count on quantum graphs while analyzing eigenvalues of the magnetic Schrödinger operator. In particular, we show a relation between the stability of eigenvalues of the magnetic Schrödinger operator with respect to magnetic flux and the number of zeros of the corresponding eigenfunctions. We also study the location of the zeros of eigenfunctions while analyzing partitions. Specifically, we show that the critical points of the energy functional are the nodal partitions corresponding to zeros of an eigenfunction and that the stability of these critical points is related to the nodal count.
Then using Floquet-Bloch theory, we study the spectrum of the Schrödinger operator on infinite periodic graphs by analyzing the eigenvalues of the magnetic Schrödinger operator on a fundamental domain. Here we consider both discrete and quantum graphs. We find a characterization of critical points of the dispersion relation that occur inside the Brillouin zone under certain conditions on the graph. In particular, we show that if the fundamental domain is a tree, then the eigenfunction corresponding to an interior critical point must be zero on a vertex.
Finally, we use the results from infinite periodic graphs to study the magnetic Schrödinger operator on a finite quantum d-mandarin graph. We find that extremal points of the dispersion surface occur inside the Brillouin zone where two surfaces touch and the corresponding eigenfunction is zero on a vertex
Semiclassical and quantum behavior of the Mixmaster model in the polymer approach for the isotropic Misner variable
We analyze the semiclassical and quantum behavior of the Bianchi IX Universe
in the Polymer Quantum Mechanics framework, applied to the isotropic Misner
variable, linked to the space volume of the model. The study is performed both
in the Hamiltonian and field equations approaches, leading to the surprising
result of a still singular and chaotic cosmology, whose Poincar\'e return map
asymptotically overlaps the standard Belinskii-Khalatnikov-Lifshitz one. In the
quantum sector, we reproduce the original analysis due to Misner, within the
revised Polymer approach and we arrive to demonstrate that the quantum numbers
of the point-Universe still remain constants of motion. This issue confirms the
possibility to have quasi-classical states up to the initial singularity. The
present study clearly demonstrates that the asymptotic behavior of the Bianchi
IX Universe towards the singularity is not significantly affected by the
Polymer reformulation of the spatial volume dynamics both on a pure quantum and
a semiclassical level.Comment: 25 pages, 10 figures. v2: more discussions/clarification adde
An Adynamical, Graphical Approach to Quantum Gravity and Unification
We use graphical field gradients in an adynamical, background independent
fashion to propose a new approach to quantum gravity and unification. Our
proposed reconciliation of general relativity and quantum field theory is based
on a modification of their graphical instantiations, i.e., Regge calculus and
lattice gauge theory, respectively, which we assume are fundamental to their
continuum counterparts. Accordingly, the fundamental structure is a graphical
amalgam of space, time, and sources (in parlance of quantum field theory)
called a "spacetimesource element." These are fundamental elements of space,
time, and sources, not source elements in space and time. The transition
amplitude for a spacetimesource element is computed using a path integral with
discrete graphical action. The action for a spacetimesource element is
constructed from a difference matrix K and source vector J on the graph, as in
lattice gauge theory. K is constructed from graphical field gradients so that
it contains a non-trivial null space and J is then restricted to the row space
of K, so that it is divergence-free and represents a conserved exchange of
energy-momentum. This construct of K and J represents an adynamical global
constraint between sources, the spacetime metric, and the energy-momentum
content of the element, rather than a dynamical law for time-evolved entities.
We use this approach via modified Regge calculus to correct proper distance in
the Einstein-deSitter cosmology model yielding a fit of the Union2 Compilation
supernova data that matches LambdaCDM without having to invoke accelerating
expansion or dark energy. A similar modification to lattice gauge theory
results in an adynamical account of quantum interference.Comment: 47 pages text, 14 figures, revised per recent results, e.g., dark
energy result
Quantum mechanics in fractional and other anomalous spacetimes
We formulate quantum mechanics in spacetimes with real-order fractional
geometry and more general factorizable measures. In spacetimes where
coordinates and momenta span the whole real line, Heisenberg's principle is
proven and the wave-functions minimizing the uncertainty are found. In spite of
the fact that ordinary time and spatial translations are broken and the
dynamics is not unitary, the theory is in one-to-one correspondence with a
unitary one, thus allowing us to employ standard tools of analysis. These
features are illustrated in the examples of the free particle and the harmonic
oscillator. While fractional (and the more general anomalous-spacetime) free
models are formally indistinguishable from ordinary ones at the classical
level, at the quantum level they differ both in the Hilbert space and for a
topological term fixing the classical action in the path integral formulation.
Thus, all non-unitarity in fractional quantum dynamics is encoded in a
contribution depending only on the initial and final state.Comment: 22 pages, 1 figure. v2: typos correcte
d-Path Laplacians and Quantum Transport on Graphs
We generalize the Schrödinger equation on graphs to include long-range interactions (LRI) by means of the Mellin-transformed d-path Laplacian operators. We find analytical expressions for the transition and return probabilities of a quantum particle at the nodes of a ring graph. We show that the average return probability in ring graphs decays as a power law with time when LRI is present. In contrast, we prove analytically that the transition and return probabilities on a complete and start graphs oscillate around a constant value. This allowed us to infer that in a barbell graph-a graph consisting of two cliques separated by a path-the quantum particle get trapped and oscillates across the nodes of the path without visiting the nodes of the cliques. We then compare the use of the Mellin-transformed d-path Laplacian operators versus the use of fractional powers of the combinatorial Laplacian to account for LRI. Apart from some important differences observed at the limit of the strongest LRI, the d-path Laplacian operators produces the emergence of new phenomena related to the location of the wave packet in graphs with barriers, which are not observed neither for the Schrödinger equation without LRI nor for the one using fractional powers of the Laplacian
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