scholarly journals Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 359 ◽  
Author(s):  
Claudio Cacciapuoti
Keyword(s):  
Spectral Properties ◽  
Infinite Length ◽  
Star Graph ◽  
Effective Hamiltonian ◽  
Central Vertex ◽  
Scale Invariant ◽  
Metric Graph ◽  
Suitable Norm ◽  
Compact Graph ◽  
Effective Hamiltonians

We consider a compact metric graph of size ε and attach to it several edges (leads) of length of order one (or of infinite length). As ε goes to zero, the graph G ε obtained in this way looks like the star-graph formed by the leads joined in a central vertex. On G ε we define an Hamiltonian H ε , properly scaled with the parameter ε . We prove that there exists a scale invariant effective Hamiltonian on the star-graph that approximates H ε (in a suitable norm resolvent sense) as ε → 0 . The effective Hamiltonian depends on the spectral properties of an auxiliary ε -independent Hamiltonian defined on the compact graph obtained by setting ε = 1 . If zero is not an eigenvalue of the auxiliary Hamiltonian, in the limit ε → 0 , the leads are decoupled.

scholarly journals Effective Hamiltonians and Wavefunctions for Electrons in Deformed Crystals

1983 ◽  
Vol 36 (3) ◽  
pp. 321 ◽  
Author(s):  
RA Brown
Keyword(s):  
Hilbert Space ◽  
Probability Density ◽  
Physical Interpretation ◽  
Wannier Function ◽  
Deformation Potential ◽  
Effective Hamiltonian ◽  
Wannier Functions ◽  
Modulating Functions ◽  
Strain Dependent ◽  
Effective Hamiltonians

An effective Hamiltonian for electrons in in homogeneously deformed crystals is derived by expanding the wavefunction in terms of Wannier functions of the homogeneously deformed crystal. The physical interpretation of the modulating functions which determine the amplitude of each Wannier function in the expansion, and which are governed by the effective Hamiltonian, is investigated. This leads to strain-dependent expressions for the probability density and current, averaged over the fluctuations within each unit cell. The operators which represent, in the Hilbert space of the . modulating functions, similarly averaged physical observables are introduced and explicit straindependent expressions for the velocity and momentum operators are obtained. Applications of the theory are foreshadowed and its relationship to previous deformation-potential theories is examined.

scholarly journals Effective Hamiltonians for Complexes of Unstable Particles

2013 ◽  
Vol 20 (03) ◽  
pp. 1340008 ◽  
Author(s):  
Krzysztof Urbanowski
Keyword(s):  
Evolution Equation ◽  
State Space ◽  
Total System ◽  
Effective Hamiltonian ◽  
Minimal Energy ◽  
Unstable Particles ◽  
The One ◽  
Unstable States ◽  
Instantaneous Energy ◽  
Effective Hamiltonians

Effective Hamiltonians governing the time evolution in a subspace of unstable states can be found using more or less accurate approximations. A convenient tool for deriving them is the evolution equation for a subspace of state space sometime called the Królikowski–Rzewuski (KR) equation. KR equation results from the Schrödinger equation for the total system under considerations. We will discuss properties of approximate effective Hamiltonians derived using KR equation for n-particle, two-particle and for one-particle subspaces. In a general case these effective Hamiltonians depend on time t. We show that at times much longer than times at which the exponential decay take place the real part of the exact effective Hamiltonian for the one-particle subsystem (that is the instantaneous energy) tends to the minimal energy of the total system when t → ∞ whereas the imaginary part of this effective Hamiltonian tends to zero as t → ∞.

scholarly journals TERAHERTZ DIELECTRIC RESPONSE AND COUPLED DYNAMICS OF FERROELECTRICS AND MULTIFERROICS FROM EFFECTIVE HAMILTONIAN SIMULATIONS

2013 ◽  
Vol 27 (22) ◽  
pp. 1330016 ◽  
Author(s):  
DAWEI WANG ◽  
JEEVAKA WEERASINGHE ◽  
ABDULLAH ALBARAKATI ◽  
L. BELLAICHE
Keyword(s):  
First Principles ◽  
Md Simulation ◽  
Functional Materials ◽  
Simulation Method ◽  
Effective Hamiltonian ◽  
Multiferroic Materials ◽  
Future Directions ◽  
Technical Details ◽  
Complex Phenomena ◽  
Effective Hamiltonians

Ferroelectric and multiferroic materials form an important class of functional materials. Over the last 20 years, first-principles-based effective Hamiltonian approaches have been successfully developed to simulate these materials. In recent years, effective Hamiltonian approaches were combined with molecular dynamics (MD) methods to investigate terahertz dynamical properties of various perovskites. With this combination, a variety of ferroelectric and multiferroic materials, including BaTiO 3, Ba(Sr, Ti)O 3, Pb(Zr, Ti)O 3, BiFeO 3 and SrTiO 3 bulks and films have been simulated, which led to the understanding of complex phenomena and discovery of novel effects. In this paper we first provide technical details about effective Hamiltonians and MD simulation method. Then, we present applications of the combination of these two techniques to different perovskites. Finally, we briefly discuss possible future directions of this approach.

scholarly journals Reconstruction of the Sturm-Liouville operators on a graph with $\delta'_s$ couplings

2011 ◽  
Vol 42 (3) ◽  
pp. 329-342 ◽  
Author(s):  
ChuanFu Yang
Keyword(s):  
Uniqueness Theorem ◽  
Liouville Operator ◽  
Dense Subset ◽  
Star Graph ◽  
Central Vertex ◽  
Inverse Nodal Problems ◽  
Sturm Liouville ◽  
The Given ◽  
The Uniqueness Theorem ◽  
Liouville Operators

Inverse nodal problems consist in constructing operators from the given zeros of their eigenfunctions. In this work, we deal with the inverse nodal problems of reconstructing the Sturm- Liouville operator on a star graph with $\delta'_s $ couplings at the central vertex. The uniqueness theorem is proved and a constructive procedure for the solution is provided from a dense subset of zeros of the eigenfunctions for the problem as a data.

scholarly journals Effective Hamiltonians in Nonrelativistic Quantum Electrodynamics

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2375
Author(s):  
Roberto Passante ◽  
Lucia Rizzuto
Keyword(s):  
Ground State ◽  
Quantum Electrodynamics ◽  
State Energy ◽  
Nonrelativistic Limit ◽  
Radiative Energy ◽  
Effective Hamiltonian ◽  
Dispersion Interactions ◽  
Energy Shell ◽  
Considerable Simplification ◽  
Effective Hamiltonians

In this paper, we consider some second-order effective Hamiltonians describing the interaction of the quantum electromagnetic field with atoms or molecules in the nonrelativistic limit. Our procedure is valid only for off-energy-shell processes, specifically virtual processes such as those relevant for ground-state energy shifts and dispersion van der Waals and Casimir-Polder interactions, while on-energy-shell processes are excluded. These effective Hamiltonians allow for a considerable simplification of the calculation of radiative energy shifts, dispersion, and Casimir-Polder interactions, including in the presence of boundary conditions. They can also provide clear physical insights into the processes involved. We clarify that the form of the effective Hamiltonian depends on the field states considered, and consequently different expressions can be obtained, each of them with a well-defined range of validity and possible applications. We also apply our results to some specific cases, mainly the Lamb shift, the Casimir-Polder atom-surface interaction, and the dispersion interactions between atoms, molecules, or, in general, polarizable bodies.

scholarly journals RENORMALIZATION GROUP APPROACH TO EFFECTIVE HAMILTONIANS

1996 ◽  
Vol 11 (27) ◽  
pp. 2233-2240 ◽  
Author(s):  
T.J. FIELDS ◽  
J.P. VARY ◽  
K.S. GUPTA
Keyword(s):  
Renormalization Group ◽  
Renormalization Scheme ◽  
Quantum Mechanical ◽  
Scale Invariant ◽  
Group Approach ◽  
Renormalization Group Approach ◽  
Β Function ◽  
Quantum Mechanical Systems ◽  
Invariant Quantum ◽  
Effective Hamiltonians

We introduce a way of implementing renormalization within the context of the theory of effective Hamiltonians. Our renormalization scheme involves manipulations at the level of the generalized G-matrix and is independent of any specific kinematics. We show how to calculate the β-function within this context and exhibit our method using simple scale-invariant quantum mechanical systems.

scholarly journals RT-Symmetric Laplace Operators on Star Graphs: Real Spectrum and Self-Adjointness

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Maria Astudillo ◽  
Pavel Kurasov ◽  
Muhammad Usman
Keyword(s):  
Quantum Mechanics ◽  
Spectral Properties ◽  
Block Structure ◽  
Circulant Matrices ◽  
Real Spectrum ◽  
Star Graph ◽  
Quantum Graphs ◽  
Star Graphs ◽  
Rotationally Symmetric ◽  
Symmetric Quantum

How ideas ofPT-symmetric quantum mechanics can be applied to quantum graphs is analyzed, in particular to the star graph. The class of rotationally symmetric vertex conditions is analyzed. It is shown that all such conditions can effectively be described by circulant matrices: real in the case of odd number of edges and complex having particular block structure in the even case. Spectral properties of the corresponding operators are discussed.

scholarly journals Effective Theories for Quark Flavour Physics

2020 ◽  
pp. 479-559 ◽  
Author(s):  
Luca Silvestrini
Keyword(s):  
Standard Model ◽  
New Physics ◽  
Effective Theory ◽  
Effective Hamiltonian ◽  
Weak Decays ◽  
The Standard Model ◽  
Quark Flavour ◽  
Meson Mixing ◽  
Effective Theories ◽  
Effective Hamiltonians

The purpose of the lectures that appear within this chapter is to provide the reader with an idea of how we can probe new physics with quark flavour observables using effective theory techniques. It begins by providing a concise review of the quark flavour structure of the standard model. Then it introduces the effective Hamiltonian for quark weak decays. Following on, it then considers the effective Hamiltonian for ?F=2 transitions in the standard model and beyond. It discusses how meson–anti–meson mixing and CP violation can be described in terms of the ?F=1 and ?F=2 effective Hamiltonians. Finally, it presents the Unitarity Triangle Analysis and discusses how very stringent constraints on new physics can be obtained from ?F=2 processes.

scholarly journals Spectral properties of Landau Hamiltonians with non-local potentials

2020 ◽  
Vol 120 (3-4) ◽  
pp. 337-371 ◽  
Author(s):  
Esteban Cárdenas ◽  
Georgi Raikov ◽  
Ignacio Tejeda
Keyword(s):  
Differential Operator ◽  
Toeplitz Operator ◽  
Asymptotic Distribution ◽  
Orthogonal Projection ◽  
Spectral Properties ◽  
Effective Hamiltonian ◽  
Pseudo Differential Operator ◽  
Weyl Symbol ◽  
Non Local ◽  
Local Potentials

We consider the Landau Hamiltonian H 0 , self-adjoint in L 2 ( R 2 ), whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues Λ q , q ∈ Z + . We perturb H 0 by a non-local potential written as a bounded pseudo-differential operator Op w ( V ) with real-valued Weyl symbol V, such that Op w ( V ) H 0 − 1 is compact. We study the spectral properties of the perturbed operator H V = H 0 + Op w ( V ). First, we construct symbols V, possessing a suitable symmetry, such that the operator H V admits an explicit eigenbasis in L 2 ( R 2 ), and calculate the corresponding eigenvalues. Moreover, for V which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of H V adjoining any given Λ q . We find that the effective Hamiltonian in this context is the Toeplitz operator T q ( V ) = p q Op w ( V ) p q , where p q is the orthogonal projection onto Ker ( H 0 − Λ q I ), and investigate its spectral asymptotics.

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