Energy bounds for the spiked harmonic oscillator

1995 ◽  
Vol 73 (7-8) ◽  
pp. 493-496 ◽  
Author(s):  
Richard L. Hall ◽  
Nasser Saad
Keyword(s):  
Lower Bound ◽  
Harmonic Oscillator ◽  
Ground State Energy ◽  
Upper Bound ◽  
Ground State ◽  
Trial Function ◽  
State Energy ◽  
Parameter Range ◽  
Complementary Energy ◽  
Energy Bounds

A three-parameter variational trial function is used to determine an upper bound to the ground-state energy of the spiked harmonic-oscillator Hamiltonian [Formula: see text]. The entire parameter range λ > 0 and α ≥ 1 is treated in a single elementary formulation. The method of potential envelopes is also employed to derive a complementary energy lower bound formula valid for all the discrete eigenvalues.

UPPER BOUND ON THE GROUND-STATE ENERGY FOR THE FRÖHLICH POLARON IN A MAGNETIC FIELD

1994 ◽  
Vol 08 (10) ◽  
pp. 629-639 ◽  
Author(s):  
A. V. SOLDATOV
Keyword(s):  
Magnetic Field ◽  
Ground State Energy ◽  
Upper Bound ◽  
Ground State ◽  
Coupling Strength ◽  
State Energy ◽  
Polaron Model

The ground-state energy of the Fröhlich polaron model in a magnetic field is investigated by means of the Wick symbols formalism. The upper bound on the ground-state energy is derived which is valid for all values of magnetic field and coupling strength.

UNIFORM UPPER BOUNDS IN THE FRÖHLICH POLARON THEORY

1993 ◽  
Vol 07 (27) ◽  
pp. 1773-1779 ◽  
Author(s):  
N.N. BOGOLUBOV ◽  
A.V. SOLDATOV
Keyword(s):  
Interaction Parameter ◽  
Ground State Energy ◽  
Upper Bound ◽  
Ground State ◽  
State Energy ◽  
Upper Bounds ◽  
Simple Method ◽  
Polaron Theory

We present a very simple method to derive the upper bound of the ground-state energy for the Fröhlich polaron theory. The obtained bounds are proved to be uniform for all values of the interaction parameter.

scholarly journals Ground-state energy of a low-density Bose gas: A second-order upper bound

2008 ◽  
Vol 78 (5) ◽  
Author(s):  
László Erdős ◽  
Benjamin Schlein ◽  
Horng-Tzer Yau
Keyword(s):  
Ground State Energy ◽  
Upper Bound ◽  
Ground State ◽  
State Energy ◽  
Bose Gas ◽  
Second Order ◽  
Low Density

scholarly journals Remarks on the boundary set of spectral equipartitions

2014 ◽  
Vol 372 (2007) ◽  
pp. 20120492 ◽  
Author(s):  
P. Bérard ◽  
B. Helffer
Keyword(s):  
Riemannian Manifold ◽  
Lower Bound ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Two Dimensional ◽  
Nodal Sets ◽  
Nodal Domains ◽  
Open Set ◽  
Ground State Energies

Given a bounded open set in (or in a Riemannian manifold), and a partition of Ω by k open sets ω j , we consider the quantity , where λ ( ω j ) is the ground state energy of the Dirichlet realization of the Laplacian in ω j . We denote by ℒ k ( Ω ) the infimum of over all k -partitions. A minimal k -partition is a partition that realizes the infimum. Although the analysis of minimal k -partitions is rather standard when k =2 (we find the nodal domains of a second eigenfunction), the analysis for higher values of k becomes non-trivial and quite interesting. Minimal partitions are in particular spectral equipartitions, i.e. the ground state energies λ ( ω j ) are all equal. The purpose of this paper is to revisit various properties of nodal sets, and to explore if they are also true for minimal partitions, or more generally for spectral equipartitions. We prove a lower bound for the length of the boundary set of a partition in the two-dimensional situation. We consider estimates involving the cardinality of the partition.

scholarly journals Extremal eigenvalues of local Hamiltonians

Quantum ◽  
2017 ◽  
Vol 1 ◽  
pp. 6 ◽  
Author(s):  
Aram W. Harrow ◽  
Ashley Montanaro
Keyword(s):  
Ground State Energy ◽  
Constraint Satisfaction ◽  
Upper Bound ◽  
Ground State ◽  
State Energy ◽  
Constraint Satisfaction Problems ◽  
Operator Norm ◽  
Product State ◽  
Classical Algorithm ◽  
Spin Hamiltonians

We apply classical algorithms for approximately solving constraint satisfaction problems to find bounds on extremal eigenvalues of local Hamiltonians. We consider spin Hamiltonians for which we have an upper bound on the number of terms in which each spin participates, and find extensive bounds for the operator norm and ground-state energy of such Hamiltonians under this constraint. In each case the bound is achieved by a product state which can be found efficiently using a classical algorithm.

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