scholarly journals A lower bound for the ground state energy of a Schrödinger operator on a loop

2006 ◽  
Vol 134 (12) ◽  
pp. 3629-3635 ◽  
Author(s):  
Helmut Linde
Keyword(s):  
Lower Bound ◽  
Ground State Energy ◽  
Schrödinger Operator ◽  
Ground State ◽  
State Energy ◽  
Schrodinger Operator

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.

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.

Lower Bound to the Ground-State Energy and Mass Polarization in Helium-Like Atoms

1956 ◽  
Vol 103 (1) ◽  
pp. 112-115 ◽  
Author(s):  
Lawrence Wilets ◽  
Ivan J. Cherry
Keyword(s):  
Lower Bound ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy
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