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Abstract
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We study the infimum of the spectrum, or ground state
energy (g.s.e.), of a discrete Schrödinger operator on
parametrized by a
potential
and a
frequency parameter
.
We relate this g.s.e. to that of a corresponding continuous semiclassical Schrödinger operator
on
with
parameter
,
arising from the same choice of potential. We show that:
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The discrete g.s.e. is at most the continuous one for continuous periodic
and irrational
.
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The opposite inequality holds up to a factor of
as
for sufficiently regular smooth periodic
.
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The opposite inequality holds up to a constant factor for every bounded
and
with the property that discrete and continuous averages of
on fundamental domains of
are comparable.
Our proofs are elementary and rely on sampling and interpolation
to map low-energy functions for the discrete operator on
to low-energy functions for the continuous operator on
, and
vice versa.
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Keywords
semiclassical Schrödinger operator, discrete Schrödinger
operator, spectrum, periodic potential, ground state energy
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Mathematical Subject Classification
Primary: 47A10
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Milestones
Received: 9 May 2024
Revised: 28 June 2024
Accepted: 6 August 2024
Published: 15 October 2024
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© 2024 MSP (Mathematical Sciences
Publishers). |
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