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We prove error estimates of optimal order for linear Schrödinger-type equations in the
(
)- and the
(
)-norm. We discretize only in time by the Crank-Nicolson method. The direct use of the reconstruction technique, as it has been proposed by Akrivis in [ 75 (2006) 511–531], leads to upper bounds that are of optimal order in the
(
)-norm, but of suboptimal order in the
(
...
We prove error estimates of optimal order for linear
Schrödinger-type equations in the
(
)- and the
(
)-norm. We discretize only in time by the
Crank-Nicolson method. The direct use of the reconstruction
technique, as it has been proposed by Akrivis in [
(2006) 511–531], leads to upper bounds that
are of optimal order in the
(
)-norm, but of
suboptimal order in the
(
...
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