In a Hilbert space
H
\mathfrak H
, consider a family of selfadjoint operators (a quadratic operator pencil)
A
(
t
)
A(t)
,
t
∈
R
t\in \mathbb {R}
, of the form
A
(
t
)
=
X
(
t
)
∗
X
(
t
)
A(t) = X(t)^* X(t)
, where
X
(
t
)
=
X
0
+
t
X
1
X(t) = X_0 + t X_1
. It is assumed that the point
λ
0
=
0
\lambda _0=0
is an isolated eigenvalue of finite multiplicity for the operator
A
(
0
)
A(0)
. Let
F
(
t
)
F(t)
be the spectral projection of the operator
A
(
t
)
A(t)
for the interval
[
0
,
δ
]
[0,\delta ]
. Approximations for
F
(
t
)
F(t)
and
A
(
t
)
F
(
t
)
A(t)F(t)
for
|
t
|
≤
t
0
|t| \leq t_0
(the so-called threshold approximations) are used to obtain approximations in the operator norm on
H
\mathfrak H
for the operator exponential
exp
(
−
i
τ
A
(
t
)
)
\exp (-i \tau A(t))
,
τ
∈
R
\tau \in \mathbb {R}
. The numbers
δ
\delta
and
t
0
t_0
are controlled explicitly. Next, the behavior for small
ε
>
0
\varepsilon >0
of the operator
exp
(
−
i
ε
−
2
τ
A
(
t
)
)
\exp (-i \varepsilon ^{-2} \tau A(t))
multiplied by the “smoothing factor”
ε
s
(
t
2
+
ε
2
)
−
s
/
2
\varepsilon ^s (t^2 + \varepsilon ^2)^{-s/2}
with a suitable
s
>
0
s>0
is studied. The obtained approximations are given in terms of the spectral characteristics of the operator
A
(
t
)
A(t)
near the lower edge of the spectrum. The results are aimed at application to homogenization of the Schrödinger-type equations with periodic rapidly oscillating coefficients.