We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem
\[
[
a
(
x
)
u
′
′
(
x
)
]
′
′
=
λ
ρ
(
x
)
u
(
x
)
,
−
∞
>
x
>
∞
,
\left [ a(x)u^{\prime \prime }(x)\right ] ^{\prime \prime }=\lambda \rho (x)u(x),\qquad -\infty >x>\infty ,
\]
where the functions
a
a
and
ρ
\rho
are periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described by
a
a
and
ρ
\rho
. Here we develop a theory analogous to the theory of the Hill operator
−
(
d
/
d
x
)
2
+
q
(
x
)
-(d/dx)^2+q(x)
. We first review some facts and notions from our previous works, including the concept of the pseudospectrum, or
ψ
\psi
-spectrum. Our new analysis begins with a detailed study of the zeros of the function
F
(
λ
;
k
)
F(\lambda ;k)
, for any given “quasimomentum”
k
∈
C
k\in \mathbb {C}
, where
F
(
λ
;
k
)
=
0
F(\lambda ;k)=0
is the Floquet-Bloch variety of the beam equation (the Hill quantity corresponding to
F
(
λ
;
k
)
F(\lambda ;k)
is
Δ
(
λ
)
−
2
cos
(
k
b
)
\Delta (\lambda )-2\cos (kb)
, where
Δ
(
λ
)
\Delta (\lambda )
is the discriminant and
b
b
the period of
q
q
). We show that the multiplicity
m
(
λ
∗
)
m(\lambda ^{\ast })
of any zero
λ
∗
\lambda ^{\ast }
of
F
(
λ
;
k
)
F(\lambda ;k)
can be one or two and
m
(
λ
∗
)
=
2
m(\lambda ^{\ast })=2
(for some
k
k
) if and only if
λ
∗
\lambda ^{\ast }
is also a zero of another entire function
D
(
λ
)
D(\lambda )
, independent of
k
k
. Furthermore, we show that
D
(
λ
)
D(\lambda )
has exactly one zero in each gap of the spectrum and two zeros (counting multiplicities) in each
ψ
\psi
-gap. If
λ
∗
\lambda ^{\ast }
is a double zero of
F
(
λ
;
k
)
F(\lambda ;k)
, it may happen that there is only one Floquet solution with quasimomentum
k
k
; thus, there are exceptional cases where the algebraic and geometric multiplicities do not agree. Next we show that if
(
α
,
β
)
(\alpha ,\beta )
is an open
ψ
\psi
-gap of the pseudospectrum (i.e.,
α
>
β
\alpha >\beta
), then the Floquet matrix
T
(
λ
)
T(\lambda )
has a specific Jordan anomaly at
λ
=
α
\lambda =\alpha
and
λ
=
β
\lambda =\beta
. We then introduce a multipoint (Dirichlet-type) eigenvalue problem which is the analogue of the Dirichlet problem for the Hill equation. We denote by
{
μ
n
}
n
∈
Z
\{\mu _n\}_{n\in \mathbb {Z}}
the eigenvalues of this multipoint problem and show that
{
μ
n
}
n
∈
Z
\{\mu _n\}_{n\in \mathbb {Z}}
is also characterized as the set of values of
λ
\lambda
for which there is a proper Floquet solution
f
(
x
;
λ
)
f(x;\lambda )
such that
f
(
0
;
λ
)
=
0
f(0;\lambda )=0
. We also show (Theorem 7) that each gap of the
L
2
(
R
)
L^{2}(\mathbb {R})
-spectrum contains exactly one
μ
n
\mu _{n}
and each
ψ
\psi
-gap of the pseudospectrum contains exactly two
μ
n
\mu _{n}
’s, counting multiplicities. Here when we say “gap” or “
ψ
\psi
-gap” we also include the endpoints (so that when two consecutive bands or
ψ
\psi
-bands touch, the in-between collapsed gap, or
ψ
\psi
-gap, is a point). We believe that
{
μ
n
}
n
∈
Z
\{\mu _{n}\}_{n\in \mathbb {Z}}
can be used to formulate the associated inverse spectral problem. As an application of Theorem 7, we show that if
ν
∗
\nu ^{*}
is a collapsed (“closed”)
ψ
\psi
-gap, then the Floquet matrix
T
(
ν
∗
)
T(\nu ^{*})
is diagonalizable. Some of the above results were conjectured in our previous works. However, our conjecture that if all the
ψ
\psi
-gaps are closed, then the beam operator is the square of a second-order (Hill-type) operator, is still open.