The periodic Euler-Bernoulli equation
Authors: Papanicolaou Vassilis 
Published: July 2005

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.