Let
Ω
be a domain in
R
n
,
Γ
be a hyperplane intersecting
Ω
,
ε
>
0
be a small parameter, and
D
k
,
ε
,
k
=
1
,
2
,
3
⋯
be a family of small “holes” in
Γ
∩
Ω
; when
ε
→
0
, the number of holes tends ...to infinity, while their diameters tends to zero. Let
A
ε
be the Neumann Laplacian in the perforated domain
Ω
ε
=
Ω
\
Γ
ε
, where
Γ
ε
=
Γ
\
(
∪
k
D
k
,
ε
)
(“sieve”). It is well-known that if the sizes of holes are carefully chosen,
A
ε
converges in the strong resolvent sense to the Laplacian on
Ω
\
Γ
subject to the so-called
δ
′
-conditions on
Γ
∩
Ω
. In the current work we improve this result: under rather general assumptions on the shapes and locations of the holes we derive estimates on the rate of convergence in terms of
L
2
→
L
2
and
L
2
→
H
1
operator norms; in the latter case a special corrector is required.
Let Γ be an arbitrary Z n -periodic metric graph, which does not coincide with a line. We consider the Hamiltonian H ε on Γ with the action −ɛ −1d2/dx 2 on its edges; here ɛ > 0 is a small parameter. ...Let m ∈ N . We show that under a proper choice of vertex conditions the spectrum σ ( H ε ) of H ε has at least m gaps as ɛ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of σ ( H ε ) as ɛ → 0 can be completely controlled through a suitable choice of coupling constants standing in those vertex conditions. We also show how to ensure for fixed (small enough) ɛ the precise coincidence of the left endpoints of the first m spectral gaps with predefined numbers.
Let ε>0 be a small parameter. We consider the domain Ωε:=Ω∖Dε, where Ω is an open domain in Rn, and Dε is a family of small balls of the radius dε=o(ε) distributed periodically with period ε. Let Δε ...be the Laplace operator in Ωε subject to the Robin condition ∂u∂n+γεu=0 with γε≥0 on the boundary of the holes and the Dirichlet condition on the exterior boundary. Kaizu (1985, 1989) and Brillard (1988) have shown that, under appropriate assumptions on dε and γε, the operator Δε converges in the strong resolvent sense to the sum of the Dirichlet Laplacian in Ω and a constant potential. We improve this result deriving estimates on the rate of convergence in terms of L2→L2 and L2→H1 operator norms. As a byproduct we establish the estimate on the distance between the spectra of the associated operators.
This paper deals with singular Schrödinger operators of the form−d2dx2+∑k∈Zγkδ(⋅−zk),γk∈R, in L2(ℓ−,ℓ+), where δ(⋅−zk) is the Dirac delta-function supported at zk∈(ℓ−,ℓ+) and (ℓ−,ℓ+) is a bounded ...interval. It will be shown that the interaction strengths γk and the points zk can be chosen in such a way that the essential spectrum and a bounded part of the discrete spectrum of this self-adjoint operator coincide with prescribed sets on the real line.
In this expository article some spectral properties of self‐adjoint differential operators are investigated. The main objective is to illustrate and (partly) review how one can construct domains or ...potentials such that the essential or discrete spectrum of a Schrödinger operator of a certain type (e.g. the Neumann Laplacian) coincides with a predefined subset of the real line. Another aim is to emphasize that the spectrum of a differential operator on a bounded domain or bounded interval is not necessarily discrete, that is, eigenvalues of infinite multiplicity, continuous spectrum, and eigenvalues embedded in the continuous spectrum may be present. This unusual spectral effect is, very roughly speaking, caused by (at least) one of the following three reasons: The bounded domain has a rough boundary, the potential is singular, or the boundary condition is nonstandard. In three separate explicit constructions we demonstrate how each of these possibilities leads to a Schrödinger operator with prescribed essential spectrum.
We consider a family {Ωε}ε>0 of periodic domains in R2 with waveguide geometry and analyse spectral properties of the Neumann Laplacian −ΔΩε on Ωε. The waveguide Ωε is a union of a thin straight ...strip of the width ε and a family of small protuberances with the so-called “room-and-passage” geometry. The protuberances are attached periodically, with a period ε, along the strip upper boundary. We prove a (kind of) resolvent convergence of −ΔΩε to a certain operator on the line as ε→0. Also we demonstrate Hausdorff convergence of the spectrum. In particular, we conclude that if the sizes of “passages” are appropriately scaled the first spectral gap of −ΔΩε is determined exclusively by geometric properties of the protuberances. The proofs are carried out using methods of homogenization theory.
Let Δ Ω ε be the Dirichlet Laplacian in the domain Ω ε : = Ω ∖ ( ⋃ i D i ε ). Here Ω ⊂ R n and { D i ε } i is a family of tiny identical holes (“ice pieces”) distributed periodically in R n with ...period ε. We denote by cap ( D i ε ) the capacity of a single hole. It was known for a long time that − Δ Ω ε converges to the operator − Δ Ω + q in strong resolvent sense provided the limit q : = lim ε → 0 cap ( D i ε ) ε − n exists and is finite. In the current contribution we improve this result deriving estimates for the rate of convergence in terms of operator norms. As an application, we establish the uniform convergence of the corresponding semi-groups and (for bounded Ω) an estimate for the difference of the kth eigenvalue of − Δ Ω ε and − Δ Ω ε + q. Our proofs relies on an abstract scheme for studying the convergence of operators in varying Hilbert spaces developed previously by the second author.
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