Acta Polytechnica


Acta Polytechnica 53(3):271–279, 2013 © Czech Technical University in Prague, 2013
available online at http://ctn.cvut.cz/ap/

SPECTRAL ANALYSIS OF SCHRÖDINGER OPERATORS WITH
UNUSUAL SEMICLASSICAL BEHAVIOR

Pavel Exnera,b, Diana Barseghyana,b,∗

a Doppler Institute for Mathematical Physics and Applied Mathematics, Břehová 7, 11519 Prague
b Nuclear Physics Institute ASCR, 25068 Řež near Prague, Czechia
∗ corresponding author: dianabar@ujf.cas.cz

Abstract. In this paper we discuss several examples of Schrödinger operators describing a particle
confined to a region with thin cusp-shaped ‘channels’, given either by a potential or by a Dirichlet
boundary; we focus on cases when the allowed phase space is infinite but the operator still has a discrete
spectrum. First we analyze two-dimensional operators with the potential |xy|p − λ(x2 + y2)p/(p+2)
where p ≥ 1 and λ ≥ 0. We show that there is a critical value of λ such that the spectrum for λ < λcrit
is below bounded and purely discrete, while for λ > λcrit it is unbounded from below. In the subcritical
case we prove upper and lower bounds for the eigenvalue sums. The second part of work is devoted to
estimates of eigenvalue moments for Dirichlet Laplacians and Schrödinger operators in regions having
infinite cusps which are geometrically nontrivial being either curved or twisted; we are going to show
how these geometric properties enter the eigenvalue bounds.

Keywords: Schrödinger operator, discrete spectrum, Lieb-Thirring inequality, cusp-shaped regions,
geometrically induced spectrum.

1. Introduction
The semiclassical method for analyzing the operator has proved itself a tremendously useful tool over the century
since it was proposed by Hermann Weyl. Nevertheless, there are cases when estimates based on phase-space
volume fail; the classical example is due to B. Simon [1] and describes a two-dimensional Schrödinger operator
with the potential |xy|p having deep ‘channels’ the width of which is shrinking with the distance from the origin.

The present paper is devoted to a discussion of several models of this type. It summarizes the presentation
of the second named author at the conference Analytic and algebraic methods in physics X (Prague, 2012) based
on the original papers [3, 4] to which we refer for details of the proofs which are sketched here.

Our first aim is to show that the effects known from the paper [1] can occur even if the potential is unbounded
from below; at the same time the model will exhibit a parameter transition between different spectral regimes.
One has to add that the first person to draw attention to the possibility of finding a discrete and below bounded
spectrum in a below unbounded potential was to our knowledge M. Znojil, who analyzed a related model in [2].
In the second part we will discuss Schödinger operators and Dirichlet Laplacians on cusp-shaped regions which
are geometrically nontrivial, being either bent or twisted, and show how their geometry is reflected in spectral
properties.

2. A model with potential unbounded from below and infinite phase
space

We are going to consider here the following class of operators,

Lp(λ) : Lp(λ)ψ = −∆ψ +
(
|xy|p −λ(x2 + y2)p/(p+2)

)
ψ , p ≥ 1 , (2.1)

on L2(R2) where x,y are Cartesian coordinates in R2. The parameter λ in the second term of the potential is
assumed to be non-negative; unless its value is important in a particular context we write simply Lp. Since
2p
p+2 < 2 the operator (2.1) is e.s.a. on C

∞
0 (R2) by the Faris-Lavine theorem — see [5, Theorems X.28, X.38]; in

the following we mean by the symbol Lp or Lp(λ) always its closure.
We are going to demonstrate the existence of a critical value of the coupling constant λ, expressed explicitly

in terms of the ground-state eigenvalue of the corresponding (an)harmonic oscillator Hamiltonian, such that the
spectrum of Lp(λ) is below bounded and purely discrete for λ < λcrit, while for λ > λcrit it becomes unbounded
from below. In the subcritical case we shall present upper and lower bounds to the sums of the first N eigenvalues
of Lp(λ).

271

http://ctn.cvut.cz/ap/


P. Exner, D. Barseghyan Acta Polytechnica

2.1. Discreteness of the spectrum
Let us first look at small values of λ. To speak quantitatively we need an auxiliary operator which will be an
(an)harmonic oscillator Hamiltonian H̃p : H̃pu = −u′′ + |t|pu on the natural domain in L2(R). Let γp be the
minimal eigenvalue of this operator; in view of the mirror symmetry we have γp = inf σ(Hp), where

Hp : Hpu = −u′′ + tpu (2.2)

on the natural domain in L2(R+) with the Neumann condition at the origin. It is well known that the quantity
γp depends smoothly on p being equal to one for p = 2 and tending to γ∞ = 14π

2 as p →∞; a numerical analysis
performed in [3] shows that the function p 7→ γp is convex and γp > 0.99 for any p ≥ 1.

Theorem 2.1. For any λ ∈ [0,λcrit), where λcrit := γp, the operator Lp(λ) with p ≥ 1 is bounded from below
and its spectrum is purely discrete.

Sketch of the proof. Fix first λ < γp. By the minimax principle we need to estimate Lp from below by a
self-adjoint operator with a purely discrete spectrum. To this aim we employ a suitable bracketing imposing
additional Neumann conditions at concentric circles of radii n = 1, 2, . . . . Using the polar coordinates, we get a
direct sum of operators acting as

L(1)n,pψ = −
1
r

∂

∂r

(
r
∂ψ

∂r

)
−

1
n2
∂2ψ

∂ϕ2
+
(r2p

2p
|sin 2ϕ|p −λr2p/(p+2)

)
ψ,

∂ψ

∂n

∣∣∣
r=n−1

=
∂ψ

∂n

∣∣∣
r=n

= 0, (2.3)

on the regions Gn :=
{

(r,ϕ) : n− 1 ≤ r < n, 0 ≤ ϕ < 2π
}
, n = 1, 2, . . . . Each of these annuli is compact and

the potential is regular on it, hence σ
(
L

(1)
n,p

)
is purely discrete. It thus suffices to check that inf σ

(
L

(1)
n,p

)
→∞ as

n →∞, because the spectrum of
⊕∞

n=1 L
(1)
n,p below any fixed value will then be purely discrete. We estimate

L
(1)
n,p from below by an operator with separated variables,

L(2)n,pψ = −
1
r

∂

∂r

(
r
∂ψ

∂r

)
−

1
n2
∂2ψ

∂ϕ2
+
((n− 1)2p

2p
|sin 2ϕ|p −λn2p/(p+2)

)
ψ,

∂ψ

∂n

∣∣∣
r=n−1

=
∂ψ

∂n

∣∣∣
r=n

= 0,

and establish that inf σ
(
L

(3)
n,p

)
→∞ as n →∞, where L(3)n,p is the angular part of L

(2)
n,p. The spectrum of L

(2)
n,p

is the ‘sum’ of the radial and angular component and the lowest radial eigenvalue is zero corresponding to a
constant eigenfunction. One the other hand, the angular part behaves as the anharmonic oscillator around the
potential minima, and the corresponding eigenvalue prevails over the negative λ-dependent term [3]. In this way
one gets inf σ

(
L

(2)
n,p

)
→∞ as n →∞, which proves by the minimax principle the same for operator L(1)n,p.

2.2. The supercritical case
For large λ the spectral behaviour is different.

Theorem 2.2. σ
(
Lp(λ)

)
, p ≥ 1, is unbounded from below if λ > λcrit.

Sketch of the proof. We use a similar technique, this time looking for an upper bound to Lp(λ) obtained by
Dirichlet bracketing: we consider the operators L̃(1)n,p acting as (2.3) on the annular domains Gn with Dirichlet
boundary conditions. The latter give a contribution of order O(1) as n →∞ and since the λ-dependent term
now prevails we conclude [3] that

inf σ
(
L̃(1)n,p

)
→−∞ as n →∞ ,

which means that σ
(
Lp(λ)

)
is unbounded from below.

2.3. Lower bounds to eigenvalue sums
Next we will show how the eigenvalue sums of operator Lp(λ) can be estimated for small values of λ. We
introduce the number

α :=
1
40
(
5 +
√

105
)2 ≈ 5.81;

it is clear that α−1 < γp. We denote by {λj,p}∞j=1 the eigenvalues of Lp(λ) arranged in ascending order; then we
have the following result.

Theorem 2.3. To any nonnegative λ < α−1 ≈ 0.172 there is a positive Cp depending on p only such that
N∑
j=1

λj,p ≥ Cp
(
1 −αλ

) N(2p+1)/(p+1)
(lnp N + 1)1/(p+1)

− cλN, N = 1, 2, . . . , (2.4)

where c = 2
(
α2

5 + 1
)
≈ 15.51.

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vol. 53 no. 3/2013 Spectral Analysis of Schrödinger Operators

Sketch of the proof. We denote by {ψj,p}∞j=1 the system of normalized eigenfunctions corresponding to {λj,p}
∞
j=1,

−∆ψj,p +
(
|xy|p −λ(x2 + y2)p)/(p+2)

)
ψj,p = λj,pψj,p, j = 1, 2, . . . ;

without loss of generality we may assume these functions to be real-valued. Our potential form hyperbolic-shaped
‘channels’ and we have first to estimate eigenfunction integrals in the corresponding parts of the plane. We
establish [3] that for any natural j and a δ > 0 one has

∫ ∞
1

∫ (1+δ)y−p/(p+2)
0

y2p/(p+2)ψ2j,p(x,y) dx dy

≤
5
2

(1 + δ)2
∫ ∞

1

∫ ∞
0

(∂ψj,p
∂x

)2
(x,y) dx dy + 2

1 + δ
δ

∫ ∞
1

∫ (1+δ)y−p/(p+2)
0

xpypψ2j,p(x,y) dx dy

and that for an arbitrary ε > 0 there is a number θ(ε) ∈ [1, 1 + δ] such that
∫ ∞

1
yp/(p+2)ψ2j,p

( θ(ε)
yp/(p+2)

,y
)

dy <
1
δ

∫ ∞
1

∫ (1+δ)y−p/(p+2)
y−p/(p+2)

xpypψ2j,p(x,y) dx dy + ε.

Using the potential symmetry we get an analogous bound for the other ‘channels’. Using next the normalization
‖ψj,p‖ = 1 and the mentioned estimates we find∫

R2
(x2 + y2)

p
p+2 ψ2j,p(x,y) dx dy ≤

(∫
|y|≥1

∫
|x|≤(1+δ)|y|−p/(p+2)

|y|2p/(p+2)ψ2j,p(x,y) dx dy

+
∫
|y|≥1

∫
|x|>(1+δ)|y|−p/(p+2)

|y|2p/(p+2)ψ2j,p(x,y) dx dy +
∫
|x|≥1

∫
|y|≤(1+δ)|x|−p/(p+2)

|x|2p/(p+2)ψ2j,p(x,y) dy dx

+
∫
|x|≥1

∫
(1+δ)|x|−p/(p+2)<|y|<1

|x|2p/(p+2)ψ2j,p(x,y) dy dx
)

+ 2

≤ (1 + δ) max
{5

2
(1 + δ),

2
δ

}(∫
R2
|∇ψj,p|

2 (x,y) dx dy +
∫
R2
|xy|pψ2j,p(x,y) dx dy + (1 + δ)

2
)

+ 2,

where we have used the inequality |xy|p > |y|2p/(p+2) valid on the domain of the second of the four integrals and
an analogous bound for the fourth integral; the factor (1 + δ)2 prevents double counting the ‘corner regions’
with |x|, |y| ≥ 1 and |y| ≤ (1 + δ)|x|−p/(p+2). Choosing δ = −5+

√
105

10 we arrive at∫
R2

(x2 + y2)
p
p+2 ψ2j,p(x,y) dx dy ≤ α

(∫
R2
|∇ψj,p||2(x,y) dx dy +

∫
R2
|xy|pψ2j,p(x,y) dx dy

)
+ c,

where c := α(1 + δ)2 + 2 = 2
(
α2

5 + 1
)
. Since λj,p is the eigenvalue corresponding to the eigenfunction ψj,p the

inequality derived above implies∫
R2
|∇ψj,p|2 dx dy +

∫
R2
|xy|pψ2j,p dx dy ≤

1
1 −αλ

(
λj,p + cλ

)
, j = 1, 2, . . . .

Next we use Plancherel’s theorem to express the gradients of ψj,p in the first integral and then apply the
following Lieb-Thirring-type inequality to the resulting orthonormal series:

Lemma 2.4. There is a constant C′p such that for any orthonormal system of real-valued function, Φ =
{ϕj}Nj=1 ⊂ L

2(R2), N = 1, 2, . . . , the inequality

∫
R2
ρ
p+1
Φ dx dy ≤ C

′
p(ln

p N + 1)
N∑
j=1

∫
R2
|ξη|p|ϕ̂j|2 dξ dη ,

holds true, where ρΦ :=
∑N
j=1 ϕ

2
j.

This claim was proved as Theorem 2 in [6] for p = 1 and it is straightforward to extend the argument to any
p ≥ 1. After a few simple manipulations for any non-negative parameter % we get

C′′p (1 + ln
p N)1/p%(2p+1)/p ≥ N%−

1
1 −αλ

N∑
j=1

(λj,p + cλ)

273



P. Exner, D. Barseghyan Acta Polytechnica

with a new positive constant C′′p .
In the final step we consider

g̃(N) = max
%≥0

(
N%−C′′p%

(2p+1)/p(1 + lnp N)1/p
)
.

Denoting now the expression in the bracket as h(%) we check easily that reaches its maximum at the point

%max =
(
p/(2p + 1)C′′p

)p/(p+1)
Np/(p+1)(1 + lnp N)−1/(p+1)

and its value there equals

g̃(N) = h(%max) = Cp
N(2p+1)/(p+1)

(1 + lnp N)1/(p+1)

with some constant Cp > 0. In this way we arrive at the bound

1
1 −αpλ

N∑
j=1

(λj,p + cpλ) ≥ h (%max) = g̃(N).

which is equivalent to the claim of the theorem.

2.4. Upper bounds
Next we estimate spectral sums of Lp(λ), p ≥ 1, from above for any subcritical λ. It will show, in particular,
that in the case λ = 0 the asymptotics given by Theorem 2.3 is exact up the value of the constant.

Theorem 2.5. To any p ≥ 1 there is a constant C̃p > 0 such that

N∑
j=1

λj,p ≤ C̃p
N(2p+1)/(p+1)

(1 + lnp N)1/(p+1)
, N = 1, 2, . . .

holds for any 0 ≤ λ < γp.

Sketch of the proof. It clearly suffices to prove the claim for λ = 0. Consider the operator Ĥp = −∆ + Q, where
Q(x,y) = |xy|p + |x|p + |y|p + 1 in L2(R2). Its spectrum is discrete by Theorem 2.1 in combination with the
minimax principle since Hp ≥ Lp, thus we have to establish the bound of the indicated type for the eigenvalues
0 ≤ β1,p ≤ β2,p ≤ ··· of the estimating operator Ĥp.

We employ Weyl asymptotics for the number of eigenvalues of below-bounded differential operators in a
version due to G. Rozenblum. Let T = −∆ + V in Rm, where the potential V (x) ≥ 1 and tends to infinity as
|x|→∞. We denote by E(λ,V ) the set

{
x ∈ Rm : V (x) < λ

}
and put

σ(λ,V ) = mes E(λ,V ).

For any unit cube D ⊂ Rm we denote the mean value of the function V in D by VD. Furthermore, given a
function f ∈ L1(D) and t ≤

√
m we define its L1-modulus of continuity by the formula

ω1(f,t,D) := sup
|z|<t

∫
x∈D,x+z∈D

∣∣f(x + z) −f(x)∣∣ dx.
Then we have the following result [7]:

Lemma 2.6. Suppose that the potential V satisfies the conditions:
(1.) There is c > 0 such that σ(2λ,V ) ≤ cσ(λ,V ) holds all λ large enough.
(2.) V (y) ≤ cV (x) holds if as |x−y| ≤ 1.
(3.) There is a continuous and monotonous function η : t ∈ [0,

√
m] → R+ with η(0) = 0 and a number

β ∈ [0, 12 ) such that for any unit square D we have

ω1(V,t,D) ≤ η(t) t2β V
1+β
D .

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vol. 53 no. 3/2013 Spectral Analysis of Schrödinger Operators

Under these assumptions the asymptotic formula

N(λ,V ) ∼ ΓmΦ(λ,V )

holds for the operator T = −4 + V , where N(λ,V ) is the number of eigenvalues of T smaller than λ, unit-ball
volume Γm = (2

√
π)−m

(
Γ(m2 + 1)

)−1, and
Φ(λ,V ) =

∫
R2

(λ−V )m/2+ dx dy .

It is straightforward to check the hypotheses of the lemma for the operator Ĥp, which gives an asymptotic
formula for spectral distribution function N(λ,Ĥp). After some simple estimates [3] we arrive at the following
upper bound on the spectrum of the operator Ĥp,

N∑
j=1

βj,p ≤ C̃p
N(2p+1)/(p+1)

(ln N + 1)p/(p+1)

with a constant C̃p depending on p only; this proves the theorem.

3. Schrödinger operators in cusps with non-trivial geometry
In the second part of the paper we shall discuss Schrödinger type operators

HΩ = −∆ΩD −V (3.1)

with a bounded measurable potential V ≥ 0 on L2(Ω), where −∆ΩD is the Dirichlet Laplacian on a region
Ω ⊂ Rd. In the spirit of the previous considerations we will be particularly interested in situations where Ω
is unbounded but HΩ still has a purely discrete spectrum. It is well known [1, 9] that for some unbounded
regions the spectrum may be purely discrete; typically this happens if Ω has cusps. The negative spectrum
of HΩ consists of a finite number of eigenvalues counted with their multiplicities; it is natural to ask about
bounds on the negative spectrum moments in terms of their geometrical properties, in the spirit the seminal
work of Lieb and Thirring [10], or in the present context referring to Berezin, Lieb, Li and Yau [11–14]. Note
that estimates of this type have been derived recently in [5] for various cusped regions; a typical example is
Ω =

{
(x,y) ∈ R2 : |xy| < 1

}
with hyperbolic ends. Our goal is to discuss situations when such infinite cusps of

Ω are geometrically nontrivial, being curved or twisted, to see in which way the geometry influences the spectral
estimates.

First we will discuss a curved planar cusp and derive estimates on negative spectrum moments which include
a curvature-induced potential. This result can be generalized for bent cusps with a circular cross section; we
refer to [4] for a description. Then, in Section 3.2, we will consider cusps of a non-circular cross section in R3
which are straight but twisted. The geometry of the region will again be involved in the obtained eigenvalue
estimates, now in a different way than for curved cusps, because the effective interaction associated with twisting
is repulsive rather than attractive.

3.1. Curved planar cusps
We consider an infinite cusp-shaped Ω ⊂ R2 assuming that its boundary is smooth, so one can describe it by
specifying its axis and the cusp width at each point of it. This will allow us to employ natural curvilinear
coordinates by analogy with the theory of quantum waveguides [15] and to ‘straighten’ the cusp, translating its
geometric properties into the coefficients of the resulting operator.

To be specific, we characterize our region by three functions: sufficiently smooth a,b : R → R2 and a positive
and continuous f : R → R+ in such a way that

Ω :=
{(
a(s) −uḃ(s),b(s) + uȧ(s)

)
: s ∈ R, |u| < f(s)

}
, (3.2)

where the dot marks the derivative with respect to s; to make the region Ω cusp-shaped we shall always suppose
that

lim
|s|→∞

f(s) = 0. (3.3)

Since the reference curve Γ =
{(
a(s),b(s)

)
: s ∈ R

}
can always be parametrized by its arc length we may

suppose without loss of generality that ȧ(s)2 + ḃ(s)2 = 1 and s is the arc length. The signed curvature γ(s) of Γ
is then given by

γ(s) = ḃ(s)ä(s) − ȧ(s)b̈(s);

275



P. Exner, D. Barseghyan Acta Polytechnica

knowing this one can reconstruct the functions a,b describing the Cartesian coordinates of the cusps axis,
modulo Euclidean transformations.

Under the condition (3.3) the region is quasi-bounded so it may have a purely discrete spectrum. This is
indeed the case; recall that the necessary and sufficient condition for the purely discrete spectral character [16,
Thm 2.8] is that we can cover Ω by a family of unit balls whose centres tend to infinity in such a way that the
volumes of their intersections with Ω tend to zero; it is not difficult to construct such a ball sequence if (3.3) is
valid.

Our main aim is to provide bounds on the eigenvalue moments; as usual when dealing with a Lieb-Thirring
type problem we restrict our attention to the negative part of the spectrum noting that it can be always made
non-empty by including a suitable constant into the potential.

Theorem 3.1. Consider the Schrödinger operator (3.1) on the region (3.2). Suppose that the curvature γ ∈ C4,
the inequality

∥∥f(·)γ(·)∥∥
L∞(R) < 1 holds true, and Ω does not intersect itself. Then for any σ ≥ 3/2 we have the

estimate

tr(HΩ)σ− ≤
∥∥1 + f|γ|∥∥−2σ∞ Lclσ,1

∫
R

∞∑
j=1

(
−
( πj

2f(s)

)2
+
∥∥1 + f|γ|∥∥2∞W−(s) + ∥∥1 + f|γ|∥∥2∞∥∥Ṽ (s, ·)∥∥∞

)σ+1/2
+

ds,

(3.4)
where ‖·‖∞ ≡‖·‖L∞(R) and Lclσ,1 is the semiclassical constant,

Lclσ,1 :=
Γ(σ + 1)

√
4π Γ(σ + 32 )

, (3.5)

and furthermore, we have introduced

W−(s) :=
γ(s)2

4
(
1 −f(s)|γ(s)|

)2 + f(s)
∣∣γ̈(s)∣∣

2
(
1 −f(s)|γ(s)|

)3 + 5f2(s)γ̇(s)24(1 −f(s)|γ(s)|)4
and Ṽ (s,u) := V

(
a(s) −uḃ(s),b(s) + uȧ(s)

)
.

Sketch of the proof. Using the mentioned ‘straightening’ transformation [15] we infer that HΩ is unitarily
equivalent to the operator H0 on L2(Ω0) acting as

(H0ψ)(s,u) = −
∂

∂s

( 1
(1 + uγ(s))2

∂ψ

∂s
(s,u)

)
−
∂2ψ

∂u2
(s,u) +

(
(W − Ṽ )ψ

)
(s,u),

where Ω0 = {(s,u) : s ∈ R, |u| < f(s)}, the curvature-induced potential is

W(s,u) := −
γ2(s)

4
(
1 + uγ(s)

)2 + uγ̈(s)2(1 + uγ(s))3 − 54 u
2γ̇2(s)(

1 + uγ(s)
)4

and Dirichlet boundary conditions are imposed at u = ±f(s). In view of the unitary equivalence it is enough to
establish inequality (3.4) for the operator H0. We employ the minimax principle: consider the operator H−0
defined on the domain H20(Ω0) in L2(Ω0) by

H−0 = −∆
Ω0
D −

∥∥1 + f|γ|∥∥2∞(W− + Ṽ ),
where −∆Ω0D is as usual the corresponding Dirichlet Laplacian; it is obvious that

H0 ≥
∥∥1 + f|γ|∥∥−2∞ H−0 (3.6)

holds true, therefore it is sufficient to get the upper bound for negative eigenvalue moments of operator H−0 .
We use a variational argument — see [17] — to estimate the negative eigenvalue moments of H−0 by the

negative eigenvalue moments of the other operator with the operator-valued potential defined on domain
H1
(
R,L2(R)

)
and given as follows

−
∂2

∂s2
⊗ IL2(R) + H

(
s, Ṽ ,W−

)
,

where H
(
s, Ṽ ,W−

)
is the negative part of the Sturm-Liouville operator

−
d2

du2
−
∥∥1 + f|γ|∥∥2∞(W− + Ṽ ).

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vol. 53 no. 3/2013 Spectral Analysis of Schrödinger Operators

Consequently,

tr (H−0 )
σ
− ≤ tr

(
−
∂2

∂ s2
⊗ IL2(R) + H

(
s, Ṽ ,W−

))σ
−

holds for any nonnegative number σ. This makes it possible to employ the version of Lieb-Thirring inequality
for operator-valued potentials [18] for operator-valued potentials, which yields

tr(H−0 )
σ
− ≤ L

cl
σ,1

∫
R

tr
(
H
(
s, Ṽ ,W−

))σ+1/2
−

ds, σ ≥ 3/2, (3.7)

with the semiclassical constant Lclσ,1. It remains to estimate the negative spectrum of the Sturm-Liouville
operator − d

2

du2 −
∥∥1 + f|γ|∥∥2∞(W−(s) + ∥∥Ṽ (s, ·)∥∥∞) which is easily done,(( πj

2f(s)

)2
−
∥∥1 + f|γ|∥∥2∞W−(s) −∥∥1 + f|γ|∥∥2∞∥∥Ṽ (s, ·)∥∥∞

)
−
;

hence in view of (3.6) and (3.7) we find that

tr(H0)σ− ≤
∥∥1 + f|γ|∥∥−2σ∞ Lclσ,1

∫
R

∞∑
j=1

(
−
( πj

2f(s)

)2
+
∥∥1 + f|γ|∥∥2∞W−(s) + ∥∥1 + f|γ|∥∥2∞∥∥Ṽ (s, ·)∥∥∞

)σ+1/2
+

ds,

which proves the theorem.

We finish this section with two remarks. First we note that while the standard phase-space-volume estimates
give correct high-energy behaviour one can find finite regions Ω for which there exists an intermediate energy
region where the bound (3.4) is much stronger than the Berezin-Li-Yau inequality. An example is given in
paper [4], to which we refer also for the mentioned generalization of Theorem 3.1 to higher dimensions.

3.2. Twisted cusps of non-circular cross section in R3
Let us now look at another type of nontrivial cusp geometry. As before we will suppose that its cross section
changes along the curve playing role of the axis, however, now we allow it to be non-circular. Consider an open
connected set ω0 ⊂ R2 and a positive function f : R → R satisfying the condition (3.3), and set

ωs := f(s)ω0 , (3.8)

where we use the conventional shorthand αA :=
{

(αx,αy) : (x,y) ∈ A
}
for α > 0 and A ⊂ R2. Using (3.8) we

define a straight cusped region determined by ω0 and the function f as Ω0 :=
{

(s,x,y) : s ∈ R, (x,y) ∈ ωs
}
.

Next we twist the region Ω0. We fix a C1-smooth function θ : R → R with a bounded derivative,
∥∥θ̇∥∥∞ < ∞,

and introduce the set Ωθ as the image
Ωθ := Lθ(Ω0), (3.9)

where the map Lθ : R3 → R3 is given by

Lθ(s,x,y) :=
(
s,x cos θ(s) + y sin θ(s),−x sin θ(s) + y cos θ(s)

)
. (3.10)

We are interested primarily in nontrivial situations, assuming that
(1.) the function θ is not constant ,
(2.) ω0 is not rotationally symmetric with respect to the origin in R2.

To formulate the result of this section, we need a few more preliminaries. First of all, we introduce % :=
sup(x,y)∈ω0

√
x2 + y2 and assume that

%
∥∥f(·)θ̇(·)∥∥∞ < 1. (3.11)

Next we set Ṽ (s,x,y) := V
(
Lθ(s,x,y)

)
by analogy with the corresponding definitions in the previous sections,

and finally, we introduce the operator

Ltrans := −i
(
x
∂

∂y
−y

∂

∂x

)
, Dom(Ltrans) = H10(ω0),

describing the angular momentum component canonically associated with rotations in the transverse plane. We
have the following claim:

277



P. Exner, D. Barseghyan Acta Polytechnica

Theorem 3.2. Let HΩθ be the operator (3.1) referring to the region Ωθ defined by (3.9) and (3.10) with a
potential V ≥ 0 which is bounded and measurable. Under the assumption (3.11) for the negative spectrum of
HΩθ the inequality

tr
(
H

Ωθ
D

)σ
− ≤ L

cl
σ,1
(
1 −%

∥∥fθ̇∥∥∞)σ
∫
R

∞∑
j=1

(
−
λ0,j(s)
f2(s)

+
∥∥Ṽ ∥∥∞

1 −%
∥∥fθ̇∥∥∞

)σ+1/2
+

ds

holds true for σ ≥ 3/2, where Lclσ,1 is the constant (3.5) and λ0,j(s), j = 1, 2, . . . , are the eigenvalues of the
operator

Hf,θ(s) := −∆ω0D + f
2(s)θ̇2(s)L2trans

defined on the domain H20(ω0) in L2(ω0).

Sketch of the proof. As before, we employ suitable curvilinear coordinates, this time to ‘untwist’ the region. We
define a unitary operator from L2(Ωθ) to L2(Ω0) by Uθψ := ψ ◦ Lθ which allows us to pass from HDΩθ to the
operator

H0 := Uθ
(
HDΩθ

)
U−1θ

in L2(Ω0). From paper [19] we know that H0 is the self-adjoint operator associated with the quadratic form

Q0 : Q0[ψ] :=
∥∥∂sψ + iθ̇Ltransψ∥∥2 + ∥∥∇′ψ∥∥2 −∫

Ω0

(
Ṽ |ψ|2

)
(s,x,y) ds dx dy

defined on H10, where ∇′ := (∂x,∂y) is the transversal gradient and the norms refer to L2(Ω0). For any function
ψ ∈ H10 (ωs) we have

|Ltransψ| ≤ %f(s)|∇′ψ|

and applying the Cauchy-Schwarz inequality we infer

2
∣∣∣∣
∫

Ω0
θ̇∂sψLtransψ ds dx dy

∣∣∣∣ ≤ %∥∥fθ̇∥∥∞(∥∥∂sψ∥∥2L2(Ω0) + ∥∥∇′ψ∥∥2L2(Ω0)),
thus Q0[ψ] can be estimated from below by(

1 −%
∥∥fθ̇∥∥∞)(∥∥∇ψ∥∥2L2(Ω0) + ∥∥θ̇Ltransψ∥∥2L2(Ω0))−

∫
Ω0

∥∥Ṽ (s, ·)∥∥∞|ψ|2 ds dx dy.
We introduce the operator

H−0 = −∆
Ω0
D + θ̇

2L2trans −
1

1 −%
∥∥fθ̇∥∥∞

∥∥Ṽ (s, ·)∥∥∞
defined on H20 (ω0), then the above estimate implies

H0 ≥
(

1 −%
∥∥fθ̇∥∥∞)H−0 , (3.12)

hence by the minimax principle and the condition (3.11) it is enough to establish the upper estimate for the
negative spectrum of operator H−0 .

Using again the variational technique one can estimate the negative eigenvalue moments of operator H−0
by the moments of the operator with separated variables, − ∂

2

∂ s2
⊗ IL2(R2) + H(s, Ṽ ), defined on the domain

H1
(
R,L2(R2)

)
, i.e.

tr
(
H−0

)σ
− ≤ tr

(
−
∂2

∂s2
⊗ IL2(R2) + H(s, Ṽ )

)σ
−
, σ ≥ 0.

In view of the Lieb-Thirring inequality for operator valued potentials this implies

tr
(
H−0

)σ
− ≤ L

cl
σ,1

∫
R

tr H
(
s, Ṽ

)σ+1/2
− d s, for any σ ≥ 3/2 (3.13)

with the semiclassical constant Lclσ,1. Then, by virtue of unitary equivalence of operators H
Ωθ
D and H0, inequalities

(3.12), (3.13) and the condition (3.11) we get

tr
(
H

Ωθ
D

)σ
− ≤ L

cl
σ,1

(
1 −%

∥∥fθ̇∥∥∞)σ
∫
R

tr H
(
s, Ṽ

)σ+1/2
− ds for σ ≥ 3/2. (3.14)

278



vol. 53 no. 3/2013 Spectral Analysis of Schrödinger Operators

It is easy to see that the eigenvalues of operator H(s, Ṽ ) are(λ0,j(s)
f2(s)

−
1

1 −%
∥∥fθ̇∥∥∞

∥∥Ṽ (s, ·)∥∥∞)−, j = 1, 2, . . . , (3.15)
where λ0,j(s), j = 1, 2, . . . are eigenvalues of the operator

Hf,θ(s) := −∆ω0D + f
2(s)θ̇2(s)L2trans.

From inequalities (3.14) and (3.15) it follows that tr
(
H

Ωθ
D

)σ
− is estimated by

Lclσ,1

(
1 −%

∥∥fθ̇∥∥∞)σ
∫
R

∞∑
j=1

(
−
λ0,j(s)
f2(s)

+
1

1 −%
∥∥fθ̇∥∥∞

∥∥Ṽ (s, ·)∥∥∞)σ+1/2+ ds,
for any σ ≥ 3/2 which proves the theorem.

Acknowledgements
The research presented here was supported by the Czech Science Foundation within project P203/11/0701. The authors
are grateful to the referees for reading the manuscript carefully and pointing out some minor slips of the pen.

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279

http://arxiv.org/abs/1203.2098

	Acta Polytechnica 53(3):271–279, 2013
	1 Introduction
	2 A model with potential unbounded from below and infinite phase space
	2.1 Discreteness of the spectrum
	2.2 The supercritical case
	2.3 Lower bounds to eigenvalue sums
	2.4 Upper bounds

	3 Schrödinger operators in cusps with non-trivial geometry
	3.1 Curved planar cusps
	3.2 Twisted cusps of non-circular cross section in Rˆ3

	Acknowledgements
	References