A Quantum Dot with Impurity in the Lobachevsky Plane

0.2 MB·English

by  V. Geyler

A Quantum Dot with Impurity in the Lobachevsky
Plane
8
0
0
2 V. Geyler, P. Sˇˇtov´ıˇcek and M. Tuˇsek

n
a
Abstract. The curvature effect on a quantum dot with impurity is investi-
J
gated. The model is considered on the Lobachevsky plane. The confinement
4
and impurity potentials are chosen so that the model is explicitly solvable.

The Green function as well as theKrein Q-function are computed.
]
h
p Keywords. quantum dot,Lobachevsky plane, point interaction, spectrum.
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h
t
a
m
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1. Introduction

3
v Physically, quantum dots are nanostructures with a charge carriers confinement
0
in all space directions. They have an atom-like energy spectrum which can be
9
modified by adjusting geometricparametersof the dots as wellas by the presence
7
2 of an impurity. Thus the study of these dependencies may be of interest from the
. point of view of the nanoscopic physics.
9
0 Adetailedanalysisofthree-dimensionalquantumdotswithashort-rangeim-
7 purityinthe Euclideanspacecanbe foundin[1].Therein,theharmonicoscillator
0 potentialwasusedtointroducetheconfinement,andtheimpuritywasmodeledby
:
v a pointinteraction(δ-potential).The startingpointof the analysiswasderivation
i of a formula for the Green function of the unperturbed Hamiltonian (i.e., in the
X
impurityfreecase),andapplicationoftheKreinresolventformulajointlywiththe
r
a notion of the Krein Q-function.
The current paper is devoted to a similar model in the hyperbolic plane.
The nontrivial hyperbolic geometry attracts regularly attention, and its influence
on the properties of quantum-mechanical systems has been studied on various
models (see, for example, [2, 3, 4]). Here we make use of the same method as in
[1] to investigate a quantum dot with impurity in the Lobachevskyplane.We will
introduce an appropriate Hamiltonian in a manner quite analogous to that of [1]
andderiveanexplicitformulaforthe correspondingGreenfunction.Inthissense,
2 V. Geyler, P. Sˇˇtov´ıˇcek and M. Tuˇsek
our model is solvable, and thus its properties may be of interest also from the
mathematical point of view.
Duringthecomputationstofollow,thespheroidalfunctionsappearnaturally.
Unfortunately, the notation in the literature concerned with this type of special
functions is not yet uniform (see, e.g., [5] and [6]). This is why we supply, for the
reader’s convenience, a short appendix comprising basic definitions and results
related to spheroidal functions which are necessary for our approach.
2. A quantum dot with impurity in the Lobachevsky plane
2.1. The model
Denote by (̺,φ), 0 < ̺ < , 0 φ < 2π, the geodesic polar coordinates on the
∞ ≤
Lobachevsky plane. Then the metric tensor is diagonal and reads
̺
(g )=diag 1,a2sinh2
ij
a
(cid:16) (cid:17)
wherea,0<a< ,denotesthe socalledcurvatureradiuswhichisrelatedtothe
∞
scalarcurvature bythe formulaR= 2/a2.Furthermore,the volumeformequals
−
dV = asinh(̺/a)d̺ dφ. The Hamiltonian for a free particle of mass m = 1/2
∧
takes the form
1 1 ∂ ∂ 1
H0 = ∆ + = √ggij
− LB 4a2 −√g∂xi ∂xj − 4a2
(cid:18) (cid:19)
where ∆ is the Laplace-Beltrami operator and g =detg . We have set ~=1.
LB ij
The choice of a potential modeling the confinement is ambiguous. We nat-
urally require that the potential takes the standard form of the quantum dot
potential in the flat limit (a ). This is to say that, in the limiting case, it
→ ∞
becomes the potential of the isotropic harmonic oscillator V = 1ω2̺2. However,
∞ 4
this condition clearly does not specify the potential uniquely. Having the freedom
of choice let us discuss the following two possibilities:
a) V (̺)= 1a2ω2tanh2 ̺, (2.1)
a 4 a
b) U (̺)= 1a2ω2sinh2 ̺. (2.2)
a 4 a
Potential V is the same as that proposed in [7] for the classical harmonic
a
oscillator on the Lobachevsky plane. With this choice, it has been demonstrated
in [7] that the model is superintegrable, i.e., there exist three functionally inde-
pendent constants of motion. Let us remark that this potential is bounded, and
so it represents a bounded perturbation to the free Hamiltonian. On the other
hand, the potential U is unbounded. Moreover, as shown below, the stationary
a
Schr¨odingerequationforthispotentialleads,afterthepartialwavedecomposition,
tothedifferentialequationofspheroidalfunctions.Thecurrentpaperconcentrates
exclusively on case b).
The impurity is modeled by a δ-potential which is introduced with the aid
of self-adjoint extensions and is determined by boundary conditions at the base
A Quantum Dot with Impurity in the Lobachevsky Plane 3
point. We restrictourselvesto the case when the impurity is located in the center
of the dot (̺=0). Thus we start from the following symmetric operator:
∂2 1 ̺ ∂ 1 ̺ ∂2 1 1 ̺
H = + coth + sinh−2 + + a2ω2sinh2 ,
− ∂̺2 a a ∂̺ a2 a ∂φ2 4a2 4 a
(cid:18) (cid:16) (cid:17) (cid:16) (cid:17) ̺(cid:19) (cid:16) (cid:17)
Dom(H)=C∞((0, ) S1) L2 (0, ) S1,a sinh d̺dφ .
0 ∞ × ⊂ ∞ × a
(cid:16) (cid:16) (cid:17) (cid:17) (2.3)
2.2. Partial wave decomposition
Substituting ξ =cosh(̺/a) we obtain
1 ∂2 ∂ ∂2 a4ω2 1 1
H = (1 ξ2) 2ξ +(1 ξ2)−1 + (ξ2 1) =: H˜,
a2 − ∂ξ2 − ∂ξ − ∂φ2 4 − − 4 a2
(cid:20) (cid:21)
Dom(H)=C∞((1, ) S1) L2 (1, ) S1,a2dξdφ .
0 ∞ × ⊂ ∞ ×
(2.4)
(cid:0) (cid:1)
Using the rotationalsymmetry which amounts to a Fourier transform in the vari-
able φ, H˜ may be decomposed into a direct sum as follows
∞
H˜ = H˜ ,
m
m=−∞
M
∂ ∂ m2 a4ω2 1
H˜ = (ξ2 1) + + (ξ2 1) ,
m −∂ξ − ∂ξ ξ2 1 4 − − 4
(cid:18) (cid:19) −
Dom(H˜ )=C∞(1, ) L2((1, ),dξ).
m 0 ∞ ⊂ ∞
Note that H˜ is a Sturm-Liouville operator.
m
Proposition2.1. H˜ isessentiallyself-adjoint form=0,H˜ hasdeficiencyindices
m 0
6
(1,1).
Proof. The operator H˜ is symmetric and semibounded, and so the deficiency
m
indices are equal. If we set
a4ω2 1
µ= m, 4θ = , λ= z ,
| | − 4 − − 4
then the eigenvalue equation
∂ ∂ m2 a4ω2 1
(ξ2 1) + + (ξ2 1) ψ =zψ (2.5)
−∂ξ − ∂ξ ξ2 1 4 − − 4
(cid:18) (cid:18) (cid:19) − (cid:19)
takes the standard form of the differential equation of spheroidal functions (A.1).
According to chapter 3.12, Satz 5 in [6], for µ = m N a fundamental system
0
∈
y , y of solutions to equation (2.5) exists such that
I II
{ }
y (ξ)=(1 ξ)m/2P (1 ξ), P (0)=1,
I 1 1
− −
y (ξ)=(1 ξ)−m/2P (1 ξ)+A y (ξ)log(1 ξ),
II 2 m I
− − −
4 V. Geyler, P. Sˇˇtov´ıˇcek and M. Tuˇsek
where, for ξ 1 < 2, P ,P are analytic functions in ξ, λ, θ; and A is a
1 2 m
| − |
polynomial in λ and θ of total order m with respect to λ and √θ; A = 1/2.
0
Supposethatz C R.Form=0,everysolutionsto(2.5)issquarein−tegrable
∈ \
near1;while form=0,y is the onlyone solution,up to afactor,whichis square
I
6
integrable in a neighborhood of 1. On the other hand, by a classical analysis due
to Weyl, there exists exactly one linearly independent solution to (2.5) which is
square integrable in a neighborhood of , see Theorem XIII.6.14 in [8]. In the
∞
caseofm=0this obviouslyimplies thatthe deficiencyindices are(1,1).Ifm=0
6
then, by Theorem XIII.2.30 in [8], the operator H˜ is essentially self-adjoint. (cid:3)
m
Define the maximal operator associated to the formal differential expression
∂ ∂ a4ω2 1
L= (ξ2 1) + (ξ2 1)
−∂ξ − ∂ξ 4 − − 4
(cid:18) (cid:19)
as follows
Dom(H )= f L2((1, ),dξ): f,f′ AC((1, )),
max
∈ ∞ ∈ ∞
(cid:26)
∂ ∂f a4ω2
(ξ2 1) + (ξ2 1)f L2((1, ),dξ) ,
− ∂ξ − ∂ξ 4 − ∈ ∞
(cid:18) (cid:19) (cid:27)
H f =Lf.
max
According to Theorem 8.22 in [9], H =H˜†.
max 0
Proposition 2.2. Let κ ( , ]. The operator H˜ (κ) defined by the formulae
0
∈ −∞ ∞
Dom(H˜ (κ))= f Dom(H ): f =κf , H˜ (κ)f =H f,
0 max 1 0 0 max
{ ∈ }
where
f(ξ) 1
f := 4πa2 lim , f := lim f(ξ)+ f log 2a2(ξ 1 ),
0 − ξ→1+ log(2a2(ξ 1)) 1 ξ→1+ 4πa2 0 −
−
(cid:0) (cid:1)
is a self-adjoint extension of H˜ . There are no other self-adjoint extensions of H˜ .
0 0
Proof. The methods to treat δ like potentials are now well established [10]. Here
we follow an approach described in [11], and we refer to this source also for the
terminology and notations. Near the point ξ = 1, each f Dom(H ) has the
max
∈
asymptotic behavior
f(ξ)=f F(ξ,1)+f +o(1) as ξ 1+
0 1
→
where f ,f C and F(ξ,ξ′) is the divergent part of the Green function for the
0 1
∈
Friedrichs extension of H˜ . By formula (2.11) which is derived below, F(ξ,1) =
0
1/(4πa2)log 2a2(ξ 1) . Proposition 1.37 in [11] states that (C,Γ ,Γ ), with
1 2
− −
Γ f =f and Γ f =f , is a boundary triple for H .
1 0 2 1 max
(cid:0) (cid:1)
According to theorem 1.12 in [11], there is a one-to-one correspondence be-
tween all self-adjoint linear relations κ in C and all self-adjoint extensions of H˜
0
A Quantum Dot with Impurity in the Lobachevsky Plane 5
given by κ H˜ (κ) where H˜ (κ) is the restriction of H to the domain of
0 0 max
←→
vectors f Dom(H ) satisfying
max
∈
(Γ f,Γ f) κ. (2.6)
1 2
∈
Everyself-adjointrelationinCisoftheformκ=Cv C2 forsomev R2,v =0.
If (with some abuse of notation) v =(1,κ), κ R, th⊂en relation (2.6)∈means 6that
∈
f =κf . If v =(0,1) then (2.6) means that f =0 which may be identified with
1 0 0
the case κ= . (cid:3)
∞
Remark. Letq betheclosureofthequadraticformassociatedtothesemibounded
0
symmetric operator H˜ . Only the self-adjoint extension H˜ ( ) has the property
0 0
∞
that all functions from its domain have no singularity at the point ξ = 1 and
belongtothe formdomainofq .ItfollowsthatH˜ ( )isthe Friedrichsextension
0 0
∞
of H˜ (see, for example, Theorem X.23 in [12] or Theorems 5.34 and 5.38 in [9]).
0
2.3. The Green function
LetusconsidertheFriedrichsextensionoftheoperatorH˜ inL2 (1, ) S1,dξdφ
∞ ×
which was introduced in (2.4). The resulting self-adjoint operator is in fact the
(cid:0) (cid:1)
Hamiltonian for the impurity free case. The corresponding Green function is
z
G
the generalized kernel of the Hamiltonian, and it should obey the equation
∞
(H˜ z) (ξ,φ;ξ′,φ′)=δ(ξ ξ′)δ(φ φ′)= 1 δ(ξ ξ′)eim(φ−φ′).
z
− G − − 2π −
m=−∞
X
If we suppose to be of the form
z
G
∞
(ξ,φ;ξ′,φ′)= 1 m(ξ,ξ′)eim(φ−φ′), (2.7)
Gz 2π Gz
m=−∞
X
then, for all m Z,
∈
(H˜ z) m(ξ,ξ′)=δ(ξ ξ′). (2.8)
m− Gz −
Let us consider an arbitrary fixed ξ′, and set
a4ω2 1
µ=m, 4θ = , λ= z .
− 4 − − 4
Then for all ξ = ξ′ equation (2.8) takes the standard form of the differential
6
equation of spheroidal functions (A.1). As one can see from (A.8), the solution
which is square integrable near infinity equals S|m|(3)(ξ, a4ω2/16).Furthermore,
ν
−
the solution which is square integrable near ξ = 1 equals Ps|m|(ξ, a4ω2/16) as
ν
−
one may verify with the aid of the asymptotic formula
Γ(ν+m+1)
Pm(ξ) (ξ 1)m/2 as ξ 1+, for m N .
ν ∼ 2m/2m!Γ(ν m+1) − → ∈ 0
−
6 V. Geyler, P. Sˇˇtov´ıˇcek and M. Tuˇsek
We conclude that the mth partial Green function equals
1 a4ω2 a4ω2
m(ξ,ξ′)= Ps|m| ξ , S|m|(3) ξ ,
Gz −(ξ2 1)W(Ps|m|,S|m|(3)) ν < − 16 ν > − 16
ν ν (cid:18) (cid:19) (cid:18) (cid:19)
−
(2.9)
where the symbol W(Ps|m|,S|m|(3)) denotes the Wronskian, and ξ , ξ are re-
ν ν < >
spectively the smaller and the greater of ξ and ξ′. By the generalSturm-Liouville
theory, the factor (ξ2 1)W(Ps|m|,S|m|(3)) is constant. Since m = −m decom-
− ν ν Gz Gz
position (2.7) may be simplified,
∞
1 1
(ξ,φ;ξ′,φ′)= 0(ξ,ξ′)+ m(ξ,ξ′)cos[m(φ φ′)]. (2.10)
Gz 2π Gz π Gz −
m=1
X
2.4. The Krein Q-function
The Krein Q-function plays a crucial role in the spectralanalysis of impurities. It
is defined at a point of the configuration space as the regularized Green function
evaluated at this point. Here we deal with the impurity located in the center of
the dot (ξ =1, φ arbitrary), and so, by definition,
Q(z):= reg(1,0;1,0).
Gz
Due to the rotational symmetry,
1
(ξ):= (ξ,φ;1,0)= (ξ,φ;1,φ)= (ξ,0;1,0)= 0(ξ,1),
Gz Gz Gz Gz 2π Gz
and hence
(H˜ z) (ξ)=0, for ξ (1, ).
0 z
− G ∈ ∞
Letusnotethatfromtheexplicitformula(2.9),onecandeducethatthecoefficients
m(ξ,1) in the series in (2.10) vanish for m = 1,2,3,.... The solution to this
Gz
equation is
a4ω2
(ξ) S0(3) ξ, .
Gz ∝ ν − 16
(cid:18) (cid:19)
The constant of proportionality can be determined with the aid the following
theorem which we reproduce from [13].
Theorem 2.3. Let d(x,y) denote the geodesic distance between points x,y of a
two-dimensional manifold X of bounded geometry. Let
n
U (X):= U : U :=max(U,0) Lp0 (X), U :=max( U,0) Lpi(X)
∈P + ∈ loc − − ∈
n Xi=1 o
for an arbitrary n N and 2 p . Then the Green function of the
i U
∈ ≤ ≤ ∞ G
Schro¨dinger operator H = ∆ +U has the same on-diagonal singularity as
U LB
−
that for the Laplace-Beltrami operator itself, i.e.,
1 1
(ζ;x,y)= log + reg(ζ;x,y)
GU 2π d(x,y) GU
where reg is continuous on X X.
GU ×
A Quantum Dot with Impurity in the Lobachevsky Plane 7
Letusdenoteby H andQH(z)theGreenfunctionandtheKreinQ-function
Gz
forthe FriedrichsextensionofH,respectively.SinceH˜ =a2H and(H˜ z) =δ,
z
− G
we have
GzH(ξ,φ;ξ′,φ′)=a2Ga2z(ξ,φ;ξ′,φ′), QH(z)=a2Q(a2z).
One may verify that
1
logd(̺,0;~0)=log̺=log(aargcoshξ)= log 2a2(ξ 1) +O(ξ 1)
2 − −
(cid:0) (cid:1)
as ̺ 0+ or, equivalently, ξ 1+. Finally, for the divergent part
→ →
F(ξ,ξ′):= (ξ,φ;ξ′,φ) reg(ξ,φ;ξ′,φ)= (ξ,0;ξ′,0) reg(ξ,0;ξ′,0)
Gz −Gz Gz −Gz
of the Green function we obtain the expression
z
G
1
F(ξ,1)= log 2a2(ξ 1) . (2.11)
−4πa2 −
(cid:0) (cid:1)
From the above discussion, it follows that the Krein Q-function depends on the
coefficients α, β in the asymptotic expansion
a4ω2
S0(3) ξ, =αlog(ξ 1)+β+o(1) as ξ 1+, (2.12)
ν − 16 − →
(cid:18) (cid:19)
and equals
β log(2a2)
Q(z)= + . (2.13)
−4πa2α 4πa2
To determine α, β we need relation(A.10) for the radial spheroidalfunction
ofthethirdkind.Forν andν+1/2beingnon-integer,formula(A.12)impliesthat
sin(νπ)
S0(1)(ξ,θ)= e−iπ(ν+1)K0(θ)Qs0 (ξ,θ),
ν π ν −ν−1
(2.14)
sin(νπ)
S0(1) (ξ,θ)= eiπνK0 (θ)Qs0(ξ,θ).
−ν−1 π −ν−1 ν
Applying the symmetry relation (A.5) for expansion coefficients, we derive that
∞
Qs0 (ξ,θ)= ( 1)ra0 (θ)Q0 (ξ)
−ν−1 − −ν−1,r −ν−1+2r
r=−∞
X
∞
= ( 1)ra0 (θ)Q0 (ξ).
− ν,r −ν−1−2r
r=−∞
X
Using the asymptotic formulae (see [5])
1 ξ 1
Q0(ξ)= log − +Ψ(1) Ψ(ν+1)+O((ξ 1)log(ξ 1)),
ν −2 2 − − −
8 V. Geyler, P. Sˇˇtov´ıˇcek and M. Tuˇsek
the series expansion in (A.11) and formulae (2.14), we deduce that, as ξ 1+,
→
sin(νπ)
S0(1)(ξ,θ) e−iπ(ν+1)K0(θ)
ν ∼− π ν
1 ξ 1
s0(θ)−1 log − Ψ(1)+πcot(νπ) +Ψs (θ) ,
× ν 2 2 − ν
(cid:20) (cid:18) (cid:19) (cid:21)
sin(νπ)
S0(1) (ξ,θ) eiπνK0 (θ)
−ν−1 ∼− π −ν−1
1 ξ 1
s0(θ)−1 log − Ψ(1) +Ψs (θ) ,
× ν 2 2 − ν
(cid:20) (cid:18) (cid:19) (cid:21)
where the coefficients sµ(θ) are introduced in (A.7),
n
∞
Ψs (θ):= ( 1)ra0 (θ)Ψ(ν+1+2r),
ν − ν,r
r=−∞
X
and where we have made use of the following relation for the digamma function:
Ψ( z)=Ψ(z+1)+πcot(πz).
−
We conclude that
S0(3)(ξ,θ) αlog(ξ 1)+β+O((ξ 1)log(ξ 1)) as ξ 1+,
ν ∼ − − − →
where
itan(νπ)
α= eiπνK0 (θ) e−iπ(2ν+3/2)K0(θ) ,
2πs0(θ) −ν−1 − ν
ν
(cid:16) (cid:17)
β =α log2 2Ψ(1)+2Ψs (θ)s0(θ) +e−2iπνs0(θ)−1K0(θ).
− − ν ν ν ν
The substitution(cid:0) for α, β into (2.13) yields (cid:1)
1 a4ω2 a4ω2
Q(z)= log2 2Ψ(1)+2Ψs s0
− 4πa2 − − ν − 16 ν − 16
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
1 K0 ( a4ω2) −1 log(2a2) (2.15)
+ eiπ(3ν+3/2) −ν−1 − 16 1 +
2a2tan(νπ) Kν0(−a41ω62) − ! 4πa2
where ν is chosen so that
a4ω2 1
λ0 = z . (2.16)
ν − 16 − − 4
(cid:18) (cid:19)
For ν = n being an integer, we can immediately use the known asymptotic
formulae for spheroidal functions (see Section 16.12 in [5]) which yield
is0(θ) is0(θ)log2
S0(3)(ξ,θ)= n log(ξ 1) n
n 4√θK0(θ) − − 4√θK0(θ)
n n
is0(θ)2 K0(θ)
+ n ( 1)ra0 (θ)h + n +O(ξ 1),
2√θK0(θ) − n,r n+2r s0(θ) −
n 2r≥−n n
X
as ξ 1+. Here, h =1,h =1/1+1/2+...+1/k. By (2.13), one can calculate
0 k
→
the Q-function in this case, too.
A Quantum Dot with Impurity in the Lobachevsky Plane 9
2.5. The spectrum of a quantum dot with impurity
The Green function of the Hamiltonian describing a quantum dot with impurity
is given by the Krein resolvent formula
1
H(χ)(ξ,φ;ξ′,φ′)= H(ξ,φ;ξ′,φ′) H(ξ,0;1,0) H(1,0;ξ′,0)
Gz Gz − QH(z) χGz Gz
−
(recall that, due to the rotational symmetry, H(ξ,φ;1,0) = H(ξ,0;1,0)). The
Gz Gz
parameter χ := a2κ ( , ] determines the corresponding self-adjoint exten-
∈ −∞ ∞
sion H(χ) of H. In the physical interpretation, this parameter is related to the
strength of the δ interaction. Recall that the value χ = corresponds to the
∞
Friedrichs extension of H representing the case with no impurity. This fact is also
apparent from the Krein resolventformula.
The unperturbed Hamiltonian H( ) describes a harmonic oscillator on the
∞
Lobachevsky plane. As is well known (see, for example, [14]), for the confinement
potential tends to infinity as ̺ , the resolvent of H( ) is compact, and the
→ ∞ ∞
spectrum of H( ) is discrete and semibounded. The eigenvalues of H( ) are
∞ ∞
solutions of a scalar equation whose introduction also relies heavily on the theory
ofspheroidalfunctions.We arescepticaboutthepossibilityofderivinganexplicit
formula for the eigenvalues.But the equationturned outto be convenientenough
to allow for numerical solutions. A more detailed discussion jointly with a basic
numerical analysis is provided in a separate paper [15].
A similar observation about the basic spectral properties (discreteness and
semiboundedness) is also true for the operators H(χ) for any χ R since, by the
∈
Krein resolvent formula, the resolvents for H(χ) and H( ) differ by a rank one
∞
operator.Moreover,themultiplicitiesofeigenvaluesofH(χ)andH( )maydiffer
∞
at most by 1 (see [9, Section 8.3]).
±
Amoredetailedandrathergeneralanalysiswhichisgivenin[1]canbecarried
overtoourcasealmostliterally.DenotebyσthesetofpolesofthefunctionQH(z)
depending on the spectral parameter z. Note that σ is a subset of spec(H( )).
∞
Consider the equation
QH(z)=χ. (2.17)
Theorem2.4. ThespectrumofH(χ)isdiscreteandconsistsoffournonintersecting
parts S , S , S , S described as follows:
1 2 3 4
1. S is the set of all solutions to equation (2.17) which do not belong to the
1
spectrum of H( ). The multiplicity of all these eigenvalues in the spectrum
∞
of H(χ) equals 1.
2. S is the set of all λ σ that are multiple eigenvalues of H( ). If the mul-
2
∈ ∞
tiplicity of such an eigenvalue λ in spec(H( )) equals k then its multiplicity
∞
in the spectrum of H(χ) equals k 1.
−
3. S consistsofallλ spec(H( )) σthatarenotsolutionstoequation(2.17).
3
∈ ∞ \
the multiplicities of such an eigenvalue λ in spec(H( )) and spec(H(χ)) are
∞
equal.
10 V. Geyler, P. Sˇˇtov´ıˇcek and M. Tuˇsek
4. S consists of all λ spec(H( )) σ that are solutions to equation (2.17).
4
∈ ∞ \
If the multiplicity of such an eigenvalue λ in spec(H( )) equals k then its
∞
multiplicity in the spectrum of H(χ) equals k+1.
HencetheeigenvaluesofH(χ),χ R,differentfromthoseoftheunperturbed
∈
Hamiltonian H( ) are solutions to (2.17). As far as we see it, this equation can
∞
be solvedonly numerically.We havepostponedasystematicnumericalanalysisof
equation (2.17) to a subsequent work.Note that the Krein Q-function (2.15) is in
fact a function of ν, and hence dependence (2.16) of the spectral parameter z on
ν is fundamental. In this context, it is quite useful to know for which values of ν
the spectral parameter z is real. A partial answer is given by Proposition A.1.
3. Conclusion
We have proposed a Hamiltonian describing a quantum dot in the Lobachevsky
plane to which we added an impurity modeled by a δ potential. Formulas for the
corresponding Q- and Green functions have been derived. Further analysis of the
energy spectrum may be accomplished for some concrete values of the involved
parameters (by which we mean the curvature a and the oscillator frequency ω)
with the aid of numerical methods.
Appendix: Spheroidal functions
Here we follow the source [5]. Spheroidal functions are solutions to the equation
∂2ψ ∂ψ
(1 ξ2) 2ξ + λ+4θ(1 ξ2) µ2(1 ξ2)−1 ψ =0, (A.1)
− ∂ξ2 − ∂ξ − − −
(cid:2) (cid:3)
where all parameters are in general complex numbers. There are two solutions
thatbehavelikeξν times asingle-valuedfunctionandξ−ν−1 times asingle-valued
function at . The exponent ν is a function of λ, θ, µ, and is called the charac-
∞
teristic exponent. Usually, it is more convenient to regard λ as a function of ν, µ
and θ. We shall write λ = λµ(θ). If ν or µ is an integer we denote it by n or m,
ν
respectively. The functions λµ(θ) obey the symmetry relations
ν
λµ(θ)=λ−µ(θ)=λµ (θ)=λ−µ (θ). (A.2)
ν ν −ν−1 −ν−1
A first group of solutions (radial spheroidal functions) is obtained as expan-
sions in series of Bessel functions,
∞
Sµ(j)(ξ,θ)=(1 ξ−2)−µ/2sµ(θ) aµ ψ(j) (2θ1/2ξ), (A.3)
ν − ν ν,r ν+2r
r=−∞
X