On the existence of impurity bound excitons in one-dimensional systems with zero range interactions
Jonas Have, Hynek Kovarik, Thomas G. Pedersen, Horia D. Cornean
OOn the existence of impurity bound excitons in one-dimensionalsystems with zero range interactions
Jonas Have , Hynek Kovařík , Thomas G. Pedersen , and Horia D. Cornean Department of Mathematical Sciences, Aalborg University Department of Physics and Nanotechnology, Aalborg University DICATAM, Sezione di Matematica, Università degli studi di Brescia
Abstract
We consider a three-body one-dimensional Schrödinger operator with zero range potentials, whichmodels a positive impurity with charge κ > interacting with an exciton. We study the existenceof discrete eigenvalues as κ is varied. On one hand, we show that for sufficiently small κ there existsa unique bound state whose binding energy behaves like κ , and we explicitly compute its leadingcoefficient. On the other hand, if κ is larger than some critical value then the system has no boundstates. In this paper we consider a system of three one-dimensional non-relativistic quantum particles with zerorange interactions. The system models an impurity interacting with an exciton, which is a pair made ofan electron and a hole in either a semiconductor or an insulator. We want to give a rigorous descriptionof the existence of bound states in the cases where the impurity has either a small or a large charge. Inthe small charge case we prove the existence of a non-degenerate groundstate, we explicitly compute itsleading order behavior and compare it to numerical calculations. In the case of a large impurity chargewe prove the existence of a critical charge above which the discrete spectrum is absent, and we computeit numerically. The proofs of our main results are based on a combined application of the Feshbachinversion formula and the Birman-Schwinger principle.The bound states of a helium like system with two negatively charged particles and a positivelycharged nucleus interaction through zero range potentials were previously examined in [1] and in [2],while the bound states of a system with a negatively charged particle and two positively charged particleswith infinite mass were examined in [3]. Also, the spectral properties of the similar, but more realistic,three-body Coulomb systems in three dimensions have been examined in [4, 5, 6].The choice of Coulomb interaction potential in one-dimensional systems is a non-trivial one. TheSchrödinger operator for the one-dimensional hydrogen atom with the / | x | Coulomb potential is notessentially self-adjoint but has an infinite number of self-adjoint extensions, and the choice of extensionand corresponding spectral properties are still the subject of active research[7, 8, 9]. Other options areto modify the Coulomb potential to get rid of the singularity[10] or use zero range interactions, as usedin the present paper. One-dimensional systems and zero range interactions might seem unphysical, butin many cases they can be used as toy models in order to avoid complicated numerical computations. Infact some three-dimensional Coulomb systems and one-dimensional systems with zero range interactionsshare important spectral properties. A classical example is the analogy between the one-dimensionalhydrogen atom and the true three-dimensional hydrogen atom as described in [11].Also, such simplified models naturally emerge as effective models for higher-dimensional systemssubmitted to various forms of confinement, like for example the one-dimensional effective models forexcitons in carbon nanotubes in [12, 13] , one-dimensional models of optical response in one-dimensionalsemiconductors in [14], and the effective model for atoms in strong magnetic fields in [15, 16]. In asimilar fashion, the system we consider in this paper can be interpreted as a model for impurity boundexcitons in a one-dimensional semiconductor using the Wannier model. Excitonic effects are knownto have a significant impact on the optical properties of semiconductors[17], especially in one- and two-dimensional semiconductors where the reduced screening leads to large exciton binding energies compared1 a r X i v : . [ m a t h - ph ] M a y o the bandgap. For a thorough introduction to systems with zero range potentials we refer to the bookin [18].The paper is structured as follows. In Section 2 we present the model and comment on the main resultsof the paper. In Section 3 we specify the framework and introduce some important notation. In Section4 we prove our first main result, namely that there exists a single discrete eigenvalue for sufficiently smallimpurity charge. In Sections 5 and 6 we prove our second main result about the disappearance of thediscrete spectrum if κ becomes supercritical. Consider the system of two equal but oppositely charged particles with charge ± and mass m , and animpurity with charge κ and mass M . Let σ = m/ ( m + M ) denote the mass fraction, ≤ σ < . Usingrelative atomic coordinates and removing its center of the mass, the system is formally described by theSchrödinger operator H κ,σ = −
12 ∆ − σ∂ x ∂ y − δ ( x − y ) + κδ ( x ) − κδ ( y ) , (2.1)on L ( R ) , where ∆ is the two-dimensional Laplace operator, δ is the Dirac delta distribution.The discrete spectrum of H κ,σ corresponds to impurity localized excitons. In the following we stateour results regarding the discrete spectrum of H κ,σ and prove them in Secs. 4, 5, and 6. The situationis sketched in Figure 1a where we see the ground state energy and the essential spectrum for σ = 0 .The essential spectrum of H κ, will be derived in Section 3, but as illustrated by the shaded area in thefigure, its bottom stays equal to − / on the closed interval [0 , / √ , while for larger κ it equals − κ / .The first result concerns the existence and behaviour of a discrete eigenvalue of H κ, =: H κ when κ ∈ (0 , / √ . Theorem 2.1. If κ > is sufficiently small, the operator H κ has precisely one discrete eigenvalue andits leading order behaviour is: E ( κ ) = − − (cid:18) π − (cid:19) κ + O ( κ ) . (2.2) Furthermore, the energy E ( κ ) is non-degenerate and decreasing if κ ∈ (0 , / √ , hence the operator H κ has at least one discrete eigenvalue on this interval. The behaviour κ of the leading order of E ( κ ) (for κ sufficiently small) equals the weak couplingasymptotic of the ground state energy of one-dimensional Schrödinger operators with zero-mean poten-tials as was shown in [19]. Also, the binding requirement (that κ should be sufficiently small) is similarto one of the two binding requirements that were found in [5] for the three-dimensional Coulomb system.In Figure 1b the leading behavior of the discrete eigenvalue given in Theorem 2.1 is compared toa numerical calculation of the smallest discrete eigenvalue of H κ . The numerical calculations are doneusing a similar method to what was presented in [1]. The figure shows that they agree well for κ below . .The results can be generalized to hold for < σ < as well. If κ is sufficiently small the operator H κ,σ has a single discrete eigenvalue, and the leading behavior of this discrete eigenvalue E is calculatedto be E ( κ ) = − − σ ) − β ( σ ) κ + O ( κ ) , where β ( σ ) := 4 (cid:104) σ √ − σ − (2 − σ ) σπ − σ cos − (cid:16) √ σ √ (cid:17) + tan − (cid:16) σ (1 − σ )1 − σ (cid:17)(cid:105) (1 + σ )(1 − σ ) π σ (2.3)when < σ < / √ . The solution can be extended to the range / √ ≤ σ < by choosing anotherbranch of tan − .For κ ≥ / √ we have the following results. Theorem 2.2.
Let H κ, ˜ κ be the self-adjoint operator formally described by H κ, ˜ κ = −
12 ∆ − δ ( x − y ) + ˜ κδ ( x ) − κδ ( y ) . (2.4)2 / √ κ c − − . − − . κ σ ess ( H κ ) E ( κ ) (a) . . . . − . − . − . − . − . − . κ Numerical simulationLeading term (b)
Figure 1: In (a) a plot of the ground state energy is given as a function of the impurity charge κ . Figure(b) is a comparison of the leading term of the discrete eigenvalue, and the numerically calculated discreteeigenvalue. on L ( R ) . Given any ˜ κ > there exists κ M such that H κ, ˜ κ has no discrete eigenvalues for all κ ≥ κ M .Furthermore, given any < ˜ κ < there exists some κ M such that H κ, ˜ κ has at least one discrete eigenvaluefor all κ ≥ κ M . As a consequence of the previous two theorems we will also prove the following corollary:
Corollary 2.3.
Let H κ be the operator in (2.1) . Then there exists a critical charge of the impurity,which we will denote κ c , such that the discrete spectrum of H κ is non-empty for all < κ < κ c andempty for κ ≥ κ c . Using numerical simulations to calculate the smallest discrete eigenvalue of H κ we see that at κ ≈ . the ground state energy hits the essential spectrum. Thus, we expect that the true κ c is close to . . In Figure 2 a numerical calculation of the critical charge κ c is plotted against the mass fraction σ . We see that as the mass of the impurity decreases the critical charge is increased, and thus boundstates exists for impurities with larger charges. We have also plotted the coefficient β in (2.3) againstthe mass fraction, and we see that the coefficient decreases as the mass of the impurity decreases. . . . . . . . . . . . . Mass fraction - σ C r i t i c a l C h a r g e - κ c . . . . . C o e ffi c i e n t - β Figure 2: Plot of the critical charge and the κ coefficient of the discrete eigenvalue against the massfraction σ . 3 The Framework
In this section we introduce the framework we use to study the discrete spectrum of H κ,σ . This frameworkhas been used in [20, 2], and we refer to those papers for more details.We define H κ,σ as the unique self-adjoint operator associated to the sesqui-linear form Q ( f, g ) = 12 (cid:104)∇ f, A ∇ g (cid:105) L ( R ) − (cid:104) f ( x, x ) , g ( x, x ) (cid:105) L ( R ) + κ (cid:104) f (0 , y ) , g (0 , y ) (cid:105) L ( R ) − κ (cid:104) f ( x, , g ( x, (cid:105) L ( R ) , (3.1)on H ( R ) × H ( R ) , where H ( R ) is the Sobolev space of first order and A ∈ R × is the matrix A = (cid:20) σσ (cid:21) . (3.2)Let ψ ∈ H ( R ) and let e ∈ R be a unit vector. We define the trace operator τ e : H ( R ) → L ( R ) by ( τ e ψ )( s ) := ψ ( se ) . Let us write τ := ( τ e , τ e , τ e ) as an operator defined on H ( R ) with values in [ L ( R )] := ⊕ i =1 L ( R ) , where { e , e } is the canonical basis in R and e = 1 / √ e + e ) . Then H κ,σ is H κ,σ = −
12 ∆ − σ ∂ ∂x∂y + τ ∗ gτ, (3.3)where g := diag {− κ, κ, − } ∈ R × .As a direct application of the Hunziker - Van Winter - Zhislin (HVZ) theorem[21] and a consequenceof the signs of the potential terms in (2.1) the following lemma holds. Lemma 3.1.
The essential spectrum of H κ,σ is (cid:20) min (cid:26) − − σ ) , − κ (cid:27) , ∞ (cid:19) . The essential spectrum of H κ, is illustrated by the shaded area in Figure 1a. Write the operator in(2.1) as H κ,σ = H ,σ − V κ , where H ,σ := −
12 ∆ − σ ∂ ∂x∂y , V κ := δ ( x − y ) − κδ ( x ) + κδ ( y ) . If R ( z ) denotes the full resolvent operator ( H κ,σ − z ) − and R ( z ) denotes the resolvent ( H ,σ − z ) − ,then by Krein’s formula R ( z ) = R ( z ) − R ( z ) τ ∗ ( g − + τ R ( z ) τ ∗ ) − τ R ( z ) , z ∈ ρ ( H ,σ ) ∩ ρ ( H κ,σ ) . (3.4)Define: G κ,σ ( z ) := g − + τ R ( z ) τ ∗ . (3.5)It can be shown that E < inf σ ess ( H κ,σ ) belongs to the discrete spectrum of H κ if and only if G κ,σ ( E ) is not invertible. Note that G κ,σ ( z ) is a × operator valued matrix which acts on [ L ( R )] and itsentries are z dependent. We will denote the elements of τ R ( z ) τ ∗ by τ R ( z ) τ ∗ = T ,σ T ,σ T ∗ ,σ T ,σ T ,σ T ∗ ,σ T ,σ T ,σ T ,σ . (3.6)The integral kernel of R ( z ) is R ( x , y , z ) = 12 π (cid:90) R e i k · ( x − y ) | k | + 2 σk k − z d k d k . (3.7)Using the integral kernel of R ( z ) in the Fourier representation we can explicitly calculate the integralkernels of the elements in τ R ( z ) τ ∗ (the first and the the last operators are multiplication operators in4ourier space): ˆ T ,σ ( s ) = 1 (cid:112) (1 − σ ) s − z (3.8) ˆ T ,σ ( s, t ) = 1 π s + t + 2 σst − z (3.9) ˆ T ,σ ( s, t ) = 1 π t + ( s − t ) + 2 σt ( s − t ) − z (3.10) ˆ T ,σ ( s ) = 1 (cid:112) (1 − σ ) s − (1 − σ )4 z . (3.11)From these expressions it is easy to see that the operators in (3.6) are bounded if Re( z ) < , and theirnorms go to zero when Re( z ) → −∞ . We are now ready to prove the first of our main results, i.e. the existence of a single discrete eigenvalueof H κ = H κ, when κ becomes sufficiently small. In the following we will also denote T i, by T i .Assume that κ < / √ . In that case it follows from Lemma 3.1 that any discrete eigenvalues E ∈ R must satisfy E < − / . Moreover, E is a discrete eigenvalue of H κ if and only if the operator G κ ( E ) isnot invertible. Define ˜ G κ ( E ) := κ − G κ ( E ) for κ > , then ˜ G κ ( E ) is invertible when G κ ( E ) is invertible.In matrix representation we can write ˜ G κ ( E ) as ˜ G κ ( E ) = −
00 0 0 + κ T T T ∗ T T T ∗ T T − + T , (4.1)where denotes the identity operator on L ( R ) . In order to find the values E where the inverse of ˜ G κ ( E ) does not exist, we use Feshbach’s formula (see Equations (6.1) and (6.2) in [22]) to reduce the dimensionof the operator pencil we are trying to invert.Let Π be the orthogonal projection such that Π[ L ( R )] is isomorphic to L ( R ) , and Π ˜ G κ ( E )Π ∼ = − κ + κT . The congruence symbol simply means that Π ˜ G κ ( E )Π can be identified with − κ + κT on L ( R ) . Let Π ⊥ := − Π correspond to the orthogonal subspace Π ⊥ [ L ( R )] which is isomorphic to [ L ( R )] . Then, we get Π ⊥ ˜ G κ ( E )Π ⊥ ∼ = (cid:20) − (cid:21) + κ (cid:20) T T T T (cid:21) . (4.2)The next Lemma gives conditions under which the inverse of Π ⊥ ˜ G κ ( E )Π ⊥ exists as an operator on Π ⊥ [ L ( R )] . Lemma 4.1.
There exists
K > such that R ( E ) := [Π ⊥ ˜ G κ ( E )Π ⊥ ] − exists in Π ⊥ [ L ( R )] for all E < − / and < κ < K .Proof. We rewrite Π ⊥ ˜ G κ ( E )Π ⊥ as Π ⊥ ˜ G κ ( E )Π ⊥ ∼ = (cid:18)(cid:20) (cid:21) + κ (cid:20) T T T T (cid:21) (cid:20) − (cid:21)(cid:19) (cid:20) − (cid:21) . (4.3)The operators T and T are uniformly bounded on L ( R ) for E < − / . Thus, we can choose a constant K > such that (cid:13)(cid:13)(cid:13)(cid:13) κ (cid:20) T T T T (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) < , (4.4)for all E < − / and < κ < K . Then the inverse Π ⊥ ˜ G κ ( E )Π ⊥ exists for all < κ < K and E < − / .Additionally, we can write R ( E ) as a Neumann series R ( E ) ∼ = (cid:20) − (cid:21) + ∞ (cid:88) j =1 ( − j κ j (cid:20) − (cid:21) (cid:18)(cid:20) T T T T (cid:21) (cid:20) − (cid:21)(cid:19) j , (4.5)for all < κ < K . 5y Feshbach’s formula and Lemma 4.1 there exists K sufficiently small such that if < κ < K and E < − / , the inverse of ˜ G κ ( E ) exists if and and only if the inverse of S W ( E ) = Π ˜ G κ ( E )Π − Π ˜ G κ ( E )Π ⊥ R ( E )Π ⊥ ˜ G κ ( E )Π , (4.6)exists as an operator restricted to the proper subspace. Using the matrix representation we can write S W ( E ) as S W ( E ) ∼ = − T − κ (cid:2) T T (cid:3) R ( E ) (cid:20) T ∗ T ∗ (cid:21) , on L ( R ) . (4.7)Note that the contribution to S W ( E ) from the first term of R ( E ) in (4.5) is zero. To find the values wherethe inverse of S W ( E ) does not exist on L ( R ) we use the following version of the Birman-Schwinger[23]principle. Proposition 4.2.
Let
E < − / and let S W ( E ) be given by (4.7) . There exist two bounded operators V : [ L ( R )] → L ( R ) and V : L ( R ) → [ L ( R )] such that S W ( E ) − exists if and only if the inverse of − κV ( − T ) − V (4.8) exists on [ L ( R )] , where is the identity operator on [ L ( R )] . We call the operator in (4.8) for theBirman-Schwinger operator.Proof. Let Ψ ∈ L ( R ) and define V : L ( R ) → [ L ( R )] as V Ψ = R ( E ) (cid:20) T ∗ Ψ T ∗ Ψ (cid:21) . (4.9)By the boundedness of R ( E ) and T ∗ it follows that V is a bounded operator. Furthermore, let Ψ =[Ψ , Ψ ] ∈ [ L ( R )] and define the operator V : [ L ( R )] → L ( R ) by V Ψ = T Ψ + T Ψ . (4.10)The operator V is bounded since T is bounded. Using V and V it is possible to rewrite the operator S W ( E ) on L ( R ) as S W ( E ) = − T − κV V = ( − κV V ( − T ) − )( − T ) , since the bounded inverse of − T exists on L ( R ) for all E < − / . Consequently S W ( E ) exists ifand only if ( − κV V (1 − T ) − ) − exists on L ( R ) . But for any fixed κ we can choose E sufficientlynegative such that (cid:107) κV V ( − T ) − (cid:107) < and we can expand in a Neumann series ( − κV V ( − T ) − ) − = (cid:88) j =0 κ j [ V V ( − T ) − ] j . Using resummation, we obtain that if E is sufficiently negative we have S W ( E ) − = ( − T ) − + κ ( − T ) − V (cid:0) − κV ( − T ) − V (cid:1) − V ( − T ) − . (4.11)Both the left-hand and the right-hand side define meromorphic functions for Re( E ) < − / , hence wecan use the right-hand side to extend S W ( E ) − everywhere where the Birman-Schwinger operator exists.This proves one implication.Conversely, if we define A := κV ( − T ) − V , equation (4.11) implies: κV S W ( E ) − V = A + A ( − A ) − A = − + ( − A ) − or ( − A ) − = + κV S W ( E ) − V . (4.12)Now we can extend ( − A ) − using the right-hand side. This concludes the proof.6et V and V be as in the above proof. Then the discrete eigenvalues E of H κ for < κ < K arethose E < − / for which the inverse of the Birman-Schwinger operator (4.8) does not exist on [ L ( R )] .In Fourier representation the operator ( − T ) − is given by multiplication with √ π (cid:18) − √ s − E (cid:19) − = 1 √ π s − E − √ π + 1 √ π √ s − E + 1 . (4.13)The first term on the right hand side has a singularity at E = − / . As κ becomes small any possiblediscrete eigenvalues will be close to − / , and thus we expect the singular term to be the significantcontribution. To simplify notation we define ε := − E − > . Taking the Fourier transform of eachterm on the right-hand side of (4.13) we get the integral kernel of ( − T ) − : ( − T ) − ( x, y ) = 1 √ ε e −√ ε | x − y | + δ ( x − y ) + 12 π (cid:90) R e is ( x − y ) √ s + ε + 1 + 1 d s = 1 √ ε − (cid:90) | x − y | e −√ εs d s + δ ( x − y ) + 12 π (cid:90) R e is ( x − y ) √ s + ε + 1 + 1 d s. (4.14)From (4.14) we see that there are four contributions to V ( − T ) − V . We will show that the operatorsthat we get from the three last terms in (4.14) are uniformly bounded for ε > . Only the second termmay pose problems due to its linear growth, while the third term is the distribution kernel of the identityoperator and the fourth term is multiplication by a uniformly bounded function in Fourier space for ε > .We show that the operator corresponding to the second term is uniformly bounded. By the con-struction of V and V the contribution that might be problematic is the operator with the integralkernel ≤ C ( x, y ) = (cid:90) R T ∗ ( x, t ) (cid:32)(cid:90) | t − t (cid:48) | e −√ εs d s (cid:33) T ( t (cid:48) , y ) d t d t (cid:48) , (4.15)since the other factors from V and V are bounded. We will show that C ( x, y ) is the integral kernel of aHilbert-Schmidt operator. To do that, we need the following result which is based on the Paley-Wienertheorem[24]. Lemma 4.3.
There exists α > sufficiently small such that the kernels T ( x, y ) e α | x | , T ∗ ( x, y ) e α | y | , T ( x, y ) e α | y | and T ( x, y ) e α | x | are in L ( R ) uniformly in ε > .Proof. We will show that T ( x, y ) e α | x | ∈ L ( R ) using the Paley-Wiener theorem. The proofs for theother integral kernels are similar and therefore not included. To apply Paley-Wiener we must show that ˆ T ( s, t ) can be analytically continued to a subset of the type { ξ ∈ C : | Im( ξ ) | < a } ⊂ C , for some a > . Write s = s + is and t = t + it , with s , s , t , t ∈ R , then ˆ T ( s, t ) = 1 π t − t + 2 it t + ( s − t ) − ( s − t ) + 2 i ( s − t )( s − t ) + ε + 1 . This function has no poles for t and s satisfying t + ( s − t ) < , and is analytic on the subset (cid:26) ξ ∈ C : | Im( ξ ) | < (cid:27) ⊂ C . Take η = ( s , t ) ∈ R such that | η | < / and define δ := ε + 1 − t − ( s − t ) . By the choice of η weget δ > , and the norm (cid:107) ˆ T ( · + iη ) (cid:107) L ( R ) ≤ π (cid:90) R t + ( s − t ) + δ ) d s d t = 1 πδ < ∞ . Thus, (cid:107) ˆ T , ( · + iη ) (cid:107) L ( R ) < ∞ for all such η ∈ R . Then the Paley-Wiener theorem implies that e α √ x + y T ( x, y ) ∈ L ( R ) for all α < / . This concludes the proof of T ( x, y ) e α | x | ∈ L ( R ) .7e are now ready to show that C ( x, y ) is an integral kernel of a Hilbert-Schmidt operator. To dothat we use the following inequality C ( x, y ) ≤ (cid:90) R T ∗ ( x, t ) | t | T ( t (cid:48) , y ) d t d t (cid:48) + (cid:90) R T ∗ ( x, t ) | t (cid:48) | T ( t (cid:48) , y ) d t d t (cid:48) , (4.16)which follows from the definition of C ( x, y ) and the inequality (cid:90) | t − t (cid:48) | e −√ εs d s ≤ | t − t (cid:48) | ≤ | t | + | t (cid:48) | . We will show that the last the term in (4.16) is in L ( R ) (the proof that the first term is also in L ( R ) is identical). Note that the integral is separable and (cid:90) R T ∗ ( x, t ) | t (cid:48) | T ( t (cid:48) , y ) d t d t (cid:48) =: F ( x ) G ( y ) . We will show that
F, G ∈ L ( R ) . Applying the Cauchy-Schwarz inequality with respect to the t -integraland using Lemma 4.3 we find (cid:107) F (cid:107) L ( R ) = (cid:90) R (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R T ∗ ( x, t ) d t (cid:12)(cid:12)(cid:12)(cid:12) d x = (cid:90) R (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R T ∗ ( x, t ) e α | t | e − α | t | d t (cid:12)(cid:12)(cid:12)(cid:12) d x ≤ C α (cid:90) R | T ∗ ( x, t ) | e α | t | d t d x < ∞ , (4.17)for α > sufficiently small. Similarly: (cid:107) G (cid:107) L ( R ) = (cid:90) R (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R | t (cid:48) | T ( t (cid:48) , y ) d t (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) d y ≤ C α (cid:90) R (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R e α | t (cid:48) | T ( t (cid:48) , y ) d t (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) d y = C α (cid:90) R (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R e − α | t (cid:48) | e α | t (cid:48) | T ( t (cid:48) , y ) d t (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) d y ≤ ˜ C α (cid:90) R e α | t (cid:48) | | T ( t (cid:48) , y ) | d t (cid:48) d y < ∞ , again for α > sufficiently small. We conclude that C ( x, y ) ∈ L ( R ) uniformly in ε > .Using the expansion in (4.14) the integral kernel of the Birman-Schwinger operator is − κ (cid:90) R V ( x, t )[( − T ) − ]( t, t (cid:48) ) V ( t (cid:48) , y ) d t d t (cid:48) = − κ √ ε | Ψ (cid:105)(cid:104) Φ | + κB ε ( x, y ) , (4.18)where B ε ( x, y ) is the integral kernel of the uniformly bounded operator for ε > that comes from thenon-singular terms of (4.14). Also: Φ( y ) := (cid:90) R V ( x, y ) d x, Ψ( x ) := (cid:90) R V ( x, y ) d y. (4.19)The functions Ψ and Φ are in L ( R ) and let us prove this for Ψ . From the above definition and from(4.9) we see that it is enough to prove that (cid:82) R T ∗ ( x, t ) dt belongs to L ( R ) . But this is exactly what wedid in (4.17).By the usual factorization trick, the operator in (4.18) is invertible if and only if − κ √ ε | Ψ (cid:105)(cid:104) Φ | ( + κB ε ) − is invertible. The later operator is not invertible if and only if ε is a zero of the following function (0 , ∞ ) (cid:51) ε (cid:55)→ − κ √ ε (cid:104) Φ | ( + κB ε ) − | Ψ (cid:105) ∈ R . Introduce the new variable r = ε . The above function has a positive root ε if and only if the map [ − , (cid:51) r (cid:55)→ f κ ( r ) := κ (cid:104) Φ | ( + κB r ) − | Ψ (cid:105) ∈ [ − , has a positive fixed point r > and r = √ ε . 8t is not difficult to extend the methods we used for proving that B ε was uniformly bounded in ε > in order to show that actually all the ε dependent quantities are norm differentiable with globallybounded derivatives on ε > . Thus f κ becomes a contraction if κ is small enough and its unique fixedpoint r can be computed by iteration starting from r = 0 .Using the definitions of V and V (in which we put ε = 0 or equivalently E = − / ) we can calculatethe inner product (cid:104) Φ , Ψ (cid:105) to get (cid:104) Φ , Ψ (cid:105)| ε =0 = 8 κ (cid:18) π − (cid:19) + O ( κ ) > . Thus r ∼ κ (cid:104) Φ , Ψ (cid:105) ∼ κ > if κ is small enough which leads to ε = r ∼ κ (cid:104) Φ , Ψ (cid:105) ∼ κ . Consequently,the leading order behaviour of the discrete eigenvalue E ( κ ) of H κ for κ sufficiently small is E ( κ ) = − − (cid:18) π − (cid:19) κ + O ( κ ) . where we used the formula ε = − E − > . This concludes the first part of the proof of Theorem 2.1.We will now prove that the ground state energy is always non-degenerate (when it exists) by firstshowing that the heat semigroup e − tH κ is positivity improving. Some key formulas from [25] give theexplicit expression of the heat kernel of − d / d y + κδ ( y ) from which we conclude that the integral kernelof e − t ( − ∆+ κδ ( y )) ( x, y ; x (cid:48) , y (cid:48) ) = e t d d x ( x, x (cid:48) ) e − t ( − d d y + κδ ( y )) ( y, y (cid:48) ) , t > , is positive and point-wise smaller than e t ∆ ( x, y ; x (cid:48) , y (cid:48) ) . Applying the analogue of the Dyson formula be-tween e − tH κ and e − t ( − ∆+ κδ ( y )) (one has to be careful when deriving it due to the singularity of the delta"potentials") we see that the integral kernel of e − tH κ is larger or equal than that of e − t ( − ∆+ κδ ( y )) , henceit is also positivity improving. The Perron-Frobenius theorem[26] then guarantees the non-degeneracyof the lowest eigenvalue of H κ , provided that such an eigenvalue exists.In order to prove that a discrete eigenvalue exists for all κ ∈ (0 , / √ we first need to extend ourprevious analysis to negative κ ’s. It is not difficult to see from the expression of H κ that the previousexistence result also holds for small negative κ (cid:54) = 0 as well. The family H κ is analytic of type B in thesense of Kato. The regular analytic perturbation theory allows one to extend the construction of a realanalytic ground state energy E ( κ ) from a neighborhood of κ (cid:54) = 0 to some maximal open intervals I ± respectively included in (0 , / √ and ( − / √ , . The only reason for which the right endpoint of I + might not go all the way to / √ is that E ( κ ) might start increasing and eventually hit the bottom ofthe essential spectrum (i.e. − / ) at some κ + < / √ . We will show that this is not possible.Fix (cid:15) > small enough for which we know that E ( ± (cid:15) ) exist. Then we can construct two families ofreal analytic normalized eigenvectors Ψ κ on I ± , starting from some given eigenvectors at κ = ± (cid:15) .The operator which implements the interchange of x with y is denoted by U and acts as ( U f )( x, y ) = f ( y, x ) . It is unitary and U = U − . Moreover, we have U H κ U − = H − κ , H κ U − Ψ − κ = E ( − κ ) U − Ψ − κ . This shows that E ( − κ ) is also an eigenvalue for H κ , hence E ( κ ) ≤ E ( − κ ) . By a similar argument wealso obtain that E ( − κ ) ≤ E ( κ ) , hence E ( κ ) = E ( − κ ) as long as they exist. Moreover, there must exista unimodular complex number e iφ ( k ) (the phase can be chosen to be smooth on | κ | > (cid:15) ) such that Ψ κ ( x, y ) = e iφ ( k ) Ψ − κ ( y, x ) , κ ∈ I ± . (4.20)All the quantities defined above are smooth if κ (cid:54) = 0 , but the eigenvectors are not a-priori κ -differentiable in the H ( R ) norm, only in L ( R ) . We can formally apply the Feynman-Hellmannformula to the quadratic form and get: E (cid:48) ( κ ) = (cid:90) R ( | Ψ κ ( x, | − | Ψ κ (0 , x ) | ) dx. (4.21)The rigorous proof of this identity is based on the following identity
11 + αE ( κ ) = (cid:104) Ψ κ , (1 + αH κ ) − Ψ κ (cid:105) , < α (cid:28) ,
9n which we now can differentiate with respect to κ in the norm topology and after that take the limit α ↓ .We will now show that there cannot exist a κ ∈ I + such that E (cid:48) ( κ ) > . Assume the contraryand consider such a κ . Define the vector Φ( x, y ) = Ψ − κ ( y, x ) and choose κ (cid:48) ∈ I + with κ (cid:48) > κ . Φ isa normalized vector which belongs to the form domain of H κ (cid:48) . First using the min-max principle andsecond (4.20) we have: E ( κ (cid:48) ) ≤ (cid:104) Φ , H κ (cid:48) Φ (cid:105) = E ( κ ) + ( κ (cid:48) − κ ) (cid:90) R ( | Φ( t, | − | Φ(0 , t ) | ) dt. Taking the limit κ (cid:48) ↓ κ in ( E ( κ (cid:48) ) − E ( κ )) / ( κ (cid:48) − κ ) leads to: E (cid:48) ( κ ) ≤ (cid:90) R ( | Φ( t, | − | Φ(0 , t ) | ) dt. (4.22)Due to (4.20) we have | Φ( t, | = | Ψ κ (0 , t ) | and | Φ(0 , t ) | = | Ψ κ ( t, | , hence (4.21) implies: (cid:90) R ( | Φ( t, | − | Φ(0 , t ) | ) dt = − (cid:90) R ( | Ψ κ ( t, | − | Ψ κ (0 , t ) | ) dt = − E (cid:48) ( κ ) . (4.23)Introducing this identity back into (4.22) we obtain E (cid:48) ( κ ) ≤ . We conclude that E (cid:48) ( κ ) ≤ for all κ ∈ I + , hence E ( κ ) ≤ E ( (cid:15) ) < − / for κ ∈ I + which insures the existence of a positive minimaldistance between E ( κ ) and the essential spectrum. Consequently, the right endpoint κ + of I + cannot besmaller than / √ because in that case E ( κ + ) := lim κ ↑ κ + E ( κ ) would be an eigenvalue, thus I + couldbe extended a bit to the right of κ + by analytic perturbation theory. Hence the operator H κ must haveat least one eigenvalue for < κ ≤ / √ . This concludes the proof of Theorem 2.1. In this section we prove the second main result, namely that if ˜ κ > is fixed, then H κ, ˜ κ has no discreteeigenvalues for κ in a connected neighborhood of + ∞ . The proof is based on a similar method as usedin Section 4.Since H κ and H κ, ˜ κ only differ in the positive interaction term while the bottom of the essentialspectrum is given by the negative interaction terms, we have that σ ess ( H κ, ˜ κ ) = σ ess ( H κ ) . We assumethat κ ≥ / √ . Then Lemma 3.1 implies: σ ess ( H κ, ˜ κ ) = (cid:20) − κ , ∞ (cid:19) . The framework described in Section 3 is easily generalized to the operator H κ, ˜ κ . Consequently, E < − κ / is a discrete eigenvalue of H κ, ˜ κ if and only if the inverse of the operator G κ, ˜ κ ( E ) does not existon [ L ( R )] , where G κ, ˜ κ ( E ) is given by G κ, ˜ κ ( E ) = g − + τ R ( E ) τ ∗ , and τ R ( E ) τ ∗ is as before but g is changed to diag {− κ, ˜ κ, − } . To study when the operator G κ, ˜ κ ( z ) isinvertible, we scale it using the unitary operator U κ which acts on L ( R ) by [ U κ f ]( x ) = √ κf ( κx ) . Wehave: [ U κ ˆ T ( E ) U ∗ κ f ]( x ) = 1 πκ (cid:90) R x + y − Eκ f ( y ) d y. Define a rescaled energy ε := − E/κ > . Thus U κ ˆ T ( E ) U ∗ κ = κ ˆ T ( − ε ) . Equivalent results hold for ˆ T , ˆ T , ˆ T ∗ and ˆ T . Consequently, the operator G κ, ˜ κ ( E ) is unitarily equivalent to the operator: G κ, ˜ κ ( − ε ) := − κ κ
00 0 − + 1 κ T ( − ε ) T ( − ε ) T ∗ ( − ε ) T ( − ε ) T ( − ε ) T ∗ ( − ε ) T ( − ε ) T ( − ε ) T ( − ε ) . (5.1)As mentioned the strategy we apply to show the absence of discrete eigenvalues is basically the sameas in Section 4, i.e. some applications of Feshbach’s formula and the Birman-Schwinger principle. So webegin by choosing the orthogonal projection Π on [ L ( R )] which satisfies Π G κ, ˜ κ ( − ε )Π ∼ = 1 κ (cid:20) − + T ( − ε ) T ( − ε ) T ( − ε ) κ ˜ κ + T ( − ε ) (cid:21) , (5.2)10n Π[ L ( R )] . We will also need the projection on the orthogonal subspace of Π[ L ( R )] , which is definedby Π ⊥ := − Π . Lemma 5.1.
Let G κ, ˜ κ ( − ε ) be given by (5.1) . Then R ( ε ) := [Π ⊥ G κ, ˜ κ ( − ε )Π ⊥ ] − exists as a boundedoperator on the proper subspace for all ε > and κ > / √ .Proof. By the definition of Π ⊥ we have Π ⊥ G κ, ˜ κ ( − ε )Π ⊥ ∼ = − + κ − T ( − ε ) . We need to check theinvertibility of − κ − T ( − ε ) on L ( R ) . In the Fourier representation, this operator is a multiplicationoperator with the function (cid:18) − κ √ s + 2 ε (cid:19) − . (5.3)Thus, the norm (cid:107) κ − T (cid:107) < / ( κ √ ≤ for all κ ≥ / √ and ε > . Consequently, Π ⊥ G κ ( − ε )Π ⊥ isinvertible on L ( R ) for all κ > / √ and ε > .By Feshbach’s formula and Lemma 5.1 the inverse of G κ, ˜ κ ( − ε ) exists if the inverse of S W ( ε ) := Π G κ, ˜ κ ( − ε )Π − Π G κ, ˜ κ ( − ε )Π ⊥ R ( ε )Π ⊥ G κ, ˜ κ ( − ε )Π (5.4)exists as an operator on [ L ( R )] . In order to simplify notation, we stop writing the explicit dependenceon ε of the various T -operators. We get the following expression for S W ( ε ) : S W ( ε ) ∼ = (cid:20) − + T T T κ ˜ κ + T (cid:21) + 1 κ (cid:20) T ∗ T ∗ (cid:21) (cid:18) − κ T (cid:19) − (cid:2) T T (cid:3) . (5.5)To find the conditions for the inverse of S W ( ε ) to exist on [ L ( R )] , we apply Feshbach’s formula again.Consequently, we need to define another pair of orthogonal projections ˜Π and ˜Π ⊥ := − ˜Π on [ L ( R )] such that ˜Π S W ( ε ) ˜Π ∼ = − + T + 1 κ T ∗ ( − κ − T ) − T on L ( R ) . (5.6) Lemma 5.2.
Let S W ( ε ) be given by (5.5) , and let ˜Π ⊥ be the orthogonal projection on [ L ( R )] such that ˜Π ⊥ S W ( ε ) ˜Π ⊥ ∼ = κ ˜ κ + T + 1 κ T ∗ ( − κ − T ) − T , (5.7) on L ( R ) . Then ˜ R ( ε ) := [ ˜Π ⊥ S W ( ε ) ˜Π ⊥ ] − exists on the proper subspace for all κ ≥ / √ and ε > .Proof. The proof follows from the fact that T and κ − T ∗ ( − κ − T ) − T are bounded and positive forall κ ≥ / √ and ε > .Lemma 5.2 and Feshbach’s formula implies that the inverse of G κ, ˜ κ ( ε ) exists if the inverse of ˜ S W ( ε ) ∼ = − T − κ T ∗ ( − κ − T ) − T + W κ, ˜ κ ( ε ) (5.8)exists as an operator on L ( R ) , where W κ, ˜ κ ( ε ) := D (cid:32) κ ˜ κ + T + 1 κ T ∗ (cid:18) − κ T (cid:19) − T (cid:33) − D, (5.9) D := T + 1 κ T ∗ (cid:18) − κ T (cid:19) − T . (5.10)The idea is to apply the Birman-Schwinger principle to study for which values of ε > and κ ≥ / √ the inverse of ˜ S W ( ε ) does not exist on L ( R ) . Before we do that we rewrite ˜ S W ( ε ) a bit. Factorizing κ/ ˜ κ in W κ, ˜ κ ( ε ) we can write ˜ S W ( ε ) ∼ = − T + 1 κ (cid:102) W κ, ˜ κ , (5.11)where (cid:102) W κ, ˜ κ = − T ∗ (cid:18) − κ T (cid:19) − T + ˜ κD (cid:32) + ˜ κκ T + ˜ κκ T ∗ (cid:18) − κ T (cid:19) − T (cid:33) − D. (5.12)We are now ready to construct the Birman-Schwinger operator for ˜ S W ( ε ) given by (5.11).11 roposition 5.3. Let ˜ S W ( ε ) be as in (5.11) , and let ˜ κ > be fixed, κ ≥ / √ and ε > . Then thereexists bounded operators V : [ L ( R )] → L ( R ) and V : L ( R ) → [ L ( R )] such that ˜ S W ( ε ) is invertibleif and only if + 1 κ V ( − T ) − V (5.13) is invertible on [ L ( R )] .Proof. The proof is almost identical to the proof of Theorem 4.2, so we will only describe the constructionof V : [ L ( R )] → L ( R ) and V : L ( R ) → [ L ( R )] . We need V and V to have the property that V V = ˜ W κ, ˜ κ . Let Ψ ∈ L ( R ) and let D be as in (5.10). Define the operator V : L ( R ) → [ L ( R )] by V Ψ = − (cid:0) − κ T (cid:1) − T Ψ (cid:16) + ˜ κκ T + ˜ κκ T ∗ (cid:0) − κ T (cid:1) − T (cid:17) − / D Ψ . (5.14)Similarly, let Φ = [Φ , Φ ] T ∈ [ L ( R )] . We define the operator V : [ L ( R )] → L ( R ) by V Φ = (cid:20) T ∗ (cid:0) − κ T (cid:1) − , ˜ κD (cid:16) + ˜ κκ T + ˜ κκ T ∗ (cid:0) − κ T (cid:1) − T (cid:17) − / (cid:21) (cid:20) Φ Φ (cid:21) = T ∗ (cid:18) − κ T (cid:19) − Φ + ˜ κD (cid:32) + ˜ κκ T + ˜ κκ T ∗ (cid:18) − κ T (cid:19) − T (cid:33) − / Φ . (5.15)For Ψ ∈ L ( R ) we find that V V Ψ is given by V V Ψ = ˜ W κ, ˜ κ Ψ and we have our factorization.The strategy to show an absence of discrete eigenvalues is to find a necessary condition which anyeigenvalue must satisfy, and then show that for every fixed ˜ κ > and for any κ larger than some value κ M (depending on ˜ κ ) the above necessary condition cannot be satisfied.The first important remark is that both V and V have finite limits when κ → ∞ , uniformly in (cid:15) > .Thus the operator in (5.13) is always invertible if (cid:15) is larger than some value ε κ > . Moreover, this ε κ converges to when κ goes to infinity. Therefore we know a priori that the points where (5.13) mightnot be invertible on [ L ( R )] must obey ε ∈ (1 , if κ is larger than some value κ . Let us expand theintegral kernel of ( − T ) − around the threshold ε = 1 and introduce the variable λ (see below) to findthe following ( − T ) − ( x, y ) = 1 λ − | x − y | + δ ( x − y ) + 12 π (cid:90) R e is ( x − y ) √ s + 1 + 1 d s + O ( λ ) , λ := √ ε − . (5.16)Using this expansion of the integral kernel, The Birman-Schwinger operator (5.13) can be written as + 1 κ | Ψ (cid:105)(cid:104) Φ | λ + 1 κ B, (5.17)where the operator B is given by the product of V , the non-singular terms of (5.16) and V . Usingthe same approach as in Sec. 4, we can show that B is uniformly bounded for λ > and κ ≥ / √ .Furthermore, | Ψ (cid:105) and (cid:104) Φ | in (5.17) is given by | Ψ (cid:105) := (cid:90) R V ( x, x (cid:48) ) d x (cid:48) , (cid:104) Φ | := (cid:90) R V ( y (cid:48) , y ) d y (cid:48) , (5.18)and Ψ and Φ can be shown to be in L ( R ) using Lemma 4.3. Let us rewrite the Birman-Schwingeroperator in (5.17): + 1 κ | Ψ (cid:105)(cid:104) Φ | λ + 1 κ B = (cid:32) + 1 κ | Ψ (cid:105)(cid:104) Φ | λ (cid:20) + 1 κ B (cid:21) − (cid:33) (cid:18) + 1 κ B (cid:19) . (5.19)12ut since B is uniformly bounded in both λ > and κ > / √ , there exists some κ ≥ κ > / √ suchthat if κ > κ we have that (cid:0) + κ − B (cid:1) − exists on [ L ( R )] for all λ > . Consequently, for κ > κ the inverse of the Birman-Schwinger operators exists at λ ∈ (0 , if and only if (cid:32) + 1 κ | Ψ (cid:105)(cid:104) Φ | λ (cid:20) + 1 κ B (cid:21) − (cid:33) − , κ > κ (5.20)exists. Using Feshbach’s formula with a rank- projection constructed from | Ψ (cid:105) we get the only valuesof < λ < where (5.20) might not exist are those which solve λ + 1 κ (cid:42) Φ , (cid:20) + 1 κ B (cid:21) − Ψ (cid:43) = 0 , κ > κ . (5.21)Thus if κ > κ , any discrete eigenvalue of H κ, ˜ κ has to have a corresponding λ ∈ (0 , which is a solutionto (5.21).Let us define the function: f ( λ, κ ) := (cid:42) Φ , (cid:20) + 1 κ B (cid:21) − Ψ (cid:43) , λ ∈ [0 , , κ ≥ κ . We are interested in finding possible values of λ ∈ (0 , where the graphs of f ( λ, κ ) and − κλ cross eachother. The function f is jointly uniformly continuous. Moreover, by explicit computation we obtain: lim κ →∞ f (0 , κ ) = 2 π (cid:18) ˜ κ (cid:90) R ˆ T (0 , s ) d s − (cid:90) R ˆ T ∗ (0 , s ) ˆ T ( s, d s (cid:19) (cid:12)(cid:12)(cid:12) λ =0 = ˜ κ − . (5.22)Thus there exists κ > κ such that f (0 , κ ) ≥ (˜ κ − / , κ > κ . From the uniform continuity of f we obtain the existence of some δ ∈ (0 , such that f ( λ, κ ) ≥ (˜ κ − / > , λ ∈ (0 , δ ) , κ > κ . (5.23)Moreover, | f ( λ, κ ) | is bounded by some constant K for all λ and κ . This implies that if κ > κ , thevalue of f ( · , κ ) is positive on (0 , δ ) and is larger than − K on [ δ, . At the same time, − λκ is negativeon (0 , δ ) and less than − κδ on [ δ, . Define κ M = max { κ , K/δ } . Then the two graphs cannot intersecteach other if κ > κ M and this completes the proof of absence of eigenvalues.Now let us consider the case < ˜ κ < . All our previous considerations remain true up to andincluding the identity (5.22) where now ˜ κ − < , hence f (0 , κ ) ≤ (˜ κ − / < , κ > κ . (5.24)Also, as before, f ( λ, κ ) ≥ − K for all λ and κ .Consider the function g ( λ, κ ) = λκ + f ( λ, κ ) with λ ∈ [0 , . We have g (0 , κ ) = f (0 , κ ) < while g (1 , κ ) = κ + f (1 , κ ) ≥ κ − K > provided κ > K . Thus g ( · , κ ) must have a zero in (0 , , and thisproves the existence of discrete spectrum for all κ > K . We can now prove the final result, i.e. the existence of a critical charge κ c which has the property thatfor every < κ < κ c the operator H κ has at least one discrete eigenvalue, while if κ ≥ κ c the discretespectrum is empty.The proof has three steps. First, we show that there exists some κ ≥ / √ such that H κ has nodiscrete spectrum. Second, we show that given such a κ , the operator H κ has empty discrete spectrumfor all κ ≥ κ . Third, we show that κ c is the smallest of all such κ . Step 1 . Let κ > / √ and consider the operator H κ, , i.e. with ˜ κ = 2 > . Theorem 2.2 implies theexistence of a κ M > / √ such that H κ, has no discrete eigenvalues if κ > κ M .13e know that the operators H κ and H κ, have the same essential spectrum. Additionally, we havethat H κ ≥ H κ, if κ ≥ , (6.1)where the inequality should be understood in the sense of quadratic forms. If κ = κ M + 1 , the operator H κ , has no discrete spectrum, hence (6.1) and the min-max principle imply that the discrete spectrumof H κ is empty. Step 2 . We will now show that the discrete spectrum of H κ with κ ≥ κ is also empty. Define theunitary operator U κ : L ( R ) → L ( R ) by ( U κ Ψ)( x, y ) = κ Ψ( κx, κy ) . Then by direct calculation U − κ H κ U κ = κ (cid:101) H κ , ˜ H κ := −
12 ∆ − δ ( y ) + δ ( x ) − κ δ ( x − y ) . Using the HVZ theorem we can prove that for κ ≥ / √ the essential spectrum of ˜ H κ is [ − / , ∞ ) .Additionally, due to the sign of the κ -dependent term we have (cid:101) H κ ≥ (cid:101) H κ , if κ ≥ κ . The operator (cid:101) H κ = κ − U − κ H κ U κ has no discrete spectrum. Since the bottom of the essential spec-trum of (cid:101) H κ is constant in κ and equals − / , the min-max principle implies that (cid:101) H κ has no discretespectrum and the same holds true for H κ . Step 3 . The set S consisting of all the κ ’s considered in the previous two steps is bounded from belowby / √ due to Theorem 2.1. Let κ (cid:48) ≥ / √ be the infimum of S . Assume that κ (cid:48) does not belong to S . Then there would exist a ground state with energy E ( κ (cid:48) ) < − κ (cid:48) / . Using the analytic perturbationtheory we could extend this ground state energy to a small interval centered at κ (cid:48) , thus κ (cid:48) would notbelong to the closure of S , contradiction.Thus S = [ κ (cid:48) , ∞ ) and κ c = κ (cid:48) . In fact, this proof provides us with an alternative characterisation of κ c , i.e. κ c is the right endpoint of the open interval of κ ’s for which a ground state exists. In this paper we considered the discrete spectrum of the Schrödinger operator for a one-dimensionalthree-body system with Dirac delta potentials, which models an impurity interacting with an exciton.We have proven that for κ close to zero there exists a single non-degenerate bound state which behaveslike κ , and we have explicitly calculated the coefficient of the leading term. The ground state surviveswhen κ ∈ (0 , / √ , but for some charge κ c > / √ the ground state energy hits the essential spectrum,and no bound states of the system exists for κ ≥ κ c . We cannot give an explicit value for κ c , butnumerical calculations indicate that κ c ≈ . .A future project is to study a related system of an impurity and two oppositely charged particles withmultiplicative potentials in both one and two dimensions. While the results are expected to be somehowsimilar, the technical tools one needs to use are quite different. Acknowledgements
J.H. and T.G.P. are supported by the QUSCOPE Center, which is funded by the Villum Foundation.H.C. was partially supported by the Danish Council of Independent Research | Natural Sciences, GrantDFF-4181-00042. H.K. was partially supported by the MIUR-PRIN2010-11 grant for the project “Calcolodelle variazioni” .
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