On the number of isolate eigenvalues of a pair of particles in a quantum wire arxiv:1812.11804v1 [math-ph] 31 Dec 2018 Joachim Kerner 1 Department of Mathematics an Computer Science FernUniversität in Hagen 58084 Hagen Germany Abstract In this note we consier a pair of particles moving on the positive half-line R + with the pairing generate by a har-wall potential. This moel was first introuce in [KM17] an later applie to investigate conensation of pairs of electrons in a quantum wire [Ker, Ker18]. For this, a etaile spectral analysis prove necessary anasapartofthisitwasshownin[ker]that, inaspecial case, theiscretespectrum of the Hamiltonian consists of a single eigenvalue only. It is the aim of this note to prove that this is generally the case. 1 E-mail aress: Joachim.Kerner@fernuni-hagen.e
1 Introuction In this note we consier an interacting system of two particles with the positive half-line R + = (0, ) as one-particle configuration space. More explicitly, the Hamiltonian shall be given by with har-wall potential, > 0, H = 2 x 2 +v( x y ) (1) 2 y2 v(x) := { 0 x <, else. Note that, through the potential v, the two particles actually form a pair with spatial extension characterise by > 0. The two-particle moel with Hamiltonian (1) an potential (2) was introuce in [KM17]. Its investigation grew out of stuying many-particle quantum chaos on quantum graphs [BK13a, BK13b] taking into account recent results in theoretical physics [QU16]. More generally, ue to the technical avances in the last ecaes an especially in the realm on nanotechnology, it has become pivotal to stuy the properties of interacting particle systems in one imension which may iffer greatly from those of systems in higher imension [Gam04, Gia16]. Also, since the pairing of electrons (Cooper pairs) in metals is the key mechanism in the formation of the erconucting phase in type-i erconuctors [Coo56, BCS57], an investigation of the Hamiltonian (1) is also interesting from a soli-state physics point of view. An inee, the conensation of pairs of electrons with Hamiltonian (1) was stuie in [Ker, Ker18]; in [Ker] the electrons forming a pair have same spin an in [Ker18] the electrons of a pair have opposite spin as it is the case with Cooper pairs. In this note we are intereste in spectral properties of the H. More explicitly, we are intereste in characterising the iscrete part of the spectrum. It was the key observation in [KM17] that the iscrete spectrum of H is non-trivial, i.e., there exist eigenvalues below the botton of the essential spectrum. Since this is not the case if one changes the oneparticle configuration space to be the whole real line R, the existence of a iscrete spectrum is irectly linke to the geometry of the one- an two-particle configuration space. Implementing exchange symmetry, the authors of [Ker] were able to show that the iscrete spectrum actually exists of one eigenvalue only. The main purpose of this note is to show that exchange symmetry is inee not necessary an that the iscrete spectrum always consists of one eigenvalue only. Finally, we want to raw attention to the recent paper[sep] in which spectral properties of (1) foralargeclass of interactionpotentials v : R + R where stuie. The authorsalso foun that the iscrete spectrum is non-empty an contains only finitely many eigenvalues. However, no bouns on the number of isolate eigenvalues were erive. (2) 2
2 The moel an main results Due to the formal nature of the interaction potential (2), H cannot be irectly realise as a self-ajoint operator on L 2 (R 2 + ). However, we see that this choice for v means that the two-particle configuration space is actually given by Ω = {(x,y) R 2 + : x y < }. (3) Base on Ω we then introuce the Hilbert spaces L 2 (Ω) := L 2 0 (Ω) as well as L 2 s(ω) := {ϕ L 2 (Ω) : ϕ(x,y) = ϕ(y,x)}, L 2 a(ω) := {ϕ L 2 (Ω) : ϕ(x,y) = ϕ(y,x)}. When escribing two istinguishable particles one focusses on L 2 (Ω) while the versions L 2 s/a (Ω) are use if one implements exchange symmetry between the two particles. For example, in [Ker] the authors consiere a pair of two electrons with same spin which implies that one has to work on L 2 a (Ω) since electrons are fermions. Contrary to this, in [Ker18] the electrons were assume to be of opposite spin which then requires to work on L 2 s (Ω). Now, on any of those Hilbert spaces, H is rigorously realise via its associate(quaratic) form, j {0,s,a}, q j [ϕ] := ϕ 2 x (5) with form omain D j = {ϕ H 1 (Ω) : ϕ L 2 j (Ω) an ϕ Ω D = 0} where Ω Ω D := {(x,y) R 2 + : x y = }. (6) Note here that q j [ ] obviously is a close positive form with a ense form omain [BHE08]. Also note that we write H j for the realisation of H associate with the corresponing form q j [ ]. Remark 1. It is clear that the self-ajoint operator associate with q j [ ] is nothing else than a version of the two-imensional Laplacian [GT83]. In orer to formulate our main result we recall the following statement which was prove in [KM17, Ker, Ker18]. We enote by σ ( ) the iscrete spectrum. Proposition 1. For every j {0,s,a} one has σ (H j ). It is our goal in this note to prove the following result. Theorem 1. For every j {0,s,a} one has σ (H j ) = {E j } with some E j 0 which is an eigenvalue of multiplicity one. In other wors, the iscrete spectrum consists of one eigenvalue only. 3 (4)
3 Proof of Theorem 1 In this section we establish a proof of Theorem 1. We note that this was alreay prove for j = a in [Ker] but for the sake of completeness we also inclue a proof thereof. In a first step we establish an auxiliary result: We efine the omain Ω := {(x,y) R 2 : x y < } {(x,y) R 2 : x+y < } (7) an introuce on L 2 ( Ω) the two-imensional Laplacian with Dirichlet bounary conitions along Ω. We enote this operator by (D). Note that the quaratic form associate with (D) is given by q [ϕ] := ϕ 2 x with form omain D := {ϕ H 1 ( Ω) : ϕ Ω = 0}. Proposition2. Theiscrete spectrum of the self-ajointoperator (D) one eigenvalue with multiplicity one. Ω consists of exactly Proof. Since the Laplacian is invariant uner rotations as well as translations, we may prove the statement by consiering the Dirichlet Laplacian on a rotate version of Ω. Namely, we consier the Dirichlet Laplacian on the cross-shape omain Ω 0 :={(x,y) R 2 : < y <, / 2 < x < +/ 2} {(x,y) R 2 : < x <, / 2 < y < +/ 2}. We enote this operator by (0) D. We then employ a bracketing argument an for this we introuce the irect sum of Laplacians 1 2 ; here 1 is the two-imensional Laplacian efine on the boune omain (square) Ω 1 := ( / 2,+/ 2) ( / 2,+/ 2) with Neumann bounary conitions along Ω 1. 2 enotes the two-imensional Laplacian on the omain Ω 2 := (Ω 0 \Ω 1 ) with Neumann bounary conitions along the bounary segments ajacent to Ω 1 an Dirichlet bounary conitions elsewhere. Most importantly, in terms of operators we obtain the inequality 1 2 (0) D which implies N( (0) D,E) N( 1 2,E) with N(,E) enoting the counting function that counts the number of eigenvalues up to energy E < infσ ess ( 1 2 ). Here σ ess ( ) enotes the essential spectrum. 4
From the efinition of Ω 2 it reaily follows that N( 1 2,E) = N( 1,E) whenever E < infσ ess ( 1 2 ). The reason for this is that infσ ess ( 1 2 ) = infσ ess ( 2 ) = infσ( 2 ) = π2,see[evv89,ker]formoreetails(notethatthespectrum 2 2 of 1 is purely iscrete an Ω 2 consists of four rectangular parts for which a separation of variables can be employe to etermine the (essential) spectrum irectly). Now, in orer to stuy N( 1,E) we take avantage of the fact that Ω 1 is a square. Hence, we can employ a separation of variables which allows us to etermine the eigenvalues of 1 explicitly. Namley, } σ ( 1 ) = {0, π2 π2 22, 2,.... Hence, it follows that N( 1,E) = 1 for all E < π2 which proves the statement taking 2 2 into account that infσ ess ( (0) D ) = π2, see [Ker]. 2 2 Proof. (of Theorem 1) We first consier the cases j {0,s}: We introuce the (injective) linear map I j : D j D, where I j ϕ is constructe as follows: one takes ϕ D j an then reflects it across the y-axis. This new function (consisting of the original ϕ an the new reflecte part) is then reflecte another time across the x-axis, finally yieling an element of D. Now, from the min-max principle we then conclue that, n N, µ n (H j ) = inf W n D j 0 ϕ W n q j [ϕ] ϕ 2 L 2 (Ω) q [I j ϕ] inf W n I j D j 0 I j ϕ W n I j ϕ 2 L 2 ( Ω) inf W n D 0 ϕ W n q [ϕ] ϕ 2 L 2 ( Ω) = µ n ( (D) ), where µ n ( ) enotes the n-th min-max eigenvalue [BHE08, Non]. Also, W n refers to n- imensional subspaces. FromProposition2it followsthat µ 1 ( (D) ) is the only eigenvalue in the iscrete spectrum an µ n ( (D) ) = infσ ess ( (D) ) = π2 for n > 1. Hence, by 2 2 Proposition 1 we conclue that only µ 1 (H j ) < π2 which yiels the statement since both 2 2 essential spectra start at π2 as shown in [KM17, Ker18]. 2 2 In a next step we consier the case j = a: Again we want to make use of the min-max principle an hence introuce the (injective) linear map I a : D a D /2 5
which acts as follows: to obtain I a ϕ one first restricts ϕ D a to {(x,y) R 2 + : x y < an y > x}. (8) Thisrestrictionisthenreflecteacrosstheaxisx = 0anthenbothsegmentsaretranslate in the negative y-irection by /2. Finally, we can exten this (translate) function by zero to obtain an element of D /2. Now, employing the min-max principle shows that µ n (H a ) = inf q a [ϕ] W n D a 0 ϕ W n ϕ 2 L 2 (Ω) inf W n I ad a inf q [I a ϕ] 0 I aϕ W n I a ϕ 2 L 2 ( Ω) q [ϕ] W n D 0 ϕ W n ϕ 2 L 2 ( Ω) = µ n ( (D) /2 ). For n > 1, µ n ( (D) /2 ) = infσ ess( (D) /2 ) = 2π2 2 an since it was shown in [Ker] that also infσ ess (H a ) = 2π2 2, we conclue the statement. Acknowlegement It is a great pleasure to thank S. Egger (Haifa) for useful remarks an interesting iscussions. References [BCS57] J. Bareen, L. N. Cooper, an J. R. Schrieffer, Theory of erconuctivity, Phys. Rev. 108 (1957), 1175 1204. [BHE08] J. Blank, M. Havliček, an P. Exner, Hilbert space operators in quantum physics, Springer, 2008. [BK13a] J. Bolte an J. Kerner, Quantum graphs with singular two-particle interactions, J. Phys. A 46 (2013), 045206. [BK13b], Quantum graphs with two-particle contact interactions, J. Phys. A 46 (2013), 045207. [Coo56] L. N. Cooper, Boun electron pairs in a egenerate Fermi gas, Phys. Rev. 104 (1956), 1189 1190. [Evv89] P. Exner, P. Šeba, an P. Šťovíček, On existence of a boun state in an L-shape waveguie, Czechoslovak Journal of Physics B (1989), 1181 1191. 6
[Gam04] T. Gamarchi, Quantum Physics in One Dimension, Oxfor Unversity Press, 2004. [Gia16] T. Giamarchi, One-imensional physics in the 21st century, Comptes Renus Physique 17 (2016), no. 3, 322 331. [GT83] D. Gilbarg an N. S. Truinger, Elliptic partial ifferential equations of secon orer, Springer, 1983. [Ker] [Ker18] J. Kerner, On boun electron pairs in a quantum wire, preprint, arxiv:1708.03753, to appear in Reports on Mathematical Physics., On pairs of interacting electrons in a quantum wire, Journal of Mathematical Physics 59 (2018), no. 6, 063504. [KM17] J. Kerner an T. Mhlenbruch, On a two-particle boun system on the half-line, Reports on Mathematical Physics 80 (2017), no. 2, 143 151. [Non] S. Nonnemacher, Lecture notes: Spectral theory of selfajoint operators, Lectures at University of Cariff: https://cariffmicrolocal.files.worpress.com/2017/06/stephanespectral-theory.pf. [QU16] F. Queisser an W. G. Unruh, Long-live resonances at mirrors, Phys. Rev. D 94 (2016), 116018. [SEP] J. Kerner S. Egger an K. Pankrashkin, Boun states of a pair of particles on the half-line with a general interaction potential, preprint, arxiv:1812.06500. 7