Bound states of a pair of particles on the half-line with a general interaction potential
aa r X i v : . [ m a t h - ph ] D ec BOUND STATES OF A PAIR OF PARTICLESON THE HALF-LINE WITH A GENERAL INTERACTIONPOTENTIAL
SEBASTIAN EGGER, JOACHIM KERNER, AND KONSTANTIN PANKRASHKIN
Abstract.
In this paper we study an interacting two-particle systemon the positive half-line R + . We focus on spectral properties of theHamiltonian for a large class of two-particle potentials. We characterizethe essential spectrum and prove, as a main result, the existence ofeigenvalues below the bottom of it. We also prove that the discretespectrum contains only finitely many eigenvalues. Introduction
In this paper we are concerned with spectral properties of an interactingtwo-particle system moving on the half-line R + := (0 , ∞ ). More specifically,we consider the (two-particle) Hamiltonian in L ( R + × R + ) given by H = − ∂ ∂x − ∂ ∂x + v (cid:18) | x − x |√ (cid:19) , (1)with an interaction potential v : R + → R belonging to a large class coveringall physically meaningful potentials including, e.g., quadratic and Lennard-Jones-type potentials. Note that the factor √ in the argument of v is onlychosen for further convenience. Very informally, our main result is thatif the potential v creates a bound state for the respective one-dimensionalSchr¨odinger operator on the half-line, then it creates at least one eigenvalueof H with a strictly lower energy.The present work is a far-reaching extension of the previous work [KM] inwhich a similar result was obtained for a specific class of hard wall potentials v . As described in [K17, K18], the presence of a discrete spectrum leads toa (Bose-Einstein) condensation of pairs in a gas of bosonic, non-interactingpairs with each pair described by (1). A condensation of pairs of electrons,on the other hand, is the key mechanism in the formation of the supercon-ducting phase in type-I superconductors [C, BCS]. Hence, the extension ofthe model discussed in this paper is expected to have also interesting appli-cations in solid-state physics. One should emphasize on the fact that onlyvery few two-particle problems admit an explicit solution, see e.g. [BERW],so qualitative results are of a particular importance.Let us introduce some notions used throughout the paper: To keep thenotation as simple as possible, we will work with real-valued Hilbert spaces.For a self-adjoint and semi-bounded operator A we denote by D ( A ) its do-main and by D [ A ] the domain of the associated bilinear form (which willoften referred to as the form domain of A ). The bilinear form itself will be denoted as A [ · , · ], the spectrum and the essential spectrum of A will bedenoted by σ ( A ) and σ ess ( A ) respectively.Let v be a real-valued potential on R + with the following properties:(A) v ∈ L ( R + ) and max {− v, } ∈ L ∞ ( R + ),(B) The one-particle Schr¨odinger operator h := − d d x + v ( x )in L ( R + ), which is rigorously defined through its form h [ ϕ, ϕ ] = Z R + (cid:0) ( ϕ ′ ( x ) + v ( x ) ϕ ( x ) (cid:1) d x , D [ h ] = n ϕ ∈ H ( R + ) : Z R + v ( x ) ϕ ( x ) d x < + ∞ o , is such that the bottom of the spectrum inf σ ( h ) =: ε is an isolatedeigenvalue,(C) The bottom eigenvalue is strictly lower than the values of v at infin-ity, i.e. it holds ε < lim inf x →∞ v ( x ) := v ∞ .The assumption (C) is to avoid potentials with a pathological behavior, andit holds for the physically reasonable cases. It is well known that that theassumptions (B) and (C) are satisfied in two important cases:(a) for v ∞ = + ∞ ,(b) v ∞ < ∞ and v − v ∞ ∈ L ( R + ) with Z R + (cid:0) v ( x ) − v ∞ (cid:1) d x < v which are sufficiently regular near 0, for example, for v | (0 , ∈ L (0 , h correspondsto the Neumann condition ϕ ′ (0) = 0 at the origin. In general, the operator h can be in the limit point case at 0 (if v diverges very fast at zero) in whichcase the characterization of boundary conditions is more involved. However,this subtlety is of no importance for our constructions in the following.The associated two-particle Schr¨odinger operator H = − ∆ + v (cid:18) | x − x |√ (cid:19) in L ( R ) is rigorously defined through its form, H [ ϕ, ϕ ] = Z Z R (cid:16)(cid:12)(cid:12) ∇ ϕ ( x , x ) (cid:12)(cid:12) + v (cid:18) | x − x |√ (cid:19) ϕ ( x , x ) (cid:17) d x d x , D [ H ] = (cid:8) ϕ ∈ H ( R ) : H [ ϕ, ϕ ] < ∞ (cid:9) ;note the factor √ in the argument of v which is chosen for convenience inorder to have less factors in later computations. Our results are summarizedas follows: Theorem 1.1.
The essential spectrum of H is [ ε , + ∞ ) , and its discretespectrum is non-empty and finite. OUND STATES OF A PAIR OF PARTICLES 3
We remark that the presence of a non-empty discrete spectrum is proba-bly the most important result. It relies on a rather involved construction ofa test function whose structure was proposed in [LP] for a different probleminvolving specific potentials with explicitly known ground states, and it alsoappeared in e.g. [HM, P]. So we propose another extension to rather gen-eral operators and hope that it can be used beyond our framework (See e.g.Remark 2.2 below.) The proof of the finiteness of the discrete spectrum es-sentially follows the scheme of [MT] for another specific operator and essen-tially represents a realization of the Feshbach projection method, which wasalso used in [KP]. A new ingredient is delivered by the fact that some newproperties of the ground state of h should be established first. The fact thatwe work with rather singular potentials v , which can be non-integrable near0, brings a number of technical subtleties concerning the regularity of func-tions, and we collect the respective results on one-dimensional Schr¨odingeroperators in Section A.2. Proof of Theorem 1.1
Reductions by symmetries.
Let us first perform some standard re-ductions in order to deal with a model case. DenoteΩ := { ( x , x ) ∈ R × R + : | x | < x } and consider the diffeomorphism (rotation by π/ → R , Φ( x , x ) = 1 √ x + x , x − x ) , and the unitary transform (pull back) U : L ( R ) → L (Ω), U ϕ = ϕ ◦ Φ .Using the standard change of variables one easily checks that U D [ H ] = D [ Q ] , H [ ϕ, ϕ ] = Q [ U ϕ, U ϕ ] , with Q being the operator in L (Ω) given by its form Q [ ϕ, ϕ ] = Z Z Ω (cid:0) |∇ ϕ ( x , x ) | + v (cid:0) | x | (cid:1) ϕ ( x , x ) (cid:1) d x d x , D [ Q ] = (cid:8) ϕ ∈ H (Ω) : Q [ ϕ, ϕ ] < ∞ (cid:9) , which is then unitarily equivalent to H . To use the parity with respect to x we consider the right half of Ω,Ω := (cid:8) ( x , x ) ∈ R : 0 < x < x (cid:9) , and the unitary transformΘ : L (Ω) ∋ ϕ ( U + ϕ, U − ϕ ) ∈ L (Ω ) × L (Ω ) ,U ± ϕ ( x , x ) = ϕ ( x , x ) ± ϕ ( − x , x ) √ . If one introduces self-adjoint operators Q ± in L (Ω ) given by Q ± [ ϕ, ϕ ] = Z Z Ω (cid:0) |∇ ϕ ( x , x ) | + v ( x ) ϕ ( x , x ) (cid:1) d x d x , D ( Q + ) = (cid:8) ϕ ∈ H (Ω ) : Q + [ ϕ, ϕ ] < ∞ (cid:9) , D ( Q − ) = (cid:8) ϕ ∈ H (Ω ) : Q − [ ϕ, ϕ ] < ∞ (cid:9) , EGGER, KERNER, AND PANKRASHKIN then one easily checks that D [ Q ± ] := U ± D [ Q ] , Q [ ϕ, ϕ ] = Q + [ U + ϕ, U + ϕ ] + Q − [ U − ϕ, U − ϕ ] . It follows that Q (hence, also H ) is unitarily equivalent to Q + ⊕ Q − . As thebilinear form of Q + is an extension of that for Q − , it follows by the min-max principle that λ := inf σ ess ( Q + ) ≤ inf σ ess ( Q − ) and that the number ofeigenvalues of Q − below λ does not exceed that for Q + .Therefore, inf σ ess ( H ) = min (cid:8) inf σ ess ( Q − ) , inf σ ess ( Q + ) (cid:9) = inf σ ess ( Q + ),and the non-emptyness and finiteness of the discrete spectrum of Q + willimply the non-emptyness and finiteness of the discrete spectrum of H . Thisshows that Theorem 1.1 becomes a consequence of the following assertion,whose proof will be given in the rest of the section: Proposition 2.1.
The essential spectrum of the operator Q + is [ ε , + ∞ ) ,and its discrete spectrum is non-empty and finite. Remark 2.2.
It is clear that the above operators Q ± correspond to therestrictions of the initial operator H to the symmetric/anti-symmetric func-tions, i.e. ϕ ( x , x ) = ± ϕ ( x , x ). While the operator Q − is “dominated”by the operator Q + (in the sense that the qualitative spectral picture for H is determined by that of Q + only), it can be studied on its own, and theanalog of Proposition 2.1 has then the following form: Proposition 2.3.
Let h be the operator in L ( R + ) with h [ ϕ, ϕ ] = Z R + (cid:0) ( ϕ ′ ( x ) + v ( x ) ϕ ( x ) (cid:1) d x , D [ h ] = n ϕ ∈ H ( R + ) : Z R + v ( x ) ϕ ( x ) d x < + ∞ o . (2) If the bottom of the spectrum inf σ ( h ) =: ε ∗ is an isolated eigenvalue with ε ∗ < lim inf x →∞ v ( x ) := v ∞ , then the essential spectrum of Q − is [ ε ∗ , + ∞ ) and the discrete spectrum is non-empty and finite. This can be proved by a literal repetition of the proof of Proposition 2.1given in the following three subsections (see also Remark A.11 in Appendixconcerning h ).2.2. Essential spectrum.
Let us show the equality σ ess ( Q + ) = [ ε , + ∞ )by establishing separately the inclusions in both directions. The construc-tions of this section are very standard and are given to render a self-containedpresentation.In a first step, let us prove first that σ ess ( Q + ) ⊂ [ ε , ∞ ) employing an op-erator bracketing argument: For that, we partition Ω into three subdomainsΩ j , j = 1 , ,
3, using the straight lines x = L and x = L with L > := (cid:8) ( x , x ) ∈ Ω : x < L (cid:9) is the bounded triangle,Ω := (cid:8) ( x , x ) ∈ Ω : x < L, x > L (cid:9) is the half-infinite strip,Ω := (cid:8) ( x , x ) ∈ Ω : x > L (cid:9) is the remaining infinite sector. OUND STATES OF A PAIR OF PARTICLES 5
Define self-adjoint operators Q j in L (Ω j ), j ∈ { , , } , through their bilin-ear forms Q j [ ϕ, ϕ ] = Z Z Ω j (cid:0) |∇ ϕ ( x , x ) | + v ( x ) ϕ ( x , x ) (cid:1) d x d x , D [ Q j ] = (cid:8) ϕ ∈ H (Ω j ) : Q j [ ϕ, ϕ ] < ∞ (cid:9) . Using the canonical orthogonal projections P j : L (Ω ) → L (Ω j ), de-fined just as restrictions to Ω j , we observe that P j D [ Q ] ⊂ D [ Q j ] and that Q + [ ϕ, ϕ ] = P j =1 Q j [ P j ϕ, P j ϕ ] and, in addition, that the map I : L (Ω ) ∋ ϕ ( P ϕ, P ϕ, P ϕ ) ∈ M j =1 L (Ω j )is unitary. It follows by the min-max principle thatinf σ ess ( Q ) ≥ inf σ ess ( Q ⊕ Q ⊕ Q ) = min j ∈{ , , } inf σ ess ( Q j ) . Since Ω is a bounded Lipschitz domain, the form domain D [ Q ] ⊂ H (Ω ) is compactly embedded in L (Ω ), which implies that the spectrumof Q is purely discrete. Furthermore, we have inf σ ( Q ) ≥ inf x >L v ( x ) >ε for all L ≥ L with L chosen sufficiently large, due to to the assumption(C) on the potential v . It follows thatinf σ ess ( Q ) ≥ min (cid:8) ε , inf σ ess ( Q ) (cid:9) for L ≥ L . (3)To analyze Q we remark first that it admits a separation of variables, Q = h NL ⊗ + ⊗ q , (4)where h NL is the operator in L (0 , L ) associated with the form h NL [ ϕ, ϕ ] = Z L (cid:0) ( ϕ ′ ( x ) + v ( x ) ϕ ( x ) (cid:1) d x , D [ h NL ] = (cid:8) ϕ ∈ H (0 , L ) : Z L v ( x ) ϕ ( x ) d x < ∞ (cid:9) , while q acts in L ( L, + ∞ ), being defined via its associated form q [ ϕ, ϕ ] = Z ∞ L ϕ ′ ( x ) d x, D [ q ] = H ( L, + ∞ ) , i.e. q acts as ϕ
7→ − ϕ ′′ with the Neumann boundary condition at L , and σ ( q ) = σ ess ( q ) = [0 , + ∞ ). By (4) there holds inf σ ess ( Q ) = inf σ ( h NL ). Itis standard to see (see Proposition A.5) that lim L →∞ inf σ ( h NL ) = ε . By(3) one has inf σ ess ( Q + ) ≥ lim inf L → + ∞ min (cid:8) ε , inf σ ( q NL ) (cid:9) = ε .Now let us show the reverse inclusion [ ε , ∞ ) ⊂ σ ess ( Q + ) by constructinga suitable Weyl sequence. For that, let τ : R → R be a smooth functionwith 0 ≤ τ ≤ τ ( x ) = 1 for x ≥ τ ( x ) = 0 for x ≤
1. Pickany k ∈ [0 , ∞ ). For n ≥ ϕ n ( x , x ) = f n ( x ) g n ( x ) with f n ( x ) = ψ ( x ) τ ( n − x ) , g n ( x ) = cos ( kx ) τ ( x − n ) τ (2 n − x ) . EGGER, KERNER, AND PANKRASHKIN
Then ϕ n vanishes outside the rectangle [0 , n − × [ n + 1 , n − ⊂ Ω , and ϕ ∈ D ( Q + ). For large n one estimates, with a suitable a > k ϕ n k L (Ω ) ≥ Z n − f n ( x ) d x Z n − n +2 cos ( kx )d x ≥ an . On the other hand, ( Q + ϕ )( x , x ) = F n ( x ) g n ( x ) + f n ( x ) G n with F n ( x ) = − ψ ′′ ( x ) τ ( n − x ) + 2 ψ ′ ( x ) τ ′ ( n − x ) − ψ ( x ) τ ′′ ( n − x ) + v ( x ) ψ ( x ) τ ( n − x )= ε f n ( x ) + Φ n ( x ) ,G n ( x ) = k cos ( kx ) τ ( x − n ) τ (2 n − x )+ 2 k sin( kx ) (cid:2) τ ′ ( x − n ) τ (2 n − x ) − τ ( x − n ) τ ′ (2 n − x ) (cid:3) − cos ( kx ) h τ ′′ ( x − n ) τ (2 n − x ) + 2 τ ′ ( x − n ) τ ′ (2 n − x )+ τ ( x − n ) τ ′′ (2 n − x ) i = k g n ( x ) + Ψ n ( x ) , where with some b > | Φ n | ≤ b (cid:0) | ψ | + | ψ ′ | (cid:1) with supp Φ n ⊂ [ n − , n − , k Ψ n k ∞ ≤ b with supp Ψ n ⊂ [ n + 1 , n + 2] ∪ [2 n − , n − . One has (cid:0) Q + − ( ε + k ) (cid:1) ϕ ( x , x ) = Φ n ( x ) g n ( x ) + f n ( x )Ψ n ( x ) and (cid:13)(cid:13)(cid:13)(cid:0) Q + − ( ε + k ) (cid:1) ϕ (cid:13)(cid:13)(cid:13) L (Ω ) ≤ Z n − n − Φ n ( x ) d x Z n − n +1 g n ( x ) d x + 2 Z n − f n ( x ) d x (cid:18) Z n +2 n +1 + Z n − n − Ψ n ( x ) d x ≤ b Z n − n − (cid:0) ψ + ( ψ ′ ) (cid:1) d x Z n − n +1 cos( kx ) d x + 4 b Z n − ψ ( x ) d x ≤ b ( n − Z n − n − (cid:0) ψ + ( ψ ′ ) (cid:1) d x + 4 b . Therefore, (cid:13)(cid:13)(cid:0) Q + − ( ε + k ) (cid:1) ϕ (cid:13)(cid:13) L (Ω ) k ϕ n k L (Ω ) ≤ b ( n − Z n − n − (cid:0) ψ + ( ψ ′ ) (cid:1) d x + 4 b an = 4 b a (cid:16) − n (cid:17) Z n − n − (cid:0) ψ + ( ψ ′ ) (cid:1) d x + 4 b a n n →∞ −−−→ ψ ∈ H ( R + ). Hence, ε + k ∈ σ ( Q + ) for any k ≥
0, in other words,[ ε , ∞ ) ⊂ σ ( Q + ). As the set [ ε , ∞ ) has no isolated points, it follows that[ ε , ∞ ) ⊂ σ ess ( Q + ). OUND STATES OF A PAIR OF PARTICLES 7
Existence of discrete eigenvalues.
In this section we show that thediscrete spectrum of Q + is non-empty.Recall that the bilinear form of Q + is given by Q + [ ϕ, ϕ ] = Z Z Ω (cid:16) |∇ ϕ ( x , x ) | + v ( x ) ϕ ( x , x ) (cid:17) d x d x , D ( Q + ) = (cid:8) ϕ ∈ H (Ω ) : Q + [ ϕ, ϕ ] < ∞ (cid:9) . As inf σ ess ( Q + ) = ε , it follows by the min-max principle that the non-emptyness of the discrete spectrum follows from the existence of a function ϕ ∈ D [ Q + ] satisfying the strict inequality Q + [ ϕ, ϕ ] − ε k ϕ k L (Ω ) < ϕ in the form ϕ ( x , x ) = ψ ( x ) φ ( x ) ,with ψ being as previously the ground state of h and φ a function to bespecified. Due to the standard regularity considerations (see Appendix)there holds ψ ∈ C ( R + ) ∩ L ∞ ( R + ). With some ρ > F ( x ) := Z x ψ ( x ) d x , φ ( x ) := F ( x ) ρ . (5)It is easily checked (see Proposition A.4) that φ ∈ H (0 , a ) for any a > ρ > , which is assumed from now on. Finally we introduce asmooth cut-off function χ and the associated truncations φ n , n ∈ N , by0 ≤ χ ≤ , χ ( t ) = ( , t ≥ , , t ≤ , φ n ( x ) = φ ( x ) χ (cid:16) x n (cid:17) . The function ϕ defined by ϕ n ( x , x ) := ψ ( x ) φ n ( x ) belongs then to D [ Q + ]for any n ∈ N . A calculation then yields the following: Q + [ ϕ n , ϕ n ] = Z Z Ω (cid:0) |∇ ϕ n | + v ( x ) | ϕ n | (cid:1) d x d x = Z Z Ω h(cid:0) ψ ′ ( x ) φ n ( x ) (cid:1) + v ( x ) ψ ( x ) φ n ( x ) i d x d x + Z Z Ω (cid:0) ψ ( x ) φ ′ n ( x ) (cid:1) d x d x = Z R + (cid:18)Z x ψ ′ ( x ) ψ ′ ( x ) + v ( x ) ψ ( x ) d x (cid:19) φ n ( x ) d x + Z Z Ω (cid:0) ψ ( x ) φ ′ n ( x ) (cid:1) d x d x . An integration by parts ( which is still possible for singular potentials v , seeProposition A.2 in the appendix) gives Z x ψ ′ ( x ) ψ ′ ( x ) + v ( x ) ψ ( x ) d x = Z x ψ ( x ) (cid:16) − ψ ′′ ( x ) + v ( x ) ψ ( x ) (cid:17) d x + ψ ( x ) ψ ′ ( x ) , = ε Z x ψ ( x ) d x + ψ ( x ) ψ ′ ( x ) EGGER, KERNER, AND PANKRASHKIN and which allows us to write Q + [ ϕ n , ϕ n ] = ε k ϕ n k L (Ω ) + Z R + ψ ( x ) ψ ′ ( x ) φ n ( x ) d x + Z Z Ω ψ ( x ) φ ′ n ( x ) d x d x . (6)Integrating the middle term on the right-hand side by parts one obtains Z R + ψ ( x ) ψ ′ ( x ) φ n ( x ) d x = ( ψ φ n ) ( ∞ ) − ( ψ φ n ) (0)2 − Z R + ψ ( x ) φ n ( x ) φ ′ n ( x ) d x . One has φ n (0) = φ n ( ∞ ) = 0 and ψ ∈ L ∞ ( R + ), which shows that the firstsummand on the right-hand side vanishes, and Z R + ψ ( x ) ψ ′ ( x ) φ n ( x ) d x = − Z R + ψ ( x ) φ n ( x ) φ ′ n ( x ) d x . Taking F ′ = ψ into account one rewrites (6) as Q + [ ϕ n , ϕ n ] − ε k ϕ n k L (Ω ) = − Z R + ψ ( x ) φ ′ n ( x ) φ n ( x ) d x + Z R + F ( x ) φ ′ n ( x ) d x = Z R + (cid:16) F ( x ) φ ′ n ( x ) − φ ′ n ( x ) F ′ ( x ) φ n ( x ) (cid:17) d x =: G n . In order to show that the term G n can be made strictly negative one usesfirst the expressions for F and φ n to compute F ( x ) φ ′ n ( x ) = F ( x ) (cid:18) ρF ( x ) ρ − F ′ ( x ) χ (cid:16) x n (cid:17) + 1 n F ( x ) ρ χ ′ (cid:16) x n (cid:17)(cid:19) = ρ F ( x ) ρ − F ′ ( x ) χ (cid:16) x n (cid:17) + 2 ρn F ( x ) ρ F ′ ( x ) χ (cid:16) x n (cid:17) χ ′ (cid:16) x n (cid:17) + 1 n F ( x ) ρ +1 χ ′ (cid:16) x n (cid:17) , and φ ′ n ( x ) F ′ ( x ) φ n ( x )= (cid:18) ρF ( x ) ρ − F ′ ( x ) χ (cid:16) x n (cid:17) + 1 n F ( x ) ρ χ ′ (cid:16) x n (cid:17)(cid:19) F ′ ( x ) F ( x ) ρ χ (cid:16) x n (cid:17) = ρF ( x ) ρ − F ′ ( x ) χ (cid:16) x n (cid:17) + 1 n F ( x ) ρ F ′ ( x ) χ (cid:16) x n (cid:17) χ ′ (cid:16) x n (cid:17) , which yields, for g n ( x ) := F ( x ) φ ′ n ( x ) − φ ′ n ( x ) F ′ ( x ) φ n ( x ), g n ( x ) = ρ ( ρ − F ( x ) ρ − F ′ ( x ) χ (cid:16) x n (cid:17) + 2 ρ − n F ( x ) ρ F ′ ( x ) χ (cid:16) x n (cid:17) χ ′ (cid:16) x n (cid:17) + 1 n F ( x ) ρ +1 χ ′ (cid:16) x n (cid:17) . OUND STATES OF A PAIR OF PARTICLES 9
One then decompse the above term G n as follows: G n = Z R + g n ( x )d x = ρ ( ρ − (cid:0) A + B n (cid:1) + 2 ρ − n C n + 1 n D n , (7) A := Z R + F ( x ) ρ − F ′ ( x ) d x ,B n := Z R + F ( x ) ρ − F ′ ( x ) (cid:18) χ (cid:16) x n (cid:17) − (cid:19) d x ,C n := Z R + F ( x ) ρ F ′ ( x ) χ (cid:16) x n (cid:17) χ ′ (cid:16) x n (cid:17) d x ,D n := Z R + F ( x ) ρ +1 χ ′ (cid:16) x n (cid:17) d x . We recall that 0 ≤ F ≤ F ′ = ψ ∈ L ( R + ) ∩ L ∞ ( R + ), which en-sures the finiteness of the integrals. One easily sees that A >
0, whilelim n → + ∞ B n = 0. We then estimate | C n | ≤ k χ ′ k ∞ Z R + F ( x ) ρ F ′ ( x )d x = k χ ′ k ∞ F ( ∞ ) ρ − − F (0) ρ − ρ + 1 = k χ ′ k ∞ ρ + 1 , | D n | ≤ k χ ′ k ∞ Z nn F ( x ) ρ +1 d x ≤ n k χ ′ k ∞ , and using (7) one has lim n → + ∞ G n = ρ ( ρ − A . Hence choosing any value ρ ∈ (cid:0) , (cid:1) we have G n < n , which concludes the proof.2.4. Finiteness of the discrete spectrum.
In this section we prove that Q + has only finitely many eigenvalues in ( −∞ , ε ).We first introduce a pair of smooth functions χ , χ : R → [0 , ∞ ) suchthat χ ( t ) = 1 for t ≤ χ ( t ) = 1 for t ≥
2, and χ + χ = 1. We set, for R > j = 1 , χ Rj ( x , x ) := χ j (cid:18) x − x R (cid:19) . Then, for any ϕ ∈ D [ Q + ] we have χ Rj ϕ ∈ D [ Q + ] and, by direct computation Q + [ ϕ, ϕ ] = Q + [ χ R ϕ, χ R ϕ ] + Q + [ χ R ϕ, χ R ϕ ] − Z Z Ω W R ϕ d x d x , (8) W R ( x , x ) := |∇ χ R | + |∇ χ R | = 2 R " χ ′ (cid:18) x − x R (cid:19) + χ ′ (cid:18) x − x R (cid:19) . Consider two following (overlapping) subdomains of Ω :Ω := (cid:8) ( x , x ) ∈ Ω : x < x + 2 R (cid:9) , Ω := (cid:8) ( x , x ) ∈ Ω : x > x + R (cid:9) , and define self-adjoint operators Q j in L (Ω j ), j ∈ { , } , by their forms Q [ ϕ, ϕ ] = Z Z Ω (cid:16)(cid:12)(cid:12) ∇ ϕ ( x , x ) (cid:12)(cid:12) + (cid:0) v ( x ) − W R ( x , x ) (cid:1) ϕ ( x , x ) (cid:17) d x d x , D [ Q ] = (cid:8) ϕ ∈ H (Ω ) : Q [ ϕ, ϕ ] < ∞ , ϕ = 0 on the line x = x + 2 R (cid:9) ,Q [ ϕ, ϕ ] = Z Z Ω (cid:16)(cid:12)(cid:12) ∇ ϕ ( x , x ) (cid:12)(cid:12) + (cid:0) v ( x ) − W R ( x , x ) (cid:1) ϕ ( x , x ) (cid:17) d x d x , D [ Q ] = (cid:8) ϕ ∈ H (Ω ) : Q [ ϕ, ϕ ] < ∞ , ϕ = 0 on the line x = x + R (cid:9) . Let us return back to (8). The functions χ Rj ϕ vanish outside Ω j , j ∈{ , } , and their restrictions to Ω j belong to D [ Q j ]. In addition, one has | χ R ϕ | + | χ R ϕ | = ϕ pointwise. This allows one to rewrite (8) as Q + [ ϕ, ϕ ] = Q [ χ R ϕ, χ R ϕ ] + Q [ χ R ϕ, χ R ϕ ] . (9)Consider an auxiliary operator b Q = Q ⊕ Q defined on L (Ω ) ⊕ L (Ω ),then D [ b Q ] = D [ Q ] × D [ Q ], with b Q (cid:2) ( ϕ , ϕ ) , ( ϕ , ϕ ) (cid:3) = Q [ ϕ , ϕ ] + Q [ ϕ , ϕ ] . The linear map J : L (Ω ) ∋ ϕ ( χ R ϕ, χ R ϕ ) ∈ L (Ω ) ⊕ L (Ω ) , is isometric and, hence, injective, with J D [ Q + ] ⊂ D [ b Q ], and Eq. (9) readsthen as Q + [ ϕ, ϕ ] = b Q [ J ϕ, J ϕ ]. Hence, if one denotes be E n ( L ) the n theigenvalue of a self-adjoint operator L , then the min-max principle gives, forany n ∈ N , E n ( Q + ) = inf V n ⊂D [ Q + ] sup = ϕ ∈ V n Q + [ ϕ, ϕ ] k ϕ k L (Ω ) = inf V n ⊂D [ Q + ] sup = ϕ ∈ V n b Q [ J ϕ, J ϕ ] k J ϕ k L (Ω ) ⊎ L (Ω ) = inf U n ⊂ J D [ Q + ] sup = ψ ∈ U n b Q [ ψ, ψ ] k ψ k L (Ω ) ⊕ L (Ω ) ≥ inf U n ⊂D [ b Q ] sup = ψ ∈ U n b Q [ ψ, ψ ] k ψ k L (Ω ) ⊕ L (Ω ) = E n ( b Q ) , where V n and U n stand for n -dimensional subspaces. Hence, if for a self-adjoint operator L and λ ∈ R we denote by N ( L, λ ) the number of eigen-values of L in ( −∞ , λ ), then it follows from the above constructions that N ( Q + , ε ) ≤ N ( b Q, ε ) = N ( Q , ε ) + N ( Q , ε ) . Hence, it is sufficient to show that N ( Q j , ε ) are finite for j ∈ { , } .Let us start with N ( Q , ε ): Consider the decomposition of Ω createdby the line x = L , i.e.Ω , int := (cid:8) ( x , x ) ∈ Ω : x < x + 2 R and x < R (cid:9) , Ω , ext := (cid:8) ( x , x ) ∈ Ω : x < x + 2 R and x > R (cid:9) , OUND STATES OF A PAIR OF PARTICLES 11 and consider the operators Q , • in L ( Q , • ) with • ∈ { int , ext } , given bytheir forms Q , • [ ϕ, ϕ ] = Z Z Ω , • (cid:12)(cid:12) ∇ ϕ ( x , x ) (cid:12)(cid:12) d x d x + Z Z Ω , • (cid:0) v ( x ) − W R ( x , x ) (cid:1) ϕ ( x , x ) d x d x , D [ Q , • ] = (cid:8) ϕ ∈ H (Ω , • ) : Q , • [ ϕ, ϕ ] < ∞ (cid:9) . The bilinear form for Q , int ⊕ Q , ext is an extension of the bilinear form for Q , and the min-max principle shows that the eigenvalues of Q can notbe lower than the respective eigenvalues of Q , int ⊕ Q , ext . In terms of thecounting functions this leads to N ( Q , ε ) ≤ N ( Q , int ⊕ Q , ext , ε ) = N ( Q , int , ε ) + N ( Q , ext , ε ) . The domain Ω , int is bounded, Lipschitz and D [ Q , int ] ⊂ H (Ω , int ) is com-pactly embedded into L (Ω , • ), which implies that Q , int is with compactresolvent, and then N ( Q , int , ε ) < ∞ for any fixed R >
0. On theother hand, the upper bound k W R k ≤ c/R with some c > v imply that for sufficiently large R one has v ( x ) − W R ( x , x ) ≥ ε for all ( x , x ) ∈ Ω , ext . It follows that Q , ext hasno spectrum below ε and N ( Q , ext , ε ). Therefore, there exists R > N ( Q , ε ) < ∞ for any R > R .In order to conclude it remains to show that N ( Q , ε ) < ∞ for large R >
0; note that Ω depends on R . Due to the fact that the functions in theform domain of Q vanish at the line x = x + R they can be extended byzero to functions in H ( R + × R ). Therefore, if one considers the operator b Q in L ( R + × R ) given by b Q [ ϕ, ϕ ] := Z Z R + × R (cid:12)(cid:12) ∇ ϕ ( x , x ) (cid:12)(cid:12) d x d x + Z Z R + × R (cid:0) v ( x ) − W R ( x , x ) (cid:1) ϕ ( x , x ) d x d x , D [ b Q ] = (cid:8) ϕ ∈ H ( R + × R ) : b Q [ ϕ, ϕ ] < ∞ (cid:9) , then it follows by the min-max principle that N ( Q , ε ) ≤ N ( b Q , ε ). There-fore, it is sufficient to show that N ( b Q , ε ) < ∞ for large R .The subsequent construction is inspired by the representation b Q = ( h ⊗ + ⊗ q ) − W R , where q is f
7→ − f ′′ in L ( R ) and W R is identified with the associatedmultplication operator.Let P be the orthogonal projection on R ψ in L ( R + ), then Π := P ⊗ is the orthogonal projection on ψ ⊗ L ( R ) in L ( R + × R ), i.e.(Π ϕ )( x , x ) = ψ ( x ) f ( x ) , f ( x ) := Z R + ϕ ( x , x ) ψ ( x ) d x . (10)Notice that Π is exactly the spectral projector on { ε } for h ⊗ (due to thefact that ε is a simple eigenvalue, see Proposition A.3) and it commutes with ⊗ q . We set Π ⊥ := − Π. Taking into account that both Π ϕ andΠ ⊥ ϕ are in D [ b Q ], for ϕ ∈ D [ b Q ] we obtain b Q [ ϕ, ϕ ] = b Q [Π ϕ, Π ϕ ] + b Q [Π ⊥ ϕ, Π ⊥ ϕ ] − W R [Π ϕ, Π ⊥ ϕ ] . (11)As W R is bounded, using Cauchy-Schwarz and triangular inequalities weestimate (cid:12)(cid:12) W R [Π ϕ, Π ⊥ ϕ ] (cid:12)(cid:12) = 2 (cid:12)(cid:12) h W R Π ϕ, Π ⊥ ϕ i L ( R + × R ) (cid:12)(cid:12) ≤ k W R Π ϕ k L ( R + × R ) k Π ⊥ ϕ k L ( R + × R ) ≤ R k W R Π ϕ k L ( R + × R ) + 1 R k Π ⊥ ϕ k L ( R + × R ) . Due to the assumption (B) on v , the eigenvalue ε of h is isolated, hence, E := inf (cid:0) σ ( h ) \ { ε } (cid:1) > ε , and( h ⊗ )[Π ⊥ ϕ, Π ⊥ ϕ ] ≥ E k Π ⊥ ϕ k L ( R + × R ) . It follows that, taking into account that the operator q is non-negative, b Q [Π ⊥ ϕ, Π ⊥ ϕ ] = ( h ⊗ )[Π ⊥ ϕ, Π ⊥ ϕ ] + ( ⊗ q )[Π ⊥ ϕ, Π ⊥ ϕ ] − W R [Π ⊥ ϕ, Π ⊥ ϕ ] ≥ E k Π ⊥ ϕ k L ( R + × R ) − W R [Π ⊥ ϕ, Π ⊥ ϕ ] . Summing up all the computations after (11) yields b Q [ ϕ, ϕ ] ≥ b Q [Π ϕ, Π ϕ ] − R k W R Π ϕ k L ( R + × R ) + (cid:16) E − R (cid:17) k Π ⊥ ϕ k L ( R + × R ) − W R [Π ⊥ ϕ, Π ⊥ ϕ ] . (12)Let A be the self-adjoint operator in ran Π given by A [Φ , Φ] = b Q [Φ , Φ] − R k W R Φ k L ( R + × R ) , D [ A ] = D [ b Q ] ∩ ran Π , and B be the operator of multiplication by E − /R − W R in ran Π ⊥ , whichis bounded and self-adjoint. Considering the unitary map J : L ( R + × R ) ∋ ϕ (Π ϕ, Π ⊥ ϕ ) ∈ ran Π ⊕ ran Π ⊥ we rewrite (12) as b Q [ ϕ, ϕ ] ≥ ( A ⊕ B )[ J ϕ, J ϕ ], which due to the min-maxprinciple implies N ( b Q , ε ) ≤ N ( A ⊕ B, ε ) = N ( A, ε ) + N ( B, ε ) . (13)As E > ε is fixed and k W R k ∞ ≤ c/R , for sufficiently large R andsome c > E − /R + W R ≥ ε showing that B has no spectrum in ( −∞ , ε ) and hence N ( B, ε ) = 0. The estimate (13)takes the form N ( b Q , ε ) ≤ N ( A, ε ), and now it is sufficient to show that N ( A, ε ) < ∞ for R being sufficiently large. OUND STATES OF A PAIR OF PARTICLES 13
In order to study A we rewrite, using the convention (10), b Q [ ψ ⊗ f, ψ ⊗ f ] = Z R (cid:16) f ′ ( x ) + (cid:0) ε − U R ( x ) (cid:1) f ( x ) (cid:17) d x ,U R ( x ) := Z R + W R ( x , x ) ψ ( x ) d x , (cid:13)(cid:13) W R ( ψ ⊗ f ) (cid:13)(cid:13) L ( R + × R ) = Z R V R ( x ) f ( x ) d x ,V R ( x ) := Z R + W R ( x , x ) ψ ( x ) d x . Recall that for ϕ ∈ D [ b Q ] ≡ D [ h ⊗ + ⊗ q ] due to the spectral theoremone has Π ϕ ≡ ( P ⊗ ) ϕ ≡ ψ ⊗ f ∈ D [ ⊗ q ], i.e. f ∈ D [ q ] = H ( R ).Consequently, one has A [ ψ ⊗ f, ψ ⊗ f ] = ε k f k L ( R ) + q [ f, f ]with q being the self-adjoint operator in L ( R ) given by the form q [ f, f ] := Z R (cid:16) f ′ ( x ) − Z R ( x ) f ( x ) (cid:17) d x , Z R = U R + RV R , defined on D [ q ] = H ( R ). As the map ran Π ∋ ψ ⊗ f f ∈ L ( R )is unitary, one sees that A is unitarily equivalent to q + ε , which yields N ( A, ε ) = N ( q , q has only finitely many negative eigen-values. The task is simplified by the fact that q is a standard one-dimensional Schr¨odinger operator. Recall that, by construction one has W R ∈ L ∞ and supp W R ⊂ (cid:8) ( x , x ) : R < x − x < R (cid:9) , i.e. W R ( x , x ) vanishes except for x − R < x < x − R . Due to Z R ( x ) = Z R + ∩ [ x − R,x − R ] (cid:16) W R ( x , x ) + RW R ( x , x ) (cid:17) ψ ( x ) d x ≤ (cid:16) k W k ∞ + R k W k ∞ (cid:17) Z R + ∩ [ x − R,x − R ] ψ ( x ) d x ≤ k W k ∞ + R k W k ∞ (14)it follows that Z R is bounded, continuous, and Z R ( x ) = 0 for x ≤ R . Inview of the well-known Bargman estimate (see, e.g., Theorem 5.1 in Chapter2.5 of [BS]) in order to obtain N ( q , < ∞ it is sufficient to show Z R | x | Z R ( x ) dx ≡ Z ∞ R x Z R ( x ) dx < ∞ (15)(recall that W R ≥
0, and then Z R ≥ a > c := Z R + e ax ψ ( x ) d x < ∞ . For x ≥ R one then estimates, using (14), Z ( x ) ≤ (cid:0) k W k ∞ + R k W k ∞ (cid:1) Z x − Rx − R ψ ( x ) d x ≤ (cid:0) k W k ∞ + R k W k ∞ (cid:1) e − a ( x − R ) Z x − Rx − R e ax ψ ( x ) d x ≤ c (cid:0) k W k ∞ + R k W k ∞ (cid:1) e − a ( x − R ) = c e − ax with c := (cid:16) k W k ∞ + R k W k ∞ (cid:17) e Ra . Hence, Z ∞ R x Z R ( x ) dx = Z RR x Z R ( x ) dx + Z ∞ R x Z R ( x ) dx ≤ R k Z R k ∞ + c Z ∞ R x e − ax d x < ∞ . This proves (15) and completes the proof.
Appendix A. Some constructions for Schr¨odinger operatorswith singular potentials
In this section we recall briefly some facts related to Schr¨odinger operatorswith singular potentials. All these facts are well-known to the specialistsbut we are not aware of their presentation within a single reference and ina suitable form under our rather weak assumptions on the potential v , andwe decided to collect them here with proofs. An interested reader may refere.g. to [EGNT] for a more detailed discussion of singular potentials.For the whole of this section, we write R + = (0 , ∞ ) and let v ∈ L ( R + )be a real-valued potential with v − := max {− v, (cid:9) ∈ L ∞ ( R + ). Let h be theself-adjoint operator in L ( R + ) generated by its bilinear form h [ ϕ, ϕ ] = Z R + (cid:0) | ϕ ′ | + v | ϕ | (cid:1) d x , D [ h ] = (cid:8) ϕ ∈ H ( R + ) : Z R + v ϕ d x < ∞ (cid:9) . Recall that D [ h ] stands for the form domain, while the operator domain isdenoted by D ( h ). In other words, a function ψ belongs to the operatordomain D ( h ) of h and hψ = ψ h if and only if ψ ∈ D [ h ] and h [ ϕ, ψ ] = Z R + ϕ ψ h d x for all ϕ ∈ D [ h ] . As the preceding equality holds for all ϕ ∈ C ∞ ( R + ) ⊂ D [ h ], it follows that h acts as hψ = − ψ ′′ + vψ . We give a proof of the following technical fact: Proposition A.1.
Let ψ ∈ D ( h ) and χ ∈ C ∞ ( R ) with χ being constant ina neighborhood of , then χψ ∈ D ( h ) and h ( χψ ) = χh ( ψ ) − χ ′ ψ ′ − χ ′′ ψ .Proof. Remark first that χψ ∈ D [ h ]. Then, we simply need to show that h [ ϕ, χψ ] = Z R + ϕ ( χhψ − χ ′ ψ ′ − χ ′′ ψ ) d x (16) OUND STATES OF A PAIR OF PARTICLES 15 for any ϕ ∈ D [ h ]. On the other hand, the assumption ψ ∈ D ( h ) alreadygives h [ χϕ, ψ ] = Z R + χϕ ( − ψ ′′ + vψ ) d x . (17)Taking the difference between (16) and (17) one sees that it is sufficient toshow the equality h [ ϕ, χψ ] − h [ χϕ, ψ ] = − Z R + (2 ϕχ ′ ψ ′ + ϕχ ′′ ψ ) d x , which reads in a more detailed form as Z R + ( ϕ ′ χ ′ ψ − ϕχ ′ ψ ′ ) d x = − Z R + (2 ϕχ ′ ψ ′ + ϕχ ′′ ψ ) d x . (18)One clearly has Z R + ( ϕ ′ χ ′ ψ + ϕχ ′ ψ ′ + ϕχ ′′ ψ ) d x = Z R + ( ϕχ ′ ψ ) ′ d x = ( ϕχ ′ ψ )( ∞ ) − ( ϕχ ′ ψ )(0) = 0 . By regrouping the terms one arrives at (18), which concludes the proof. (cid:3)
For each ψ ∈ D ( h ) one has − ψ ′′ + vψ ∈ L ( R + ). Due to the inclusions D ( h ) ⊂ D [ h ] ⊂ H ( R + ) ⊂ L ∞ ( R + ) it follows that vψ ∈ L ( R + ) and then ψ ′′ ∈ L ( R + ) and ψ ′ ∈ C ( R + ). That implies that the values ψ ( y ) and ψ ′ ( y ) make sense for any y ∈ R + . Let us add some precisions on the behaviornear 0 and ∞ . Proposition A.2.
Let ψ ∈ D ( h ) , then lim x → ( ψ ′ ψ )( x ) =: ( ψψ ′ )(0) = 0 , lim x →∞ ( ψ ′ ψ )( x ) =: ( ψψ ′ )( ∞ ) = 0 , (19) and the integration-by-parts formula Z y ψ ′ ( x ) ψ ′ ( x ) d x = ψ ( y ) ψ ′ ( y ) − Z y ψ ( x ) ψ ′′ ( x ) d x holds for y ∈ R + .Proof. In view of the above regularity of ψ , for any 0 < ǫ < y one has thestandard integration by parts Z yǫ ψ ′ ( x ) d x = ( ψψ ′ )( y ) − ( ψψ ′ )( ǫ ) − Z yǫ ψ ( x ) ψ ′′ ( x ) d x , (20)and we need to show that the passage to the limit ǫ → + is possible. Bythe definition of D ( h ) one has Z R + (cid:0) ψ ′ ( x ) + v ( x ) ψ ( x ) (cid:1) d x ≡ h [ ψ, ψ ] ≡ h ψ, hψ i L ( R + ) = Z R + ψ ( − ψ ′′ + vψ ) d x = lim ǫ → + Z ǫ − ǫ ψ ( − ψ ′′ + vψ ) d x = lim ǫ → + n Z ǫ − ǫ (cid:2) ψ ′ ( x ) + v ( x ) ψ ( x ) (cid:3) d x − ( ψψ ′ )( ǫ − ) + ( ψψ ′ )( ǫ ) o implying lim ǫ → + (cid:16) ( ψψ ′ )( ǫ − ) − ( ψψ ′ )( ǫ ) (cid:17) = 0 . (21)Let χ ∈ C ∞ ( R ) such that χ = 1 near zero, then χψ ∈ D ( h ) due to Propo-sition A.1, and (21) also holds for ψ replaced by χψ . As χψ is identicallyzero at infinity and coincides with ψ near the origin, one obtainslim ǫ → + ( χψ )( ǫ )( χψ ) ′ ( ǫ ) ≡ lim ǫ → + ( ψψ ′ )( ǫ ) = 0 . Using (21) again one has lim ǫ → + ( ψψ ′ )( ǫ − ) ≡ lim x → + ∞ ( ψψ ′ )( x ) = 0. Bypassing to the limit ǫ → + in (20) one concludes the proof. (cid:3) Assume from now on that the bottom ε of the spectrum of h is aneigenvalue. Proposition A.3.
The eigenvalue ε is simple, and the corresponding eigen-function ψ can be chosen strictly positive.Proof. Let ψ ∈ ker( h − ε ) with ψ h [ ψ , ψ ] k ψ k L ( R + ) = min ψ ∈D [ h ] , ψ =0 h [ ψ, ψ ] k ψ k L ( R + ) . For | ψ | ∈ D [ h ] one has h (cid:2) | ψ | , | ψ | (cid:3) ≤ h [ ψ , ψ ] and (cid:13)(cid:13) | ψ | (cid:13)(cid:13) L ( R + ) = k ψ k L ( R + ) ,which shows that | ψ | ∈ ker( h − ε ) ⊂ D [ h ] ⊂ C ( R + ).Assume that ψ ( a ) = 0 for some a >
0, then from | ψ | ∈ C ( R + ) it followsthat ψ ′ ( a ) = 0. Let us show that this implies ψ ( x ) = 0 for all x >
0. Thatis essentially Gronwall’s lemma, but we prefer to include it for completeness.To be definite, consider x > a (the other case x < a is considered in thesame way). The fact hψ = ε ψ can be rewritten asΨ( x ) = Z xa M ( t )Ψ( t ) d t , Ψ( x ) = (cid:18) ψ ( x ) ψ ′ ( x ) (cid:19) , M ( x ) = (cid:18) v ( x ) − E (cid:19) . Then for f := | Ψ | R ≥ m := k M k ∈ L ( R + ) one has f ( x ) ≤ Z xa m ( t ) f ( t ) d t ≤ ε + Z xa m ( t ) f ( t ) d t =: Φ( x ) P unktweg for all ε > x > a . Therefore, Φ ′ ( x ) / Φ( x ) ≤ m ( x ), so by integratingbetween a and x one arrives atΦ( x ) ≤ Φ( a ) exp Z xa m ( t ) d t for x > a .Due to f ≤ Φ and Φ( a ) = ε one obtains0 ≤ f ( x ) ≤ ε exp Z xa m ( t ) d t , x > a . As ε > f ( x ) = 0 for x > a , which implies ψ ( x ) = 0 for x > a .We conclude that an eigenfunction ψ ∈ ker( h − ε ) cannot vanish, hence,up to a multiplicative factor it is strictly positive. As two strictly positivefunctions cannot be orthogonal in L , the eigenvalue ε is simple. (cid:3) OUND STATES OF A PAIR OF PARTICLES 17
For the rest of the section, let ψ be the strictly positive eigenfunction for ε , with a unit L -norm. Proposition A.4.
Let ρ > , then the function φ : y (cid:18)Z y ψ ( x ) d x (cid:19) ρ is in H (0 , a ) for any a > .Proof. Since φ ∈ L ∞ ( R + ), we only have to take care of the derivative. Adirect calculation shows that Z a φ ′ ( y ) d y = ρ Z a ψ ( y ) (cid:18)Z y ψ ( x ) d x (cid:19) ρ − d y . Let y ( · ) be the inverse of F := y Z y ψ ( x ) d x , which is a diffeomorphism due to ψ > y ′ ( s ) = 1 F ′ ( F − ( s )) = 1 ψ (cid:0) F − ( s ) (cid:1) , and consequently k φ ′ k L (0 ,a ) = ρ Z F ( a )0 ψ (cid:0) F − ( s ) (cid:1) s ρ − d s . As ψ ∈ L ∞ ( R + ), the integral is finite for ρ > . (cid:3) For the rest of the section we assume finally that v ∞ := lim inf x → + ∞ v ( x ) > ε . (22)For L >
0, define two operators h N/DL in L (0 , L ) by h N/DL [ ϕ, ϕ ] = Z L (cid:0) ( ϕ ′ ) + vϕ (cid:1) d x with form domains D [ h NL ] = n ϕ ∈ H (0 , L ) : Z L vϕ d x < ∞ o , D [ h DL ] = (cid:8) ϕ ∈ D [ h NL ] : ϕ ( L ) = 0 (cid:9) , and denote by ε ( N/D )0 ( L ) the respective lowest eigenvalues. Proposition A.5.
There holds lim L →∞ ε ( N )0 ( L ) = ε .Proof. By the min-max principle one has ε ≤ ε ( D )0 ( L ) for all L > e h NL be the operator on L ( L, ∞ ) associated with the form e h L [ ϕ, ϕ ] := Z ∞ L (cid:0) ( ϕ ′ ) + vϕ (cid:1) d x, D [ e h L ] := { ϕ ∈ H ( L, ∞ ) : e h L [ ϕ, ϕ ] < ∞} . Again, the min-max principle then implies that ε = ≡ inf σ ( h ) ≥ inf σ ( h NL ⊕ ˜ h NL ) . (23)On the other hand, inf σ ( h NL ⊕ e h NL ) = min (cid:8) ε N ( L ) , inf σ ( e h NL ) (cid:9) , and due tothe assumption (22) for sufficiently large L one has inf σ ( e h NL ) > ε . It followsfrom (23) that ε ≥ ε ( N )0 ( L ) for large L .Now let us take χ , χ ∈ C ∞ ( R ) with χ + χ = 1, that χ ( t ) = 1 for t ≤
12 and χ ( t ) = 0 for t ≥ χ Lj ( t ) := χ j ( t/L ), j = 1 ,
2. For any ϕ ∈ D [ h NL ] we obtain h NL [ ϕ, ϕ ] = h NL [ χ L ϕ, χ L ϕ ] + h NL [ χ L ϕ, χ L ϕ ] − Z L (cid:2) (( χ L ) ′ ) + (( χ L ) ′ ) (cid:3) ϕ d x ≥ h NL [ χ L ϕ, χ L ϕ ] + h NL [ χ L ϕ, χ L ϕ ] − cL k ϕ k L (0 ,L ) for some constant c >
0. Since ( χ L ϕ )( x ) = 0 for x ≥ L we conclude that χ L ϕ ∈ D [ h DL ] and then h NL [ χ L ϕ, χ L ϕ ] = h DL [ χ L ϕ, χ L ϕ ] ≥ ε ( D )0 ( L ) k χ L ϕ k L (0 ,L ) ≥ ε k χ L ϕ k L (0 ,L ) . Using the assumption (22) on v , one can choose L sufficiently large to have v ≥ ε in ( − L/ , L ). Due to supp χ L ϕ ⊂ [ L/ , L ] there holds h NL [ χ L ϕ, χ L ϕ ] ≥ Z LL/ v ( χ L ϕ ) d x ≥ ε Z LL/ ( χ L ϕ ) d x = ε k χ L ϕ k L (0 ,L ) . Therefore, for large L one has, uniformly in ϕ ∈ D [ h NL ], h NL [ ϕ, ϕ ] ≥ ε k χ L ϕ k L (0 ,L ) + ε k χ L ϕ k L (0 ,L ) − cL k ϕ k L (0 ,L ) = (cid:16) ε − cL (cid:17) k ϕ k L (0 ,L ) , which implies ε ( N )0 ( L ) ≥ ε ( L ) − c/L due to the min-max principle. Sum-ming up we obtain, for L > ε − cL ≤ ε ( N )0 ( L ) ≤ ε , which proves the statement. (cid:3) In the next result we recall an Agmon-type estimate for the ground state ψ of h . Recall that ψ was chosen strictly positive and normalized in L ( R + ). Proposition A.6 (Agmon-type estimate) . For any θ ∈ (0 , there is R > with v ( x ) ≥ ε for x ≥ R such that Z ∞ e θ Φ( x ) ψ ( x ) d x < ∞ , Φ( x ) := , x ≤ R , Z xR p v ( t ) − ε d t , x > R . OUND STATES OF A PAIR OF PARTICLES 19
Proof.
Let us take a sufficiently large
R > v ( x ) ≥ ε for x ≥ R ;the value of R will be adjusted later. Define Φ as above, and for L > φ L ( x ) := θ min { Φ( x ) , L } , then φ L ∈ L ∞ ( R + ) and (cid:12)(cid:12) φ ′ L ( x ) (cid:12)(cid:12) ≤ θ x>R ( x ) p v ( x ) − ε , where x>R standsfor the indicator function of the set (cid:8) x ∈ R + : x > R (cid:9) .Let us show first thatfor any c ∈ R one has e cφ L ψ ∈ D [ h ] . (24)By construction, e cφ L ∈ L ∞ ( R + ), so e cφ L ψ ∈ L ( R + ) and Z R + v ( e cφ L ψ ) d x < ∞ due to Z R + vψ d x < ∞ . Furthermore, ( e cφ L ψ ) ′ = cφ ′ L e cφ L ψ + e cφ L ψ , and the second summand isin L ( R + ) due to ψ ∈ H ( R + ), while the first summand is finite due to Z R + ( φ ′ L e cφ L ψ ) d x ≤ θ e cL Z ∞ R ( v − ε ) ψ d x < ∞ . Hence, the claim (24) is proved.Now we compute h [ e φ L ψ , e φ L ψ ] = Z R + (cid:16)(cid:0) φ ′ L e φ L ψ + e φ L ψ ′ (cid:1) + v ( e φ L ψ ) (cid:17) d x = Z R + ( φ ′ L ) e φ L ψ d x + Z R + (cid:16) ( e φ L ψ ) ′ ψ ′ + v e φ L ψ ψ (cid:17) d x . Due to (24) one can transform the last summand on the right-hand side as Z R + (cid:16) ( e φ L ψ ) ′ ψ ′ + v e φ L ψ ψ (cid:17) d x = h (cid:2) e φ L ψ , ψ (cid:3) = h e φ L ψ , hψ i L ( R + ) = ε (cid:10) e φ L ψ , ψ (cid:11) L ( R + ) = ε Z R + e φ L ψ d x , which yields h [ e φ L ψ , e φ L ψ ] = Z R + (cid:16) ( φ ′ L ) + ε (cid:17) e φ L ψ d x . (25)Now let us pick any δ >
0. The min-max principle applied to h NR gives Z R (cid:16)(cid:0) ( e φ L ψ ) ′ (cid:1) + v ( e φ L ψ ) (cid:17) d x ≥ ε ( N )0 ( R ) Z R e φ L ψ d x . Hence, for large
R > ε ( N )0 ( R ) ≥ ε − δ due to Proposition A.5,and h [ e φ L ψ , e φ L ψ ] = Z R + (cid:16)(cid:0) ( e φ L ψ ) ′ (cid:1) + v ( e φ L ψ ) (cid:17) d x = Z R (cid:16)(cid:0) ( e φ L ψ ) ′ (cid:1) + v ( e φ L ψ ) (cid:17) d x + Z ∞ R (cid:16)(cid:0) e φ L ψ ) ′ (cid:1) + v ( e φ L ψ ) (cid:17) d x ≥ ε ( N )0 ( R ) Z R e φ L ψ d x + Z ∞ R ve φ L ψ d x ≥ ( ε − δ ) Z R e φ L ψ d x + Z ∞ R ve φ L ψ d x . By combining this last inequality with (25) we arrive at Z R + (cid:2) ( φ ′ L ) + ε (cid:3) e φ L ψ d x ≥ ( ε − δ ) Z R e φ L ψ d x + Z ∞ R ve φ L ψ d x . This rewrites as Z R (cid:0) ( φ ′ L ) + δ ) e φ L ψ d x ≥ Z ∞ R (cid:0) v − ε − ( φ ′ L ) (cid:1) e φ L ψ d x and taking into account the above choice of R and φ L we arrive at δ Z R ψ d x ≥ Z ∞ R (cid:0) v − ε − ( φ ′ L ) (cid:1) e φ L ψ d x ≥ (1 − θ ) Z ∞ R ( v − ε ) e φ L ψ d x . As δ > δ < v ∞ − ε , then for large R one has v − ε ≥ δ in ( R, ∞ ), and it follows from the preceding inequalitythat Z R ψ d x ≥ (1 − θ ) Z ∞ R e φ L ψ d x . Consequently, Z ∞ e φ L ψ d x = Z R e φ L ψ d x + Z ∞ R e φ L ψ d x = Z R ψ d x + Z ∞ R e φ L ψ d x ≤ (cid:0) − θ ) (cid:1) Z R ψ d x ≤ − θ , or, in a detailed form, Z ∞ exp (cid:16) θ min (cid:8) Φ( x ) , L (cid:9)(cid:17) ψ d x ≤ − θ . As the constant on the right-hand side is independent of the choice of L , thestatement then follows by taking the limit L → ∞ . (cid:3) We prefer to give a simplified version of the preceding estimate, whichwill be easier to use in the main text:
Corollary A.7.
For some a > there holds Z R + e ax ψ ( x ) d x < ∞ . OUND STATES OF A PAIR OF PARTICLES 21
Proof.
Due to the assumption (22) on v , for some b > v ( x ) − ε ≥ b for large x , and then the function Φ in Proposition A.6 satisfies the inequalityΦ( x ) ≥ √ b ( x − R ) − c for all x (with a fixed c > Z R + e θ √ b x ψ d x = e θ √ b R +2 θc Z R + e θ √ b ( x − R ) − θc ψ d x ≤ e θ √ b R +2 θc Z R + e θ Φ ψ d x < ∞ , which gives the claim with a := 2 θ √ b . (cid:3) We finish this appendix by mentioning two classical cases for which theassumption (22) is satisfied. Recall that v ∞ := lim inf x → + ∞ v ( x ) . Proposition A.8.
There holds inf σ ess ( h ) ≥ v ∞ .Proof. Let e h L be the operator on L ( L, ∞ ) given by its bilinear form e h L [ ϕ, ϕ ] := Z ∞ L (cid:0) ( ϕ ′ ) + vϕ (cid:1) d x , D [ e h L ] := (cid:8) ϕ ∈ H ( L, ∞ ) : e h L [ ϕ, ϕ ] < ∞ (cid:9) . Then the min-max principle implies inf σ ess ( h ) ≥ inf σ ess ( h NL ⊕ e h L ) for any L >
0. The operator h NL has compact resolvent and an empty essentialspectrum, hence, inf σ ess ( h ) ≥ inf σ ess ( e h L ). For any a < v ∞ one can choosea large L > v ≥ a in ( L, ∞ ), which leads to inf σ ( e h L ) ≥ a . Itfollows that inf σ ess ( h ) ≥ inf σ ess ( e h L ) ≥ inf σ ( e h L ) ≥ a . As a < v ∞ is arbitrary, this gives the result. (cid:3) Proposition A.9. If v ∞ = + ∞ , then the bottom of the spectrum of h is anisolated eigenvalue ε with ε < v ∞ .Proof. In this case σ ess ( h ) = ∅ by Proposition A.8, i.e. h is with compactresolvent. Its lowest eigenvalue ε is then automatically isolated, and theinequality ε < v ∞ is just the finiteness of ε . (cid:3) Proposition A.10.
Assume that v ∞ < + ∞ and that v − v ∞ ∈ L ( R + ) with Z R + (cid:0) v ( x ) − v ∞ (cid:1) d x < , then the bottom ε of the spectrum of h is an isolated eigenvalue, and itsatisfies ε < v ∞ .Proof. In view of Proposition A.8 it is sufficient to establish the existence ofeigenvalues in ( −∞ , v ∞ ), for which it is sufficient to find a function ϕ ∈ D [ h ]with h [ ϕ, ϕ ] − v ∞ k ϕ k L ( R + ) < For δ > ϕ : x e − δx , then ϕ ∈ D [ h ] with h [ ϕ, ϕ ] − v ∞ k ϕ k L ( R + ) = δ Z R + e − δx d x + Z R + (cid:0) v ( x ) − v ∞ (cid:1) e − δx d x = δ Z R + (cid:0) v ( x ) − v ∞ (cid:1) e − δx d x , and the right-hand side converges to a strictly negative limit as δ → + . (cid:3) Remark A.11.
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Department of Mathematics, Technion-Israel Institute of Tech-nology, 629 Amado Building, Haifa 32000, Israel
E-mail address : [email protected] (J. Kerner) Department of Mathematics and Computer Science, FernUniver-sit¨at in Hagen, 58084 Hagen, Germany
E-mail address : [email protected] (K. Pankrashkin)
Laboratoire de Math´ematiques d’Orsay, Univ. Paris-Sud,CNRS, Universit´e Paris-Saclay, 91405 Orsay, France
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