On the spectrum of a pair of particles on the half-line J. Kerner - PowerPoint PPT Presentation

On the spectrum of a pair of particles on the half-line J. Kerner FernUniversit¨ at in Hagen Graz 2019 joint work with S. Egger and K. Pankrashkin

The model • Two (distinguishable) particles moving on the half-line R + = (0 , ∞ ). • On the Hilbert space L 2 ( R + × R + ) we consider the two-particle Hamiltonian � | x 1 − x 2 | � H = − ∂ 2 − ∂ 2 + v √ ∂ x 2 ∂ x 2 2 1 2

The model v a real-valued interaction potential such that: • v ∈ L 1 loc ( R + ) and max {− v , 0 } ∈ L ∞ ( R + ) • The one-particle operator h = − d 2 d x 2 + v ( x ) on L 2 ( R + ) is such that inf σ ( h ) =: ε 0 is an isolated (non-degenerate) eigenvalue • ε 0 < lim inf x →∞ v ( x ) := v ∞

Main results Theorem The essential spectrum of H is given by the interval [ ε 0 , ∞ ) . Furthermore, the discrete spectrum is non-empty and finite. • Note that the two-particle Hamiltonian has ground state energy strictly lower than that of h . • The existence of a discrete spectrum is a quantum geometrical effect. If one considers the pair on the full real line, no discrete spectrum exists.

Motivation • The pairing of electrons plays a central role in the formation of the superconducting phase in metals (Cooper pairs). • The discrete spectrum indeed leads to a Bose-Einstein condensation of non-interacting pairs of particles. Hence, geometrical effects lead to a Bose-Einstein condensation (see also Exner&Zagrebnov 2005, K. 2017/18)

Ideas on the proof: A reduction of the problem • We actually prove the theorem for the operator Q + , defined on L 2 (Ω 0 ) where Ω 0 = { ( x 1 , x 2 ) ∈ R 2 + : 0 < x 1 < x 2 } . • � � |∇ ϕ | 2 + v ( x 1 ) | ϕ | 2 � Q + [ ϕ, ϕ ] = d x 1 d x 2 Ω 0 with form domain D ( Q + ) = { ϕ ∈ H 1 (Ω 0 ) : Q + [ ϕ, ϕ ] < ∞}

Ideas on the proof: Essential spectrum In a first step one proves that inf σ ess ( H ) ≥ ε 0 : • Here one employs an operator bracketing argument, dissecting Ω 0 into three domains using the two lines x 1 = L and x 2 = L . • Due to the lim inf-condition on the potential v , only the semi-infinite rectangle Ω 2 is important. • By a separation of variables one concludes that inf σ ess (( − ∆ + v ) | Ω 2 ) = inf σ ess ( h N L ), where h N L is the finite-volume version of h . • The final step is to realise that inf σ ess ( h N L ) → ε 0 as L → ∞ .

Ideas on the proof: Essential spectrum In a second step one proves that [ ε 0 , ∞ ) ⊂ inf σ ess ( H ): • This is done by constructing a suitable Weyl sequence. We set, for any k ∈ [0 , ∞ ), ϕ n ( x 1 , x 2 ) := ψ 0 ( x 1 ) τ ( n − x 1 ) · e ikx 2 τ ( x 2 − n ) τ (2 n − x 2 ); here ψ 0 ∈ D ( h ) is the ground state of h and τ : R → R is a smooth function with 0 ≤ τ ≤ 1 and τ ( x ) = 1 for x ≥ 2 and τ ( x ) = 0 for x ≤ 1. • A direct calculation then shows that � ( − ∆ − ( ε 0 + k 2 )) ϕ n � 2 L 2 (Ω 0 ) − → 0 � ϕ n � 2 L 2 (Ω 0 ) as n → ∞ .

On the existence of a discrete spectrum The general strategy is to find a function ϕ in the form domain of Q + such that Q + [ ϕ, ϕ ] − ε 0 � ϕ � 2 L 2 (Ω 0 ) < 0 . • We define ϕ n ( x 1 , x 2 ) := ψ 0 ( x 1 ) φ n ( x 2 ) � x 2 � • Here φ n ( x 2 ) = φ ( x 2 ) χ ; χ is a smooth cut-off function n with 0 ≤ χ ≤ 1 and χ ( t ) = 1 for t ≤ 1 and χ ( t ) = 0 for t ≥ 2. • Most importantly, for ρ ∈ (1 / 2 , 1), we set � x 2 φ 1 /ρ ( x 2 ) := F ( x 2 ) := | ψ 0 ( t ) | 2 d t . 0

On the existence of a discrete spectrum • We note that ϕ n is in the form domain of Q + . • A direct calculation then shows that Q + [ ϕ n , ϕ n ] − ε 0 � ϕ n � 2 L 2 (Ω 0 ) < 0 for n large enough.

On the finiteness of the discrete spectrum The basic strategy is to reduce the two-dimensional problem to an effective one-dimensional one. This then allows one to employ well-known Bargmann estimates on the number of eigenvalue negative eigenvalues. • For R > 0, we introduce the domains Ω 1 := { ( x 1 , x 2 ) ∈ Ω 0 : x 2 < x 1 + 2 R } Ω 2 := { ( x 1 , x 2 ) ∈ Ω 0 : x 2 > x 1 + R } and, j = 1 , 2, � x 2 − x 1 � χ R j ( x 1 , x 2 ) := χ j R with χ 1 , χ 2 : R → [0 , ∞ ) such that χ 1 ( t ) = 1 for t ≤ 1, χ 2 ( t ) = 1 for t ≥ 2 as well as χ 2 1 ( t ) + χ 2 2 ( t ) = 1.

On the finiteness of the discrete spectrum • A direct calculation shows that � Q + [ ϕ, ϕ ] = Q + [ χ R 1 ϕ, χ R 1 ϕ ] + Q + [ χ R 2 ϕ, χ R W R | ϕ | 2 d x 2 ϕ ] − Ω 0 with 1 | 2 + |∇ χ R W R ( x 1 , x 2 ) := |∇ χ R 2 | 2 • Consequently, we can introduce two operators Q 1 , Q 2 on Ω 1 , Ω 2 such that Q + [ ϕ, ϕ ] = Q 1 [ χ R 1 ϕ, χ R 1 ϕ ] + Q 2 [ χ R 2 ϕ, χ R 2 ϕ ] . • The operator Q j differs from Q + on the corresponding domain Ω j by adding the effective potential − W R . Note that we impose Dirichlet boundary conditions along the defining lines of Ω j .

On the finiteness of the discrete spectrum • We denote by N ( A , λ ) the number of eigenvalues (counted with multiplicity) below λ ∈ R of the self-adjoint operator A . • From the previous relation we can compare Q + with Q 1 ⊕ Q 2 (min-max principle) to obtain N ( Q + , ε 0 ) ≤ N ( Q 1 , ε 0 ) + N ( Q 2 , ε 0 ) . Hence, it remains to show that N ( Q 1 , ε 0 ) , N ( Q 2 , ε 0 ) < ∞ .

On the finiteness of the discrete spectrum • To show that N ( Q 1 , ε 0 ) is finite, one decomposes Ω 1 using the additional straight line x 1 = L . The lim inf-condition on v shows that the operator on the “outer” part (i.e., where x 1 > L ) has no spectrum below ε 0 . On the other hand, the remaining domain is bounded and hence the corresponding operator has purely discrete spectrum, implying the statement.

On the finiteness of the discrete spectrum • Regarding N ( Q 2 , ε 0 ) we introduce another comparison operator � Q 2 for which one has N ( Q 2 , ε 0 ) ≤ N ( � Q 2 , ε 0 ) (again by min-max principle). • More explicitly, we define � Q 2 on L 2 ( R + × R ) via its form � � |∇ ϕ | 2 + ( v ( x 1 ) − W R ( x 1 , x 2 )) | ϕ | 2 � � Q 2 [ ϕ, ϕ ] := d x R + × R D ( � Q 2 ) := { ϕ ∈ H 1 ( R + × R ) : � Q 2 [ ϕ, ϕ ] < ∞} . • We introduce the projection Π via � (Π ϕ )( x 1 , x 2 ) := ψ 0 ( x 1 ) · ϕ ( x 1 , x 2 ) ψ 0 ( x 1 ) d x 1 R +

On the finiteness of the discrete spectrum • A calculation then shows that Q 2 [ ϕ, ϕ ] ≥ � � Q 2 [Π ϕ, Π ϕ ] − R � W R Π ϕ � 2 L 2 ( R + × R ) � � E 2 − 1 � Π ⊥ ϕ � 2 L 2 ( R + × R ) − W R [Π ⊥ ϕ, Π ⊥ ϕ ] , + R where E 2 := inf { σ ( h ) \ ε 0 } . • Hence, the first two terms define a self-adjoint operator A on ran Π, and the last two terms a multiplication operator on ran Π ⊥ . • Again by the min-max principle, we conclude that N ( � Q 2 , ε 0 ) ≤ N ( A , ε 0 ) + N ( B , ε 0 )

On the finiteness of the discrete spectrum • One can show that, for sufficiently large R > 0, B has no spectrum in ( −∞ , ε 0 ) and hence N ( B , ε 0 ) = 0. • Finally, A is effectively a one-dimensional Schr¨ odinger operator with some effective potential. Classical estimates (Bargmann estimates) then show that N ( A , ε 0 ) < ∞ .

On the finiteness of the discrete spectrum • We remark that no bound on the number of eigenvalues in the discrete spectrum was derived! • However, if one considers the initial case where the potential v was informally defined as, d > 0, � √ 0 for x < d / 2 , v ( x ) := ∞ else , then one can show that the discrete spectrum consists of exactly one eigenvalue.

Thank you for your attention! S. Egger, J. Kerner, and K. Pankrashkin Bound states of a pair of particles on the half-line with a general interaction potential , arXiv:1812.06500. J. Kerner On the number of isolated eigenvalues of a pair of particles in a quantum wire , arXiv:1812.11804.