Spectral analysis of a class of Schroedinger operators exhibiting a parameter-dependent spectral transition
Diana Barseghyan, Pavel Exner, Andrii Khrabustovskyi, Milos Tater
SSpectral analysis of a class of Schr¨odinger operatorsexhibiting a parameter-dependent spectraltransition
Diana Barseghyan
Department of Mathematics, University of Ostrava, 30. dubna 22, 70103 Ostrava,Czech RepublicNuclear Physics Institute, Academy of Sciences of the Czech Republic, Hlavn´ı 130,25068 ˇReˇz near Prague, Czech RepublicE-mail: [email protected]
Pavel Exner
Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Hlavn´ı 130,25068 ˇReˇz near Prague, Czech RepublicDoppler Institute, Czech Technical University, Bˇrehov´a 7, 11519 Prague, CzechRepublicE-mail: [email protected]
Andrii Khrabustovskyi
Institute of Analysis, Karlsruhe Institute of Technology, Englerstr. 2, 76131Karlsruhe, GermanyE-mail: [email protected]
Miloˇs Tater
Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Hlavn´ı 130,25068 ˇReˇz near Prague, Czech RepublicDoppler Institute, Czech Technical University, Bˇrehov´a 7, 11519 Prague, CzechRepublicE-mail: [email protected]
Abstract.
We analyze two-dimensional Schr¨odinger operators with the potential | xy | p − λ ( x + y ) p/ ( p +2) where p ≥ λ ≥
0, which exhibit an abrupt changeof its spectral properties at a critical value of the coupling constant λ . We show thatin the supercritical case the spectrum covers the whole real axis. In contrast, for λ below the critical value the spectrum is purely discrete and we establish a Lieb-Thirring-type bound on its moments. In the critical case the essential spectrum coversthe positive halfline while the negative spectrum can be only discrete, we demonstratenumerically the existence of a ground state eigenvalue. Keywords : Schr¨odinger operator, eigenvalue estimates, spectral transition a r X i v : . [ m a t h - ph ] O c t chr¨odinger operators exhibiting a spectral transition Submitted to:
J. Phys. A.: Math. Theor. chr¨odinger operators exhibiting a spectral transition
1. Introduction
One of the problems which attracted attention recently concerns Schr¨odinger operatorswith potentials dependent on a parameter which exhibit a sudden spectral transitionwhen the value of the parameter passes a critical value. The potential is typicallyunbounded from below and has narrow channels through which the particle can ‘escapeto infinity’ in the supercritical situation. Possibly the best know example of this typeis the so-called Smilansky model [12, 13, 9, 4, 7] and its regular version [2]. Anotherexample, which will be the main subject of this paper, is a modification of the well-knownpotential | xy | p in R obtained by adding a rotationally symmetric negative componentwhich becomes stronger with the growing radius, see (1.1) below. Recall that withoutthe negative component this potential and its modifications serves to demonstrate thepossibility of a purely discrete spectrum in the situation when the classically a! llowedvolume of the phase space is infinite [11, 6, 3].The mechanism of the spectral transition comes from the balance between thenegative part of the potential and the positive contribution to the energy coming fromthe transverse confinement to a channel narrowing towards infinity. This means thatthe behavior of the two potential components at large distances from the origin mustbe properly correlated. In our case this is achieved by considering the following class ofoperators, L p ( λ ) : L p ( λ ) ψ = − ∆ ψ + (cid:0) | xy | p − λ ( x + y ) p/ ( p +2) (cid:1) ψ , p ≥ , (1.1)on L ( R ), where ( x, y ) in R are the Cartesian coordinates ( x, y ) in R and the non-negative parameter λ in the second term of the potential will serve to control thetransition. Note that pp +2 <
2, and consequently, the operator (1.1) is essentially self-adjoint on C ∞ ( R ) by Faris-Lavine theorem – cf. [10], Thms. X.28 and X.38; in thefollowing the symbol L p ( λ ) will always mean its closure.We have found already some properties of these operators in [5], our aim here is topresent a deeper spectral analysis. To describe what is know we need the (an)harmonicoscillator Hamiltonian on line, H p : H p u = − u (cid:48)(cid:48) + | t | p u (1.2)on L ( R ) with the standard domain, more exactly, its principal eigenvalue γ p ; sincethe potential has a mirror symmetry and the ground state is even, we can equivalentlyconsider the ‘cut’ (an)harmonic oscillator on L ( R + ) with Neumann condition at t = 0.The eigenvalue is known exactly for p = 2 where it equals one as well as for p → ∞ wherethe potential becomes an infinitely deep rectangular well of width two and γ ∞ = π .It is easy to see that the function p (cid:55)→ γ p is continuous and positive on the interval[1 , ∞ ); a numerical solution shows that it reaches the minimum value γ p ≈ . p ≈ . λ crit = γ p : the spectrum of L p ( λ ) is purely discrete and below bounded for λ < λ crit , chr¨odinger operators exhibiting a spectral transition λ = λ crit , while for λ > λ crit it becomes unbounded frombelow. We have also derived there crude bounds on eigenvalue sums in the subcriticalcase. In the present work we are going to establish first that for λ > λ crit the spectrumof L p ( λ ) covers the whole real line. Next we shall analyze in more detail the criticalcase, λ = λ crit , showing that one has σ ess ( L p ( λ crit )) = [0 , ∞ ) . The question of existence of a negative discrete spectrum is addressed numerically.We show that there a range of values of p for which the critical operator L p ( γ p ) hasa single negative eigenvalue. Finally, we return to the subcritical case and establishLieb-Thirring-type bounds to eigenvalue moments.
2. Supercritical case
As indicated, our first main result is the following.
Theorem 2.1.
For any λ > γ p we have σ ( L p ( λ )) = R .Proof. To demonstrate that any real number µ belongs to essential spectrum of operator L p we are going to use Weyl’s criterion: we have to find a sequence { ψ k } ∞ k =1 ⊂ D ( L p )such that (cid:107) ψ k (cid:107) = 1 which contains no convergent subsequence and (cid:107) L p ψ k − µψ k (cid:107) → k → ∞ . For the sake of clarity let us first show that 0 ∈ σ ess ( L p ). We define ψ k ( x, y ) := 1 k / ( p +2) h p (cid:0) xy p/ ( p +2) (cid:1) e iβy (2 p +2) / ( p +2) χ (cid:16) yk (cid:17) , (2.1)where h p is the ground state eigenfunction of H p , χ is a smooth function withsupp χ ⊂ [1 ,
2] satisfying (cid:82) χ ( z ) d z = 1, and β > k one can achieve that (cid:107) ψ k (cid:107) L ( R ) ≥ p/ ( p +2) as the following estimatesshow, (cid:90) R (cid:12)(cid:12)(cid:12)(cid:12) k / ( p +2) h p ( xy p/ ( p +2) ) e iβy (2 p +2) / ( p +2) χ (cid:16) yk (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) d x d y = 1 k / ( p +2) (cid:90) kk (cid:90) R (cid:12)(cid:12)(cid:12) h p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17)(cid:12)(cid:12)(cid:12) d x d y = 1 k / ( p +2) (cid:90) kk (cid:90) R y p/ ( p +2) (cid:12)(cid:12)(cid:12) h p ( t ) χ (cid:16) yk (cid:17)(cid:12)(cid:12)(cid:12) d t d y = 1 k / ( p +2) (cid:90) R | h p ( t ) | d t (cid:90) kk y p/ ( p +2) (cid:12)(cid:12)(cid:12) χ (cid:16) yk (cid:17)(cid:12)(cid:12)(cid:12) d y = 1 k / ( p +2) (cid:90) kk y p/ ( p +2) (cid:12)(cid:12)(cid:12) χ (cid:16) yk (cid:17)(cid:12)(cid:12)(cid:12) d y ≥ p/ ( p +2) (cid:90) | χ ( z ) | d z = 12 p/ ( p +2) . (2.2) chr¨odinger operators exhibiting a spectral transition ε one can find k = k ( ε ) such that (cid:107) L p ψ k (cid:107) L ( R ) < ε holds. By a straightforward calculation one gets ∂ ψ k ∂x = 1 k / ( p +2) y p/ ( p +2) h (cid:48)(cid:48) p ( xy p/ ( p +2) ) e iβy (2 p +2) / ( p +2) χ (cid:16) yk (cid:17) and ∂ ψ k ∂y = 1 k / ( p +2) e iβy (2 p +2) / ( p +2) (cid:18) − px ( p + 2) y − ( p +4) / ( p +2) h (cid:48) p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17) + p x ( p + 2) y − / ( p +2) h (cid:48)(cid:48) p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17) + ip (4 p + 4) βx ( p + 2) y ( p − / ( p +2) h (cid:48) p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17) + 2 pxk ( p + 2) y − / ( p +2) h (cid:48) p ( xy p/ ( p +2) ) χ (cid:48) (cid:16) yk (cid:17) + iβ (2 p + 2) p ( p + 2) y − / ( p +2) h p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17) + 2 iβ (2 p + 2)( p + 2) k y p/ ( p +2) h p ( xy p/ ( p +2) ) χ (cid:48) (cid:16) yk (cid:17) + 1 k h p ( xy p/ ( p +2) ) χ (cid:48)(cid:48) (cid:16) yk (cid:17)(cid:19) − β (2 p + 2) ( p + 2) y p/ ( p +2) h p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17) . (2.3)Our aim is to show that choosing k sufficiently large one can make most terms at theright-hand side of (2.3) as small as we wish. Changing the integration variables, we getfor the first term the following estimate, (cid:90) R (cid:12)(cid:12)(cid:12)(cid:12) xk / ( p +2) y ( p +4) / ( p +2) h (cid:48) p ( xy p/ ( p +2) ) e iβy (2 p +2) / ( p +2) χ (cid:16) yk (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) d x d y = 1 k / ( p +2) (cid:90) kk (cid:90) R (cid:12)(cid:12)(cid:12)(cid:12) xy ( p +4) / ( p +2) h (cid:48) p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) d x d y = 1 k / ( p +2) (cid:90) kk y (5 p +8) / ( p +2) (cid:12)(cid:12)(cid:12) χ (cid:16) yk (cid:17)(cid:12)(cid:12)(cid:12) d y (cid:90) R t | h (cid:48) p ( t ) | d t ≤ k (cid:90) | χ ( z ) | d z (cid:90) R t | h (cid:48) p ( t ) | d t , where the right-hand side tends to zero as k → ∞ . In the same way we establish thatfor large enough k all the terms in (2.3) except the last one can be made small. Thelast term is not small, what is important that it asymptotically compensates with thenegative part of the potential; using the same technique one can prove that for large k the integral 1 k / ( p +2) (cid:90) R (cid:18) ( x + y ) p/ ( p +2) − y p/ ( p +2) (cid:19) h p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17) d x d y chr¨odinger operators exhibiting a spectral transition ε > k large enough such that (cid:90) R | L p ψ k | ( x, y ) d x d y = (cid:90) R (cid:12)(cid:12)(cid:12)(cid:12) − ∂ ψ k ∂x − ∂ ψ k ∂y + | xy | p ψ k − λ ( x + y ) p/ ( p +2) ψ k (cid:12)(cid:12)(cid:12)(cid:12) d x d y ≤ k / ( p +2) (cid:90) kk (cid:90) R (cid:12)(cid:12)(cid:12)(cid:12) y p/ ( p +2) h (cid:48)(cid:48) p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17) − β (2 p + 2) ( p + 2) y p/ ( p +2) h p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17) − | xy | p h p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17) + λy p/ ( p +2) h ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) d x d y + ε = 1 k / ( p +2) (cid:90) kk (cid:90) R (cid:12)(cid:12)(cid:12)(cid:12) y p/ ( p +2) (cid:18) h (cid:48)(cid:48) p ( xy p/ ( p +2) ) − | xy p/ ( p +2) | p h p ( xy p/ ( p +2) ) − β (2 p + 2) ( p + 2) h p ( xy p/ ( p +2) ) + λh p ( xy p/ ( p +2) ) (cid:19) χ (cid:16) yk (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) d x d y + ε . Combining this result with the fact that H p h p = γ p h p and choosing β = ( p + 2)2 p + 2 (cid:112) λ − γ p (2.4)we get (cid:90) R | L p ψ k | ( x, y ) d x d y ≤ ε . (2.5)To complete this part of the proof we fix a sequence { ε j } ∞ j =1 such that ε j (cid:38) j → ∞ and to any j we construct a function ψ k ( ε j ) such that the supports for different j ’s do not intersect each other; this can be achieved by choosing each next k ( ε j ) largeenough. The norms of L p ψ k ( ε j ) satisfy the inequality (2.5) with ε j on the right-handside, and by construction the sequence ψ k ( ε j ) converges weakly to zero; this yields thesought Weyl sequence for zero energy.Passing now to an arbitrary nonzero real number µ we can use the same procedurereplacing the above functions ψ k by ψ k ( x, y ) = 1 k / ( p +2) h p ( xy p/ ( p +2) ) e i(cid:15) µ ( y ) χ (cid:16) yk (cid:17) , (2.6)where (cid:15) µ ( y ) := (cid:90) y | µ | ( p +2) / p ( p +2)( p +2) /p (2 p +2)( p +2) /pβ ( p +2) /p (cid:115) (2 p + 2) β ( p + 2) t p/ ( p +2) + µ d t , and furthermore, the functions h p , χ and the number β are the same way as above. Thesecond derivatives of those functions are ∂ ψ k ∂x = 1 k / ( p +2) y p/ ( p +2) h (cid:48)(cid:48) p ( xy p/ ( p +2) ) e i(cid:15) µ ( y ) χ (cid:16) yk (cid:17) chr¨odinger operators exhibiting a spectral transition ∂ ψ k ∂y = 1 k / ( p +2) e i(cid:15) µ ( y ) (cid:18) − px ( p + 2) y − ( p +4) / ( p +2) h (cid:48) p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17) + p x ( p + 2) y − / ( p +2) h (cid:48)(cid:48) p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17) + 2 pxk ( p + 2) y − / ( p +2) h (cid:48) p ( xy p/ ( p +2) ) χ (cid:48) (cid:16) yk (cid:17) + ip (2 p + 2) ( p + 2) β y ( p − / ( p +2) (cid:18) (2 p + 2) β ( p + 2) y p/ ( p +2) + µ (cid:19) − / h p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17) + 2 ipx ( p + 2) y − / ( p +2) (cid:18) (2 p + 2) β ( p + 2) y p/ ( p +2) + µ (cid:19) / h (cid:48) p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17) + 2 ik (cid:18) (2 p + 2) β ( p + 2) y p/ ( p +2) + µ (cid:19) / h p ( xy p/ ( p +2) ) χ (cid:48) (cid:16) yk (cid:17) + 1 k h p ( xy p/ ( p +2) ) χ (cid:48)(cid:48) (cid:16) yk (cid:17) − (cid:18) (2 p + 2) β ( p + 2) y p/ ( p +2) + µ (cid:19) h p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17)(cid:19) . It is not difficult to check that for any positive ε one choose a number k large enoughto ensure that the inequality (cid:13)(cid:13)(cid:13)(cid:13) ∂ ψ k ∂y e − i(cid:15) µ ( y ) + µψ k e − i(cid:15) µ ( y ) − e − iβy (2 p +2) / ( p +2) ∂ ∂y (cid:18) ψ k e − i(cid:15) µ ( y )+ iβy (2 p +2) / ( p +2) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) < ε holds. Using further the identity ∂ ψ k ∂x e − i(cid:15) µ ( y ) = e − iβy (2 p +2) / ( p +2) ∂ ∂x (cid:0) ψ k e − i(cid:15) µ ( y )+ iβy (2 p +2) / ( p +2) (cid:1) we arrive at the estimate (cid:107) L p ψ k − µψ k (cid:107) L ( R ) = (cid:13)(cid:13)(cid:13)(cid:13) ( L p ψ k ) e − i(cid:15) µ ( y ) − µψ k e − i(cid:15) µ ( y ) (cid:13)(cid:13)(cid:13)(cid:13) L ( R ) < (cid:13)(cid:13)(cid:13)(cid:13) e iβy (2 p +2) / ( p +2) L p (cid:18) ψ k e − i(cid:15) µ ( y )+ iβy (2 p +2) / ( p +2) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) + ε ;now we can use the result of the first part of proof to establish the claim.
3. Critical case
Let us now pass to the case when the parameter value is critical, in other words, considerthe operator L p ( γ p ) = − ∆ + ( | xy | p − γ p ( x + y ) p/ ( p +2) ) , p ≥
1, on L ( R ). We shallconsider the positive and negative spectrum separately. First we are going to show that the discreteness is lost in the positive halfline once thecoupling constant reaches the critical value. chr¨odinger operators exhibiting a spectral transition Theorem 3.1.
The essential spectrum of L p ( γ p ) contains the interval [0 , ∞ ) .Proof. The argument is similar to that used in the proof of Theorem2.1, hence we presentit briefly with emphasis on the differences. As before we check first that 0 ∈ σ ess ( L p ) byconstructing a Weyl sequence, which is now of the form ψ k ( x, y ) := 1 k / ( p +2) h p (cid:0) xy p/ ( p +2) (cid:1) χ (cid:16) yk (cid:17) with h p and χ the same as before. As this nothing but (2.1) with β = 0, not surprisinglyin view of (2.4) we can repeat the reasoning with the involved expressions appropriatelysimplified.Passing now to an arbitrary nonnegative number µ we replace (2.6) by ψ k ( x, y ) = 1 k / ( p +2) h p ( xy p/ ( p +2) ) e iη µ ( y ) χ (cid:16) yk (cid:17) , where the functions h p , χ are again the same way as above and ( η (cid:48) µ ( y )) = µ . This canbe achieved for any µ ≥ η µ ( y ) = √ µy , note that the classically allowedregion is now the whole halfline instead of the interval entering the definition of (cid:15) µ ( y )above. The second derivatives of the functions ψ k obtained in this way are ∂ ψ k ∂x = 1 k / ( p +2) y p/ ( p +2) h (cid:48)(cid:48) p ( xy p/ ( p +2) ) e i √ µy χ (cid:16) yk (cid:17) and ∂ ψ k ∂y = 1 k / ( p +2) e i √ µy (cid:18) − px ( p + 2) y − ( p +4) / ( p +2) h (cid:48) p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17) + p x ( p + 2) y − / ( p +2) h (cid:48)(cid:48) p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17) + 2 pxk ( p + 2) y − / ( p +2) h (cid:48) p ( xy p/ ( p +2) ) χ (cid:48) (cid:16) yk (cid:17) + 2 i √ µpx ( p + 2) y − / ( p +2) h (cid:48) p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17) + 2 i √ µk h p ( xy p/ ( p +2) ) χ (cid:48) (cid:16) yk (cid:17) + 1 k h p ( xy p/ ( p +2) ) χ (cid:48)(cid:48) (cid:16) yk (cid:17) − µh p ( xy p/ ( p +2) ) χ (cid:16) yk (cid:17)(cid:19) . One finds easily that for any positive ε and k large enough we have (cid:13)(cid:13)(cid:13)(cid:13) ∂ ψ k ∂y e − i √ µy + µψ k e − i √ µy − ∂ ∂y (cid:18) ψ k e − i √ µy (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) < ε and using further the trivial identity ∂ ψ k ∂x e − i √ µy = ∂ ∂x (cid:0) ψ k e − i √ µy (cid:1) we arrive at (cid:107) L p ψ k − µψ k (cid:107) L ( R ) = (cid:13)(cid:13)(cid:13)(cid:0) L p ψ k − µψ k (cid:1) e − i √ µy (cid:13)(cid:13)(cid:13) L ( R ) < (cid:13)(cid:13)(cid:13) L p (cid:16) ψ k e − i √ µy (cid:1)(cid:13)(cid:13)(cid:13) L ( R ) + ε and the result of the first part of proof allows us to establish the claim. chr¨odinger operators exhibiting a spectral transition G G G Q Q Q x = α α α . . . Figure 1.
The Neumann bracketing scheme
Next we are going to show that the inclusion σ ess ( L p ( γ p )) ⊃ [0 , ∞ ) established inTheorem 3.1 is in fact an equality. Theorem 3.2.
The negative spectrum of L p ( γ p ) , p ≥ , is discrete.Proof. By the minimax principle it is sufficient to estimate L p from below by a self-adjoint operator with a purely discrete negative spectrum. To construct such a lowerbound we employ a bracketing argument, imposing additional Neumann conditions atthe rectangles G n = {− α n +1 < x < α n +1 } × { α n < y < α n +1 } , (cid:101) G n = {− α n +1 < x <α n +1 } × {− α n +1 < y < − α n } , Q n = { α n < x < α n +1 } × {− α n < y < α n } , and (cid:101) Q n = {− α n +1 < x < − α n } × {− α n < y < α n } , n = 1 , , . . . , together with centralsquare G = ( − α , α ) – cf. Fig. 1. Here { α n } ∞ n =1 is a monotone sequence such that α n → ∞ as n → ∞ which will be specified later. In this way we obtain a direct sumof operators with Neumann boundary conditions at the rectangle boundaries which wedenote as L (1) n,p = L p | G n , (cid:101) L (1) n,p = L p | (cid:101) G n , L (2) n,p = L p | Q n , (cid:101) L (2) n,p = L p | (cid:101) Q n and L p = L p | G . It is obvious that the spectra of L ( i ) n,p , ˜ L ( i ) n,p , i = 1 ,
2, and L p are purely discrete, hence one needs to check that lim n →∞ inf σ (cid:0) L ( i ) n,p (cid:1) ≥ n →∞ inf (cid:0) σ ( ˜ L ( i ) n,p (cid:1) ≥ i = 1 ,
2, since then the spectra of all the direct sums (cid:76) ∞ n =1 L ( i ) n,p and (cid:76) ∞ n =1 ˜ L ( i ) n,p , i = 1 ,
2, below any fixed negative number contain a finitenumber of eigenvalues, the multiplicity taken into account, which implies the soughtclaim. chr¨odinger operators exhibiting a spectral transition L ( i ) n,p , ˜ L ( i ) n,p , i = 1 ,
2, frombelow by operators with separated variables and prove the analogous limiting relationsfor them. We use the lower bounds H (1) n ψ = − ∆ ψ + ( α pn | x | p − γ p ( x + α n +1 ) p/ ( p +2) ) ψ (3.1)on L ( − α n +1 , α n +1 ) ⊗ L ( α n , α n +1 ) with the boundary conditions ∂ψ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = − α n +1 = ∂ψ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = α n +1 = 0 ,∂ψ∂y (cid:12)(cid:12)(cid:12)(cid:12) y = α n = ∂ψ∂y (cid:12)(cid:12)(cid:12)(cid:12) y = α n +1 = 0 . It is clear that the spectra of H (1) n,p , n = 1 , , . . . , are purely discrete; we are going tocheck that lim n →∞ inf σ (cid:0) H (1) n,p (cid:1) ≥ . (3.2)Since the lowest Neumann eigenvalue of − d d y on the interval is zero correspondingto a constant eigenfunction, the problem reduces to analysis of the operator h (1) n,p = − d d x + α pn | x | p − γ p ( x + α n +1 ) p/ ( p +2) on L ( − α n +1 , α n +1 ). Using a simple scalingtransformation, one can check that h (1) n,p is unitarily equivalent to h (2) n,p = α p/ ( p +2) n (cid:32) − d d x + | x | p − γ p α p/ ( p +2) n (cid:18) x α p/ ( p +2) n + α n +1 (cid:19) p/ ( p +2) (cid:33) (3.3)on the interval (cid:0) − α n +1 α p/ ( p +2) n , α n +1 α p/ ( p +2) n (cid:3) with Neumann boundary conditions atits endpoints. To proceed we need to specify the sequence { α n } . Let us assume that α p/ ( p +2) n +1 − α p/ ( p +2) n → n → ∞ . (3.4)Combining this assumption with the inequality γ p α p/ ( p +2) n (cid:32)(cid:18) x α p/ ( p +2) n + α n +1 (cid:19) p/ ( p +2) − α p/ ( p +2) n +1 (cid:33) ≤ γ p α p/ ( p +2) n (cid:18) x α p/ ( p +2) n (cid:19) p/ ( p +2) ≤ γ p α p ( p +1) / ( p +2) n ( | x | p + 1) , chr¨odinger operators exhibiting a spectral transition h (2) n,p = α p/ ( p +2) n (cid:18) − d d x + | x | p − γ p α p/ ( p +2) n +1 α p/ ( p +2) n − γ p α p/ ( p +2) n (cid:18)(cid:18) x α p/ ( p +2) n + α n +1 (cid:19) p/ ( p +2) − α p/ ( p +2) n +1 (cid:19)(cid:19) ≥ α p/ ( p +2) n (cid:32) − d d x + (cid:18) − γ p α p ( p +1) / ( p +2) n (cid:19) | x | p − γ p α p ( p +1) / ( p +2) n − γ p α p/ ( p +2) n +1 α p/ ( p +2) n (cid:33) ≥ α p/ ( p +2) n (cid:32) − d d x + (cid:32) − γ p α p ( p +1) / ( p +2) n (cid:33) | x | p − γ p − γ p (cid:32) α p/ ( p +2) n +1 − α p/ ( p +2) n α p/ ( p +2) n (cid:33)(cid:33) + o (1) ≥ α p/ ( p +2) n (cid:18) − d d x + (cid:18) − γ p α p ( p +1) / ( p +2) n (cid:19) | x | p − γ p (cid:19) + o (1) ≥ α p/ ( p +2) n (cid:18) − γ p α p ( p +1) / ( p +2) n (cid:19) (cid:18) − d d x + | x | p − γ p (cid:19) + o (1) , (3.5)where the corresponding Neumann (an)harmonic oscillator is restricted to the interval( − α n +1 α p/ ( p +2) n , α n +1 α p/ ( p +2) n ) . Next we need to establish the following lemma.
Lemma 3.1.
Let l k,p = − d d x + | x | p be the Neumann operator defined on the interval [ − k, k ] , k > . Then inf σ ( l k,p ) ≥ γ p + o (cid:18) k p/ (cid:19) as k → ∞ . (3.6) Proof.
The relation (3.6) is certainly valid if inf σ ( l k,p ) ≥ γ p holds for all k from somenumber on. Assume thus that we have inf σ ( l k,p ) < γ p for infinitely many numbers k .Let ψ k,p be the normalized ground-state eigenfunction of l k,p . We fix a positive δ andcheck that (cid:90) − k +1 − k (cid:0) | ψ (cid:48) k,p | + | x | p | ψ k,p | (cid:1) d x < δ , (3.7) (cid:90) kk − (cid:0) | ψ (cid:48) k,p | + | x | p | ψ k,p | (cid:1) d x < δ . Indeed, suppose that at least one of inequalities (3.7) does not hold, then (cid:90) k − − k +1 (cid:0) | ψ (cid:48) k,p | + | x | p | ψ k,p | (cid:1) d x < γ p − δ . (3.8) chr¨odinger operators exhibiting a spectral transition ψ k,p is by assumption the ground-state eigenfunction of l k,p , we haveinf σ ( l k,p ) = (cid:90) k − k (cid:0) | ψ (cid:48) k,p | + | x | p | ψ k,p | (cid:1) d x ≤ (cid:90) − (cid:0) | φ (cid:48) | + | x | p | φ | (cid:1) d x for all k ≥ φ from the domain of the operator, inparticular, for any φ from the class C ∞ ( − ,
1) such that (cid:82) − | φ | d x = 1. Consequently,for large enough k there must exist points x (1) k,p ∈ ( − k +1 , − k +2) and x (2) k,p ∈ ( k − , k − ψ k,p (cid:16) x (1) k,p (cid:17) = O (cid:16) k p/ (cid:17) and ψ k,p (cid:16) x (2) k,p (cid:17) = O (cid:16) k p/ (cid:17) as k → ∞ . Next we construct a function ϕ k,p on semi-infinite intervals ( −∞ , x (1) k,p ) and ( x (2) k,p , ∞ ) insuch a way that g k,p ( x ) := ψ k,p ( x ) χ ( x (1) k,p ,x (2) k,p ) ( x ) + ϕ k,p ( x ) χ ( −∞ ,x (1) k,p ) ∪ ( x (2) k,p , ∞ ) ( x ) ∈ H ( R )and (cid:90) x (1) k,p −∞ (cid:0) | ϕ (cid:48) k,p | + | x | p | ϕ k,p | (cid:1) d x + (cid:90) ∞ x (2) k,p (cid:0) | ϕ (cid:48) k,p | + | x | p | ϕ k,p | (cid:1) d x = O (cid:16) k p/ (cid:17) ; (3.9)this can be always achieved, one can take, e.g., the function decreasing linearly withrespect to | x − x ( j ) k,p | from the values ψ k,p (cid:16) x ( j ) k,p (cid:17) , j = 1 , , to zero. By virtue of (3.8)and (3.9) we then have (cid:90) R (cid:0) | g k,p | + | x | p | g k,p | (cid:1) d x < γ p − δ + O (cid:18) k p/ (cid:19) < γ p for large enough k , however, this is in contradiction with the fact that γ p is the ground-state eigenvalue of l k,p . This proves the validity of (3.7).Having established the validity of inequalities (3.7) we infer from them that thereare points y (1) k,p ∈ ( − k, − k + 1) and y (2) k,p ∈ ( k − , k ) such that ψ k,p ( y ( j ) k,p ) = O (cid:16) δk p/ (cid:17) , j = 1 , . Now we repeat the argument and construct a function ˜ ϕ k,p on the semi-infinite intervals( −∞ , y (1) k,p ) and ( y (2) k,p , ∞ ) in such a way that˜ g k,p ( x ) := ψ k,p ( x ) χ ( y (1) k,p ,y (2) k,p ) ( x ) + ˜ ϕ ( x ) χ ( −∞ ,y (1) k,p ) ∪ ( y (2) k,p , ∞ ) ( x ) ∈ H ( R )and (cid:90) y (1) k,p −∞ (cid:0) | ˜ ϕ (cid:48) k,p | + | x | p | ˜ ϕ k,p | (cid:1) d x + (cid:90) ∞ y (2) k,p (cid:0) | ˜ ϕ (cid:48) k,p | + | x | p | ˜ ϕ k,p | (cid:1) d x = O (cid:16) δk p/ (cid:17) . chr¨odinger operators exhibiting a spectral transition (cid:90) R | ˜ g (cid:48) k,p | d x + (cid:90) R | x | p | ˜ g k,p | d x < inf σ ( l k,p ) + O (cid:16) δk p/ (cid:17) . However, γ p is the ground-state eigenvalue, (cid:90) R | ˜ g (cid:48) k,p | d x + (cid:90) R | x | p | ˜ g k,p | d x ≥ γ p , which in combination with above inequality givesinf σ ( l k,p ) > γ p − O (cid:16) δk p/ (cid:17) , proving the claim of the lemma.It follows from Lemma 3.1 that the right-hand side of the estimate (3.5) behavesasymptotically as o (cid:18) α p ( p +1) / ( p +2) n (cid:19) α p/ ( p +2) n + o (1)which can be made arbitrarily small by choosing n is large enough; this is what weneeded to conclude the proof of Theorem 3.2. Remark 3.1.
We know from [5, Thm. 2.1] that the critical operator L p ( γ p ) is boundedfrom below. Estimating separately the contributions to the respective quadratic formcoming from the regions { ( x, y ) : | y | ≥ } , { ( x, y ) : | x | ≥ , | y | ≤ } , and the centralsquare ( − , , we can derive a lower bound to the threshold of the negative spectrumin terms of spectral properties of the one-dimensional operators with the symbol − d d t + | t | p − γ p (cid:18) t z (4 p +4) / ( p +2) + 1 (cid:19) p/ ( p +2) with z ≥
1. As such a bound is not simple and does not provide any significant insight,however, we are not going to present it here.
Theorem 3.2 tells us that the spectrum in the negative halfline can be discrete only,and as we have remarked above one can find a lower estimate to its threshold, however,neither of these results implies anything about the negative spectrum existence . Nowwe are going address this question numerically and provide an evidence of the discretespectrum nontriviality.We considering first the operator L ( γ ) — recall that γ = 1 — and imposea cutoff at a circle of radius R circled at the origin with Dirichlet and Neumannboundary condition, and find the corresponding first and second eigenvalue using theFinite Element Method. The result is shown on Fig. 2. We see, in particular, thatthe lowest Dirichlet eigenvalue is for R (cid:38) chr¨odinger operators exhibiting a spectral transition Figure 2.
The eigenvalues E j , j = 1 ,
2, of the critical operator with p=2 as functionsof the cutoff radius R . The blue and red curves correspond to the Neumann andDirichlet boundary, respectively. and negative which by an elementary bracketing argument indicates that L (1) has anegative eigenvalue. Furthermore, the difference between the Dirichlet and Neumanneigenvalue becomes negligible for large enough R which shows that true ground-stateeigenvalue in this case is E ≈ − . R .By continuity, the ground-state eigenvalue of L p ( λ ) exists in the vicinity of thepoint p = 2; one is naturally interested what one can say about a broader range of theparameter. To this aim we plot in the left part of Fig. 4 the lowest eigenvalue of thecut-off operator as the function of p and the coupling constant. The right part showsthe zero-energy cut of the surface in which the shaded region indicates the part of the( λ, p ) plane where the lowest eigenvalue of the cut-off operator is positive, as comparedto λ crit = γ p . The two curves meet at p ≈ .
392 corresponding to λ crit ≈ . chr¨odinger operators exhibiting a spectral transition Figure 3.
The ground-state eigenfunction for p = 2, view from the top. Figure 4.
Positivity of L p ( λ ) as a function of λ and p . values of p the numerical accuracy is a demanding problem, we nevertheless conjecturethat at least the Dirichlet region operator, p = ∞ , is positive. Fig. 4 also provides an chr¨odinger operators exhibiting a spectral transition L p ( λ ) depends on the coupling constant.
4. Subcritical case, eigenvalue estimates
Let us finally pass to the subcritical case, λ < γ p . According to [5, Thm. 2.1] theoperator L p ( λ ) has in this case a purely discrete spectrum. In the mentioned paper acrude bound on eigenvalue sums was established for small values of the coupling constant λ . We are going derive now a substantially stronger result, an estimate on eigenvaluemoments valid for any λ < γ p . More specifically, let µ < µ ≤ µ ≤ · · · be the set ofordered eigenvalues of (1.1); we are looking for bounds of the quantities (cid:80) ∞ j =1 (Λ − µ j ) σ + for fixed numbers Λ and σ . This is the contents of the following theorem. Theorem 4.1.
Let λ < γ p , then for any Λ ≥ and σ ≥ / the following traceinequality holds, tr (Λ − L p ( λ )) σ + (4.1) ≤ C p,σ (Λ + 1) σ +( p +1) /p ( γ p − λ ) σ +( p +1) /p (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ln (cid:18) Λ + 1 γ p − λ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + 1 (cid:19) + C p,σ C λ (cid:16) Λ + C p/ ( p +2) λ (cid:17) σ +1 , where the constant C p,σ depends on p and σ only and C λ = max (cid:26) γ p − λ ) ( p +2) / ( p ( p +1)) , γ p − λ ) ( p +2) / (4 p ( p +1)) (cid:27) . Proof.
By the minimax principle it is sufficient to estimate L p from below by a self-adjoint operator with a purely discrete spectrum for which the moments in questioncan be calculated. To construct such a lower bound we again employ a bracketing,imposing additional Neumann conditions at the rectagles G n , (cid:101) G n , and Q n , (cid:101) Q n introducedin the proof of Theorem 3.2. The sequence { α n } ∞ n =1 is monotonically increasing byconstruction; we assume again that α n → ∞ and that the rectangles get asymptoticallythinner according to (3.4), i.e. α p/ ( p +2) n +1 − α p/ ( p +2) n → n → ∞ . Then, as before, we obtain a direct sum of operators L (1) n,p , (cid:101) L (1) n,p , L (2) n,p , (cid:101) L (2) n,p and L p . Weare going to find the eigenvalue momentum estimates for those.Let us start from L (1) n,p , n = 1 , , . . . . We again find a lower bound using theoperator H (1) n given by (3.1), the spectrum of which is the sum of two one-dimensionaloperators. Since the spectrum of one-dimensional Neumann operator − d d y on theinterval ( α n , α n +1 ) is discrete and simple with the eigenvalues (cid:110) π k ( α n +1 − α n ) (cid:111) ∞ k =0 , theproblem reduces to analysis of the operator h (1) n,p = − d d x + α pn | x | p − λ ( x + α n +1 ) p/ ( p +2) on L ( − α n +1 , α n +1 ) which is unitarily equivalent to (3.3).To proceed we put κ := γ p − λ γ p + λ +2) and assume that the edge coordinates satisfy α ≥ (cid:18) λκ (cid:19) ( p +2) / (4 p ( p +1)) , (4.2) α p/ ( p +2) n +1 − α p/ ( p +2) n < κλ . (4.3) chr¨odinger operators exhibiting a spectral transition a + b ) q ≤ a q + b q holds for any positive numbers a, b and q < λα p/ ( p +2) n (cid:32)(cid:18) x α p/ ( p +2) n + α n +1 (cid:19) p/ ( p +2) − α p/ ( p +2) n +1 (cid:33) ≤ λα p/ ( p +2) n (cid:18) x α p/ ( p +2) n (cid:19) p/ ( p +2) ≤ κ ( | x | p + 1) . Next, by virtue of (4.3) and above estimate, we have h (2) n,p ( λ ) = α p/ ( p +2) n (cid:18) − d d x + | x | p − λ α p/ ( p +2) n +1 α p/ ( p +2) n − λα p/ ( p +2) n (cid:18)(cid:18) x α p/ ( p +2) n + α n +1 (cid:19) p/ ( p +2) − α p/ ( p +2) n +1 (cid:19)(cid:19) ≥ α p/ ( p +2) n (cid:32) − d d x + (1 − κ ) | x | p − κ − λα p/ ( p +2) n +1 α p/ ( p +2) n (cid:33) = α p/ ( p +2) n − d d x + (1 − κ ) | x | p − κ − λ (cid:16) α p/ ( p +2) n +1 − α p/ ( p +2) n (cid:17) α p/ ( p +2) n − λ ≥ (1 − κ ) α p/ ( p +2) n (cid:18) − d d x + | x | p − λ (cid:48) − κ (cid:48) (cid:19) − κ , (4.4)where κ (cid:48) := κ − κ , λ (cid:48) := λ − κ , and the corresponding Neumann (an)harmonic oscillator isdefined on the interval ( − α n +1 α p/ ( p +2) n , α n +1 α p/ ( p +2) n ) . (4.5)It follows from Lemma 3.1 that if the interval (4.5) is large enough, which can be achievedby choosing α α p/ ( p +2)1 > α p +1) / ( p +2)1 > K ,p (4.6)with a large enough K ,p , we have the estimate h (2) n,p ≥ (1 − κ ) α p/ ( p +2) n (cid:32) γ p − α p/ n +1 α p / (2( p +2)) n − λ (cid:48) − κ (cid:48) (cid:33) − κ . (4.7)Our aim is now to show that by choosing a suitable sequence { α n } ∞ n =1 we can achievethat for any n ≥ σ (cid:0) h (2) n,p (cid:1) ≥ (1 − κ ) α p/ ( p +2) n ( γ p − λ )2 − κ . (4.8)This is ensured, for instance, if α ≥ (cid:18) − κ ) γ p − λ − κ ( γ p + λ + 2) (cid:19) ( p +2) / ( p ( p +1)) . (4.9) chr¨odinger operators exhibiting a spectral transition α = 1 (4.10)+ (cid:34) max (cid:40) K ( p +2) / (2( p +1))0 ,p , (cid:18) − κ ) γ p − λ − κ ( γ p + λ + 2) (cid:19) ( p +2) / ( p ( p +1)) , (cid:18) λκ (cid:19) ( p +2) / (4 p ( p +1)) (cid:41)(cid:35) , where [ · ] means the entire part. Let us now return to the eigenvalue momentumestimates. One hastr (cid:0) Λ − h (2) n,p (cid:1) σ + = inf σ (cid:0) h (2) n,p − Λ (cid:1) σ − + tr (cid:48) (cid:0) h (2) n,p − Λ (cid:1) σ − , (4.11)where tr (cid:48) the summation which yields the corresponding eigenvalue moment in whichthe ground state is not taken into account. Using next inequalities (4.4), (4.8), incombination with version of Lieb-Thiring inequality suitable for our purpose [8]), weinfer from (4.11) that for any positive Λ , σ ≥ / n ≥ (cid:0) Λ − h (2) n,p (cid:1) σ + ≤ (Λ + κ ) σ + (1 − κ ) σ α pσ/ ( p +2) n L cl σ, (cid:90) α n +1 α p/ ( p +2) n − α n +1 α p/ ( p +2) n (cid:32) Λ + κ (1 − κ ) α p/ ( p +2) n − | x | p + λ (cid:48) + κ (cid:48) (cid:33) σ +1 / d x ≤ (Λ + κ ) σ + (1 − κ ) σ α pσ/ ( p +2) n L cl σ, (cid:90) R (cid:32) Λ + κ (1 − κ ) α p/ ( p +2) n − | x | p + λ (cid:48) + κ (cid:48) (cid:33) σ +1 / d x ≤ (Λ + κ ) σ + 2 α pσ/ ( p +2) n L cl σ, (cid:32) Λ + κ (1 − κ ) α p/ ( p +2) n + λ (cid:48) + κ (cid:48) (cid:33) σ +( p +2) / (2 p ) . (4.12)We further restrict the choice of the sequence { α n } ∞ n =1 demanding α n +1 − α n < π (cid:0) Λ − inf σ ( h (2) n,p )( λ ) (cid:1) − / ; (4.13)this allows us to write the following estimatetr (cid:32) Λ − ∞ (cid:77) n =1 L (1) n,p (cid:33) σ + ≤ tr (cid:32) Λ − ∞ (cid:77) n =1 H (1) n (cid:33) σ + (4.14) ≤ ∞ (cid:88) n =1 ∞ (cid:88) k =0 tr (cid:18) Λ − π k ( α n +1 − α n ) − h (2) n,p (cid:19) σ + ≤ ∞ (cid:88) n =1 tr (cid:0) Λ − h (2) n,p (cid:1) σ + . Using next the fact that inf σ (cid:16) L (1) n,p (cid:17) ≥ inf σ (cid:16) h (2) n,p (cid:17) in combination with estimates (4.8), chr¨odinger operators exhibiting a spectral transition (cid:32) Λ − ∞ (cid:77) n =1 L (1) n,p (cid:33) σ + ≤ (cid:88) ( γ p − λ ) α p/ ( p +2) n < κ )1 − κ (Λ + κ ) σ (4.15)+ 2 L cl σ, (cid:88) ( γ p − λ ) α p/ ( p +2) n < κ )1 − κ α pσ/ ( p +2) n (cid:18) Λ + κ (1 − κ ) α p/ ( p +2) n + λ (cid:48) + κ (cid:48) (cid:19) σ +( p +2) / (2 p ) ≤ (cid:88) ( γ p − λ ) α p/ ( p +2) n < κ )1 − κ (Λ + κ ) σ + 2 L cl σ, (1 − κ ) σ +( p +2) / (2 p ) (cid:88) ( γ p − λ ) α p/ ( p +2) n < κ )1 − κ α pσ/ ( p +2) n (cid:18) Λ + κα p/ ( p +2) n + λ + κ (cid:19) σ +( p +2) / (2 p ) ≤ (Λ + κ ) σ (cid:40) α n < (cid:18) κ )(1 − κ )( γ p − λ ) (cid:19) ( p +2) / (2 p ) (cid:41) + 2 L cl σ, ( λ + 1 + κ ) σ +( p +2) / (2 p ) (1 − κ ) σ +( p +2) / (2 p ) (Λ + κ ) σ +( p +2) / (2 p ) (cid:88) α n < (Λ+ κ ) ( p +2) / (2 p ) α n + 2 L cl σ, ( λ + 1 + κ ) σ +( p +2) / (2 p ) (1 − κ ) σ +( p +2) / (2 p ) (cid:88) (Λ+ κ ) ( p +2) / (2 p ) <α n < γp − λ )( p +2) / (2 p ) ( κ )1 − κ ) ( p +2) / (2 p ) α pσ/ ( p +2) n , where {·} means the cardinality of the corresponding set.Using the same technique one obtains estimates for operators (cid:101) L (1) n,p , L (2) n,p , (cid:101) L (2) n,p analogous to (4.15). Finally, the operator L p can be estimated from below by H ,p = − ∂ ∂x − ∂ ∂y − p/ ( p +2) λα p/ ( p +2)1 on G with Neumann conditions at the boundary ∂G . The spectrum of H ,p is (cid:26) π k α + π m α − p/ ( p +2) λα p/ ( p +2)1 (cid:27) ∞ k,m =0 , and thereforetr (Λ − H ,p ) σ + ≤ ∞ (cid:88) k,m =0 (cid:18) Λ + 2 p/ ( p +2) λα p/ ( p +2)1 − π k α − π m α (cid:19) σ + ≤ (cid:16) Λ + 2 p/ ( p +2) λα p/ ( p +2)1 (cid:17) σ × α (cid:113) Λ+2 p/ ( p +2) λα p/ ( p +2)1 /π (cid:88) k =0 (cid:18) α π (cid:18) Λ + 2 p/ ( p +2) λα p/ ( p +2)1 − π k α (cid:19) / + 1 (cid:19) ≤ (cid:18) α π (cid:16) Λ + 2 p/ ( p +2) λα p/ ( p +2)1 (cid:17) / + 1 (cid:19) (cid:16) Λ + 2 p/ ( p +2) λα p/ ( p +2)1 (cid:17) σ . (4.16) chr¨odinger operators exhibiting a spectral transition α is defined in (4.10) and to any ν = α , α + 1 , α + 2 , . . . define a finitesequence of numbers by β k ( ν ) = ν + k [ ν p/ ( p +2) ln ν ] , k = 0 , , . . . , (cid:2) ν p/ ( p +2) ln ν (cid:3) −
1. Thisallows us to construct a sequence { α n } ∞ n =1 of the rectangle edge coordinates using thefollowing prescription: the first term is given by (4.10) and the further ones are α = β ( α ) , . . . , α (cid:104) α p/ ( p +2)1 ln α (cid:105) = β (cid:104) α p/ ( p +2)1 ( α ) ln α (cid:105) − , α (cid:104) α p/ ( p +2)1 ln α (cid:105) +1 = β ( α +1) , . . . , etc.,where [ · ] as usual denotes the entire part. With this choice of { α n } ∞ n =1 , one can checkthat the right-hand side of (4.15) is not larger than C p,σ (cid:18) (Λ + κ ) σ ( γ p − λ ) σ max (cid:26) , (Λ + κ ) ( p +1) /p ( γ p − λ ) ( p +1) /p ln (cid:18) κ )(1 − κ )( γ p − λ ) (cid:19)(cid:27) + (Λ + κ ) σ +1 / /p max (cid:110) , (Λ + κ ) / ln (Λ + κ ) (cid:111)(cid:19) (4.17)with a constant depending on p and σ only. On the other hand, the right-hand side of(4.16) is not larger than ˜ C p,σ α (cid:16) Λ + α p/ ( p +2)1 (cid:17) σ +1 with another constant ˜ C p,σ . In this way the theorem is established. Acknowledgments
We are obliged to Ari Laptev for a useful discussion. The research has been supported bythe Czech Science Foundation (GA ˇCR) within the project 14-06818S. D.B. acknowledgesthe support of the University of Ostrava and the project “Support of Research in theMoravian-Silesian Region 2013”. The research of A.K. is supported by the GermanResearch Foundation through CRC 1173 “Wave phenomena: analysis and numerics”.
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