aa r X i v : . [ m a t h . SP ] J u l INSTABILITY INTERVALS OF THE WHITTAKER-HILLOPERATOR
XU-DAN LUO
Abstract.
The Hill operator admits a band gap structure. As a special case,like the Mathieu operator, one has only open gaps, however, the instabil-ity intervals of the Whittaker-Hill operator may be open or closed. In 2007,P. Djakov and B. Mityagin gave the asymptotics of band gaps for a specialWhittaker-Hill operator [P. Djakov and B. Mityagin, J. Funct. Anal., 242,157-194 (2007).]. In this paper, a more general Whittaker-Hill operator isconsidered and the asymptotics of the instability intervals are studied. Introduction and main results
The Floquet (Bloch) theory indicates that the spectrum of the Schr¨odinger op-erator(1.1) Lf := − f ′′ ( x ) + ν ( x ) f ( x ) , x ∈ R with a smooth real-valued periodic potential ν ( x ) has a band gap structure. If wefurther assume that ν ( x ) is of periodic π and set ν ( x ) = − ∞ X n =1 θ n cos(2 nx ) − ∞ X m =1 φ m sin(2 mx ) , where θ n and φ m are real, then (1.1) can be written as:(1.2) Lf := − f ′′ ( x ) − " ∞ X n =1 θ n cos(2 nx ) + ∞ X m =1 φ m sin(2 mx ) f ( x ) . Moreover, there are [3] two monotonically increasing infinite sequence of real num-bers λ +0 , λ +1 , λ +2 , · · · and λ − , λ − , λ − , · · · such that the Hill equation(1.3) Lf = λf has a solution of period π if and only if λ = λ + n , n = 0 , , , · · · , and a solution ofsemi-period π (i.e., f ( x + π ) = − f ( x )) if and only if λ = λ − n , n = 1 , , , · · · . The λ + n and λ − n satisfy the inequalities λ +0 < λ − ≤ λ − ≤ λ +1 ≤ λ +2 < λ − ≤ λ − < λ +3 ≤ λ +4 < · · · Mathematics Subject Classification.
Primary : 34L15; 41A60; 47E05.
Key words and phrases.
The Whittaker-Hill operator; Instability intervals; Asymptotics. and the relations lim n →∞ λ + n = ∞ , lim n →∞ λ − n = ∞ . Besides, γ n := ( λ − n +1 − λ − n ) for odd n and γ n := ( λ + n − λ + n − ) for even n are referredto as band gaps or instability intervals, where n ≥ ν ( x ) is a single trigonometric function, i.e.,(1.4) ν ( x ) = − B cos 2 x. Ince [5] proved that all instability intervals of the Mathieu operator are open,i.e., no closed gaps for the Mathieu operator. In 1963, Levy and Keller [6] gavethe asymptotics of γ n = γ n ( B ), i.e., for fixed n and real nonzero number B , when B → γ n = 8[( n − · (cid:18) B (cid:19) n (1 + O ( B )) .
18 years later, Harrell [4] gave the asymptotics of the band gaps of the Mathieuoperator for fixed B and n → ∞ , i.e.,(1.6) γ n = λ + n − λ − n = 8[( n − · (cid:18) | B | (cid:19) n (cid:18) O (cid:18) n (cid:19)(cid:19) . Compared with the Mathieu potential, the band gaps for the Whittaker-Hillpotential(1.7) ν ( x ) = − ( B cos 2 x + C cos 4 x )may be open or closed. Specifically, if B = 4 αt and C = 2 α , for any real α andnatural number t , it is already known that for odd t = 2 m + 1, all the even gaps areclosed except the first m , but no odd gap disappears; similarly, for even t = 2 m ,except for the first m , all the odd gaps are closed, but even gaps remain open (seeTheorem 11, [1] and Theorem 7.9, [7]).In 2007, P. Djakov and B. Mityagin (see [2]) gave the asymptotics of the in-stability intervals for the above special Whittaker-Hill potential, namely, for real B, C = 0, B = 4 αt and C = 2 α , they have the following results, where either both α and t are real numbers if C > α and t are pure imaginary numbers if C < Theorem 1.1 ([2]) . Let γ n be the n − th band gap of the Whittaker-Hill operator (1.8) Lf = − f ′′ − [4 αt cos 2 x + 2 α cos 4 x ] f, where either both α and t are real, or both are pure imaginary numbers. If t is fixedand α → , then for even n (1.9) γ n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α n n [( n − n/ Y k =1 ( t − (2 k − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 + O ( α )) , and for odd n (1.10) γ n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α n t n [( n − n − / Y k =1 ( t − (2 k ) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 + O ( α )) . HITTAKER-HILL 3
Theorem 1.2 ([2]) . Let γ n be the n − th band gap of the Whittaker-Hill operator (1.11) Lf = − f ′′ − [4 αt cos 2 x + 2 α cos 4 x ] f, where either both α and t = 0 are real, or both are pure imaginary numbers. Thenthe following asymptotic formulae hold for fixed α , t and n → ∞ :for even n (1.12) γ n = 8 | α | n n [( n − (cid:12)(cid:12)(cid:12) cos (cid:16) π t (cid:17)(cid:12)(cid:12)(cid:12) (cid:20) O (cid:18) log nn (cid:19)(cid:21) , and for odd n (1.13) γ n = 8 | α | n n [( n − π (cid:12)(cid:12)(cid:12) sin (cid:16) π t (cid:17)(cid:12)(cid:12)(cid:12) (cid:20) O (cid:18) log nn (cid:19)(cid:21) , where (2 m − · · · · (2 m − , (2 m )!! = 2 · · · · (2 m ) . In this paper, a more general Whittaker-Hill operator(1.14) L = − D + ( bq m cos 2 x + cq m cos 4 x )is considered and the asymptotics of the instability intervals are derived, where b , c , q , m and m are real. Our theorems are stated as follows, in particular, we candeduce P. Djakov and B. Mityagin’s results by choosing m = 1, m = 2, b = − αt and c = − α . Theorem 1.3.
Let the Whittaker-Hill operator be (1.15) Ly = − y ′′ + ( bq m cos 2 x + cq m cos 4 x ) y, where b , c and q are real. If q → and m , m are positive real parameters, thenone of the following results holds: (1) when m > m ,(i) (1.16) γ m = (cid:12)(cid:12)(cid:12)(cid:12) · ( c ) m · q m m m [( m − (cid:12)(cid:12)(cid:12)(cid:12) + O (cid:16) q m ( m − )+ m (cid:17) , (ii) (1.17) γ = | bq m | + O ( q m − m ) , γ = | bcq m + m | O ( q m + m ) ,γ m − = (cid:12)(cid:12)(cid:12) (cid:16) c (cid:17) m − · b · q m ( m − m · m · n m − · m − X i =1 (2 m − i − · (2 i − i ! · ( m − − i )! + 2(2 m − · ( m − o(cid:12)(cid:12)(cid:12) + O (cid:16) q m ( m − )+2 m (cid:17) for m ≥ when m < m , (1.19) γ = | bq m | + O ( q m − m ) , γ = (cid:12)(cid:12)(cid:12)(cid:12) cq m + b q m (cid:12)(cid:12)(cid:12)(cid:12) + O ( q m ) , (1.20) γ n = (cid:12)(cid:12)(cid:12)(cid:12) · b n · q m n n · [( n − (cid:12)(cid:12)(cid:12)(cid:12) + O ( q m n + m − m ) for n ≥ XU-DAN LUO (3) when m = m and c < ,(i) (1.21) γ = | bq m | + O ( q m ) , γ = (cid:12)(cid:12)(cid:12)(cid:12) cq m + b q m (cid:12)(cid:12)(cid:12)(cid:12) + O ( q m ) , (ii) (1.22) γ m = 8 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q mk =1 (cid:16)(cid:16) b (cid:17) + 8 c (cid:16) k − (cid:17) (cid:17) · q m · m m · [(2 m − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( q m ( m +1) ) for m ≥ , (iii) (1.23) γ m − = 32 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b Q m − k =1 (cid:16)(cid:16) b (cid:17) + 8 ck (cid:17) · q m · (2 m − m · [(2 m − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( q m (2 m +1) ) for m ≥ . Here, m is a positive integer and γ n is the n-th instability interval. Theorem 1.4.
Let the Whittaker-Hill operator be (1.24) Ly = − y ′′ + ( bq m cos 2 x + cq m cos 4 x ) y, where b , q are real, c < and m > . Then the following asymptotic formula holdsfor fixed b , c , q and n → ∞ . (1.25) γ m = q m · m · | c | m m − · [(2 m − · (cid:12)(cid:12)(cid:12)(cid:12) cos (cid:18) bπ √− c (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:20) O (cid:18) log mm (cid:19)(cid:21) , (1.26) γ m − = q m (2 m − · | c | m − · √− c m − · [(2 m − · π · (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) bπ √− c (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:20) O (cid:18) log mm (cid:19)(cid:21) . Here, m is a positive integer and γ n is the n-th instability interval. Some lemmas
Lemma 2.1 ([2]) . Let the Schr¨odinger operator (2.1) Ly = − y ′′ + v ( x ) y be defined on R , with a real-valued periodic L ([0 , π ]) -potential v ( x ) , where v ( x ) = P m ∈ Z V ( m ) exp( imx ) , V ( m ) = 0 for odd m , then k v k = P | V ( m ) | .(a) If k v k < , then for each n = 1 , , · · · , there exists z = z n such that | z | ≤ k v k , and (2.2) 2 | β n ( z ) | (1 − k v k /n ) ≤ γ n ≤ | β n ( z ) | (1 + 3 k v k /n ) , where (2.3) β n ( z ) = V (2 n ) + ∞ X k =1 X j , ··· ,j k = ± n V ( n − j ) V ( j − j ) · · · V ( j k − − j k ) V ( j k + n )( n − j + z ) · · · ( n − j k + z ) converges absolutely and uniformly for | z | ≤ , and γ n is the n-th instability interval.(b) If V (0) = π R π v ( x ) dx = 0 , then there is N = N ( v ) such that (2.2) holdsfor n ≥ N with z = z n , | z n | < . HITTAKER-HILL 5
Lemma 2.2 ([8]) . The Ince equation (2.4) (1 + a cos 2 t ) y ′′ ( t ) + b (sin 2 t ) y ′ ( t ) + ( c + d cos 2 t ) y ( t ) = 0 can be transformed into the Whittaker-Hill equation by assuming (2.5) a = 0 , b = − q, c = λ + 2 q , d = 4( m − q. Moreover, (2.6) sign( α n − β n ) = sign q · sign n − Y p = − n (2 p − m + 1) and (2.7) sign( α n +1 − β n +1 ) = sign q · sign n − Y p = − n (2 p − m ) , where a , b and d are real; α n and β n +2 are defined by the eigenvalues correspond-ing to non-trivial even and odd solutions with period π , respectively; and α n +1 and β n +1 are defined by the eigenvalues corresponding to non-trivial even and oddsolutions with semi-period π , respectively. Lemma 2.3 ([7]) . The Whittaker-Hill equation (2.8) f ′′ + [ λ + 4 mq cos 2 x + 2 q cos 4 x ] f = 0 can have two linearly independent solutions of period π or π if and only if m is an integer. If m = 2 l is even, then the odd intervals of instability on the λ axis disappear, with at most | l | + 1 exceptions, but no even interval of instabilitydisappears. If m = 2 l + 1 is odd, then at most | l | + 1 even intervals of instabilityremain but no interval of instability disappears. Lemma 2.4.
The Whittaker-Hill operator (2.9) L = − D − ( B cos 2 x + C cos 4 x ) admits all even gaps closed except the first n when ± B √ C = − n − , n ∈ Z ≥ ; andall odd gaps closed except the first n + 1 when ± B √ C = − n − , n ∈ Z ≥ .Proof. By Lemma 2.3, we obtain that the Whittaker-Hill equation(2.10) f ′′ ( x ) + ( A + B cos 2 x + C cos 4 x ) f ( x ) = 0have two linearly independent solutions of period or semi-periodic π if and only if B √ C ∈ Z . Moreover, we transform (2.10) into the Ince equation(2.11) g ′′ ( x ) + 4 r C x · g ′ ( x ) + " ( A + C ) + B + 4 r C ! cos 2 x g ( x ) = 0 . via f ( x ) = e − √ C cos 2 x · g ( x ). From Lemma 2.2, we can write the parameters q , λ and m of equation (2.4) in terms of A , B and C , i.e., q = − r C , λ = A, m = − B √ C . (1) If m = 2 n + 1, n ∈ N + ∪ { } , i.e., B √ C = − n −
1, and the solutions satisfythe periodic boundary conditions, then we deduce from Lemma 2.2 that the first2 n + 1 eigenvalues are simple, and others are double. XU-DAN LUO (2) If m = 2 n + 2, n ∈ N + ∪ { } , i.e., B √ C = − n −
2, and the solutions satisfythe semi-periodic boundary conditions, then we also deduce from Lemma 2.2 thatthe first 2 n + 2 eigenvalues are simple, and others are double.Besides, we can also transform (2.10) into the Ince equation(2.12) g ′′ ( x ) − r C x · g ′ ( x ) + " ( A + C ) + B − r C ! cos 2 x g ( x ) = 0 . via f ( x ) = e √ C cos 2 x · g ( x ). Thus, we have similar conclusions. (cid:3) In order to prove our results, let us consider all possible walks from − n to n .Each such walk is determined by the sequence of its steps(2.13) x = ( x , · · · , x ν +1 ) , or by its vertices(2.14) j s = − n + s X k =1 x k , s = 1 , · · · , ν. The relationship between steps and vertices are given by the formula(2.15) x = n + j ; x k = j k − j k − , k = 2 , · · · , ν ; x ν +1 = n − j ν . Definition 2.5.
Let X denote the set of all walks from − n to n with steps ± ±
4. For each x = ( x s ) ν +1 s =1 ∈ X and each z ∈ R , we define(2.16) B n ( x, z ) = V ( x ) · · · V ( x ν +1 )( n − j + z ) · · · ( n − j ν + z ) . Definition 2.6.
Let X + denote the set of all walks from − n to n with positivesteps equal to 2 or 4. For each ξ ∈ X + , let X ξ denote the set of all walks x ∈ X \ X + such that each vertex of ξ is a vertex of x also. For each ξ ∈ X + and µ ∈ N , let X ξ,µ be the set of all x ∈ X ξ such that x has µ more vertices than ξ . Moreover, for each µ − tuple ( i , · · · , i µ ) of integers in I n = ( n + 2 Z ) \ {± n } , we define X ξ ( i , · · · , i µ )as the set of all walks x with ν + 1 + µ steps such that ( i , · · · , i µ ) and the sequenceof the vertices of ξ are complementary subsequences of the sequence of the verticesof x .From Definition 2.6, we deduce(2.17) X ξ,µ = [ ( i , ··· ,i µ ) ∈ ( I n ) µ X ξ ( i , · · · , i µ ) . Lemma 2.7 ([2]) . If ξ ∈ X + and n ≥ , then for z ∈ [0 , − z log nn ≤ B n ( ξ, z ) B n ( ξ, ≤ − z log n n , and for z ∈ ( − , | z | log n n ≤ B n ( ξ, z ) B n ( ξ, ≤ | z | nn . Lemma 2.8 ([2]) . For each walk ξ ∈ X + and each µ − tuple ( i , · · · , i µ ) ∈ ( I n ) µ , (2.20) ♯X ξ ( i , · · · , i µ ) ≤ µ . HITTAKER-HILL 7
Lemma 2.9. If ξ ∈ X + and | z | ≤ , then there exists n such that for n ≥ n , (2.21) X x ∈ X ξ | B n ( x, z ) | ≤ | B n ( ξ, z ) | · K log nn , where K = 40 (cid:16) | b q m | + | c q m | + | b c q m − m | (cid:17) .Proof. By Definition 2.6, we have(2.22) X x ∈ X ξ | B n ( x, z ) | = ∞ X µ =1 X x ∈ X ξ,µ | B n ( x, z ) | . Moreover,(2.23) X x ∈ X ξ,µ | B n ( x, z ) | ≤ X ( i , ··· ,i µ ) X X ξ ( i , ··· ,i µ ) | B n ( x, z ) | , where the first sum on the right is taken over all µ − tuples ( i , · · · , i µ ) of integers i s ∈ n + 2 Z such that i s = ± n .Fix ( i , · · · , i µ ), if x ∈ X ξ ( i , · · · , i µ ), then(2.24) B n ( x, z ) B n ( ξ, z ) = Q k V ( x k ) Q s V ( ξ s ) · n − i + z ) · · · ( n − i µ + z ) . Note that V ( ±
2) = b q m and V ( ±
4) = c q m . If each step of ξ is a step of x , then(2.25) Q k | V ( x k ) | Q s | V ( ξ s ) | ≤ C µ , where C := | b q m | + | c q m | + | b c q m − m | . For the general case, let ( j s ) νs =1 be thevertices of ξ , and let us put for convenience j = − n and j ν +1 = n . Since eachvertex of ξ is a vertex of x , for each s , 1 ≤ s ≤ ν + 1,(2.26) ξ s = j s − j s − = X k ∈ J s x k , where x k , k ∈ J s , are the steps of x between the vertices j s − and j s . Fix an s ,1 ≤ s ≤ ν + 1. If ξ s = 2, then there is a step x k ∗ , k ∗ ∈ J s such that | x k ∗ | = 2,otherwise, ξ s would be a multiple of 4. Hence, | V ( ξ s ) | = | V ( x ∗ k ) | , which impliesthat(2.27) Q J s | V ( x k ) || V ( ξ s ) | ≤ C b s − , where b s is the cardinality of J s .If ξ s = 4, there are two possibilities. (1) If there is k ∗ ∈ J s with | x k ∗ | = 4, then | V ( ξ s ) | = | V ( x k ∗ ) | , so the above inequality holds. (2) There are k ′ , k ′′ ∈ J s suchthat | x k ′ | = | x k ′′ | = 2, hence,(2.28) | V ( x k ′ ) V ( x k ′′ ) || V ( ξ s ) | = (cid:12)(cid:12)(cid:12)(cid:12) b c q m − m (cid:12)(cid:12)(cid:12)(cid:12) ≤ C, so the above inequality also holds. Note that(2.29) X s ( b s −
1) = µ, XU-DAN LUO we get(2.30) Q k V ( x k ) Q s V ( ξ s ) ≤ C µ holds for the general case.By(2.31) 1 | n − i + z | ≤ | n − i | , where i = ± n , | z | ≤
1, we have(2.32) | B n ( x, z ) || B n ( ξ, z ) | ≤ (2 C ) µ | n − i | · · · | n − i µ | , x ∈ X ξ ( i , · · · , i µ ) . By Lemma 2.8, we derive(2.33) X x ∈ X ξ ( i , ··· ,i µ ) | B n ( x, z ) || B n ( ξ, z ) | ≤ (10 C ) µ | n − i | · · · | n − i µ | . Combining Lemma 2.7, it yields X x ∈ X ξ,µ | B n ( x, z ) || B n ( ξ, z ) | ≤ X ( i , ··· ,i µ ) (10 C ) µ | n − i | · · · | n − i µ | ≤ X i ∈ ( n +2 Z ) \{± n } C | n − i | µ ≤ (10 C ) µ (cid:18) nn (cid:19) µ ≤ (cid:18) C log nn (cid:19) µ . (2.34)Thus,(2.35) X x ∈ X ξ,µ | B n ( x, z ) | ≤ | B n ( ξ, z ) | · (cid:18) C log nn (cid:19) µ . Hence,(2.36) X x ∈ X ξ | B n ( x, z ) || B n ( ξ, z ) | ≤ ∞ X µ =1 (cid:18) C log nn (cid:19) µ . We can choose n ∈ N + such that C log nn ≤ for n ≥ n . Then(2.37) X x ∈ X ξ | B n ( x, z ) || B n ( ξ, z ) | ≤ C log nn . Therefore, there exists n such that for n ≥ n ,(2.38) X x ∈ X ξ | B n ( x, z ) | ≤ | B n ( ξ, z ) | · K log nn , where K := 40 C = 40 (cid:16) | b q m | + | c q m | + | b c q m − m | (cid:17) . (cid:3) HITTAKER-HILL 9 Proof of Theorem 1.3
Note that(3.1) V ( ±
2) = bq m , V ( ±
4) = cq m k v k = 12 (cid:16) b q m + c q m (cid:17) , by Lemma 2.1, we have(3.3) γ n = ± (cid:16) V (2 n ) + ∞ X k =1 β k ( n, z ) (cid:17)(cid:16) O ( q · min { m ,m } ) (cid:17) , where β k ( n, z ) = X j , ··· ,j k = ± n V ( n − j ) V ( j − j ) · · · V ( j k − − j k ) V ( j k + n )( n − j + z ) · · · ( n − j k + z )= X j , ··· ,j k = ± n V ( n + j ) V ( j − j ) · · · V ( j k − j k − ) V ( n − j k )( n − j + z ) · · · ( n − j k + z )(3.4)and z = O ( q ). Moreover, all series converge absolutely and uniformly for sufficientlysmall q .Note that(3.5) ( n + j ) + ( j − j ) + · · · + ( j k − j k − ) + ( n − j k ) = 2 n, and(3.6) V ( n + j ) V ( j − j ) · · · V ( j k − j k − ) V ( n − j k )( n − j + z ) · · · ( n − j k + z ) = 0when(3.7) ( n + j ) , ( j − j ) , · · · , ( j k − j k − ) , ( n − j k ) ∈ {± , ± } . We distinguish three cases to discuss.
Case 1. If m > m , then(3.8) V ( n + j ) V ( j − j ) · · · V ( j k − j k − ) V ( n − j k )is a monomial in q of degree at least(3.9) m · h | n + j | + | j − j | + · · · + | j k − j k − | + | n − j k | i . The minimum case occurs when(3.10) ( n + j ) , ( j − j ) , · · · , ( j k − j k − ) , ( n − j k ) ∈ {± } , then(3.11) ( n + j ) + ( j − j ) + · · · + ( j k − j k − ) + ( n − j k ) ∈ Z , while(3.12) ( n + j ) + ( j − j ) + · · · + ( j k − j k − ) + ( n − j k ) = 2 n. If n is even, i.e., n = 2 m , m ∈ Z > , since | n + j | + | j − j | + · · · + | j k − j k − | + | n − j k |≥ ( n + j ) + ( j − j ) + · · · + ( j k − j k − ) + ( n − j k )= 2 n = 4 m, (3.13)we obtain(3.14) V ( n + j ) V ( j − j ) · · · V ( j k − j k − ) V ( n − j k )is a monomial in q of degree at least m · m . Such monomial in q of degree m · m corresponds to a walk from − n to n with vertices j , j , · · · , j k = ± n and positivesteps of length 4. Thus,(3.15) γ m = ± P m ( t ) q m m + O (cid:16) q m ( m − )+ m (cid:17) , where(3.16) P m ( t ) q m m = 2 (cid:16) V (4 m ) + ∞ X k =1 β k (2 m, z ) (cid:17) . We have(3.17) P ( t ) q m = 2 V (4) = cq m and P m ( t ) q m m = 2 ∞ X k =1 β k (2 m, z )= 2 · (cid:16) c (cid:17) m · q m m · m − Y j =1 (cid:16) (4 m − ( − m + 4 j ) ) (cid:17) − = 32 · ( c ) m · q m m m [( m − (3.18)for m ≥
2. Therefore,(3.19) γ m = (cid:12)(cid:12)(cid:12)(cid:12) · ( c ) m · q m m m [( m − (cid:12)(cid:12)(cid:12)(cid:12) + O (cid:16) q m ( m − )+ m (cid:17) . If n is odd, i.e., n = 2 m − m ∈ Z > , since(3.20) ( n + j ) + ( j − j ) + · · · + ( j k − j k − ) + ( n − j k ) = 2 n = 4 m − , then(3.21) V ( n + j ) V ( j − j ) · · · V ( j k − j k − ) V ( n − j k )is a monomial in q of degree at least m · h | n + j | + | j − j | + · · · + | j k − j k − | + | n − j k | i + m − m ≥ m · (4 m −
2) + m − m m ( m −
1) + m . (3.22) HITTAKER-HILL 11
Such monomial in q of degree m ( m −
1) + m corresponds to a walk from − n to n with vertices j , j , · · · , j k = ± n and positive steps. Specifically, except for onestep with length 2, the others are of length 4. Thus,(3.23) γ m − = ± P m − ( t ) q m ( m − m + O (cid:16) q m ( m − )+2 m (cid:17) , where(3.24) P m − ( t ) q m ( m − m = 2 (cid:16) V (4 m −
2) + ∞ X k =1 β k (2 m − , z ) (cid:17) . We obtain(3.25) P ( t ) q m = 2 V (2) = bq m , (3.26) P ( t ) q m + m = 2 ∞ X k =1 β k (3 , z ) = 2 (cid:18) bq m (cid:19) (cid:18) cq m (cid:19) (cid:18) − + 13 − (cid:19) = bcq m + m ,P ( t ) q m +2 m = 2 ∞ X k =1 β k (5 , z )= 2 (cid:18) bq m (cid:19) (cid:18) cq m (cid:19) (cid:20) − )(5 − ) + 1(5 − )(5 − ) + 1(5 − )(5 − ) (cid:21) = bc q m +2 m · , (3.27) P m − ( t ) q m ( m − m = 2 ∞ X k =1 β k (2 m − , z )= 2 (cid:16) c (cid:17) m − · (cid:18) b (cid:19) · q m ( m − m · n m − X i =1 i Y j =1 h (2 m − − ( − m + 1 + 4 j ) i − · m − Y j = i h (2 m − − ( − m + 3 + 4 j ) i − + m − Y j =0 h (2 m − − ( − m + 3 + 4 j ) i − + m − Y j =1 h (2 m − − ( − m + 1 + 4 j ) i − o = 2 (cid:16) c (cid:17) m − · (cid:18) b (cid:19) · q m ( m − m · m · n m − · m − X i =1 (2 m − i − · (2 i − i ! · ( m − − i )!+ 2(2 m − · ( m − o . (3.28)Hence,(3.29) γ = | bq m | + O ( q m − m ) , γ = | bcq m + m | O ( q m + m ) , γ m − = (cid:12)(cid:12)(cid:12) (cid:16) c (cid:17) m − · b · q m ( m − m · m · n m − · m − X i =1 (2 m − i − · (2 i − i ! · ( m − − i )! + 2(2 m − · ( m − o(cid:12)(cid:12)(cid:12) + O (cid:16) q m ( m − )+2 m (cid:17) (3.30)for m ≥ Case 2. If m < m , then(3.31) V ( n + j ) V ( j − j ) · · · V ( j k − j k − ) V ( n − j k )is a monomial in q of degree at least(3.32) m · h | n + j | + | j − j | + · · · + | j k − j k − | + | n − j k | i . The minimum case occurs when(3.33) ( n + j ) , ( j − j ) , · · · , ( j k − j k − ) , ( n − j k ) ∈ {± } , then(3.34) ( n + j ) + ( j − j ) + · · · + ( j k − j k − ) + ( n − j k ) ∈ Z , while(3.35) ( n + j ) + ( j − j ) + · · · + ( j k − j k − ) + ( n − j k ) = 2 n. Since | n + j | + | j − j | + · · · + | j k − j k − | + | n − j k |≥ ( n + j ) + ( j − j ) + · · · + ( j k − j k − ) + ( n − j k )= 2 n, (3.36)we have(3.37) V ( n + j ) V ( j − j ) · · · V ( j k − j k − ) V ( n − j k )is a monomial in q of degree at least m · n . Such monomial in q of degree m · n corresponds to a walk from − n to n with vertices j , j , · · · , j k = ± n and positivesteps of length 2. Thus,(3.38) γ n = ± P n ( t ) q m n + O ( q m n + m − m ) , where(3.39) P n ( t ) q m n = 2 (cid:16) V (2 n ) + ∞ X k =1 β k ( n, z ) (cid:17) . We deduce(3.40) P ( t ) q m = 2 V (2) = bq m , P ( t ) q m = 2 V (4) + (cid:16) bq m (cid:17) = cq m + b q m , HITTAKER-HILL 13 P n ( t ) q m n = 2 ∞ X k =1 β k ( n, z )= 2 · (cid:16) b (cid:17) n · q m · n · n − Y j =1 ( n − ( − n + 2 j ) ) − = 2 · (cid:16) b (cid:17) n · q m · n · n − Y j =1 ( n − ( − n + 2 j ) ) − = 2 · ( b ) n · q m · n n − · [( n − = 8 · b n · q m n n · [( n − (3.41)for n ≥
3. Therefore,(3.42) γ = | bq m | + O ( q m − m ) , γ = (cid:12)(cid:12)(cid:12)(cid:12) cq m + b q m (cid:12)(cid:12)(cid:12)(cid:12) + O ( q m ) , (3.43) γ n = (cid:12)(cid:12)(cid:12)(cid:12) · b n · q m n n · [( n − (cid:12)(cid:12)(cid:12)(cid:12) + O ( q m n + m − m )for n ≥ Case 3. If m = m , then(3.44) V ( n + j ) V ( j − j ) · · · V ( j k − j k − ) V ( n − j k )is a monomial in q of degree(3.45) m · h | n + j | + | j − j | + · · · + | j k − j k − | + | n − j k | i . Since | n + j | + | j − j | + · · · + | j k − j k − | + | n − j k |≥ ( n + j ) + ( j − j ) + · · · + ( j k − j k − ) + ( n − j k )= 2 n, (3.46)we have(3.47) V ( n + j ) V ( j − j ) · · · V ( j k − j k − ) V ( n − j k )is a monomial in q of degree at least m · n , and each such monomial of degree m · n corresponds to a walk from − n to n with vertices j , j , · · · , j k = ± n andpositive steps of length 2 or 4. The minimum case occurs when n + j , j − j , · · · , j k − j k − and n − j k are of the same sign, while the second smallest degree is forone step of length 2 with opposite sign. Thus,(3.48) γ n = ± P n ( t ) q m n + O ( q m ( n +2) ) , where(3.49) P n ( t ) q m n = 2 (cid:16) V (2 n ) + ∞ X k =1 β k ( n, z ) (cid:17) . We obtain(3.50) P ( t ) q m = 2 V (2) = bq m , P ( t ) q m = 2 V (4) + (cid:16) bq m (cid:17) = cq m + b q m ,P n ( t ) q m n = 2 ∞ X k =1 β k ( n, z )= 2 · P n (cid:16) b (cid:17) · q m · n · n − Y j =1 ( n − ( − n + 2 j ) ) − = 2 · P n (cid:16) b (cid:17) · q m · n · n − Y j =1 ( n − ( − n + 2 j ) ) − = 8 · P n (cid:0) b (cid:1) · q m · n n · [( n − (3.51)for n ≥
3. Therefore,(3.52) γ = | bq m | + O ( q m ) , γ = (cid:12)(cid:12)(cid:12)(cid:12) cq m + b q m (cid:12)(cid:12)(cid:12)(cid:12) + O ( q m ) , (3.53) γ n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · P n (cid:0) b (cid:1) · q m · n n · [( n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( q m ( n +2) )for n ≥
3, where P n (cid:0) b (cid:1) is a polynomial of b with degree n and leading coefficient1. Specifically, if n is even, i.e., n = 2 m , m ∈ Z > , then(3.54) ( n + j ) + ( j − j ) + · · · + ( j k − j k − ) + ( n − j k ) = 4 m, which implies that each walk from − m to 2 m has even number of steps with length2. We have(3.55) P m (cid:16) b (cid:17) = m Y k =1 (cid:16)(cid:16) b (cid:17) − x k (cid:17) , where x k , k = 1 , · · · , m , depend on m . By Lemma 2.4, we obtain all even gapsclosed except the first k if (cid:0) b (cid:1) = − c (cid:0) k + (cid:1) , which yields(3.56) P m (cid:16) b (cid:17) = m Y k =1 (cid:16)(cid:16) b (cid:17) + 8 c (cid:16) k − (cid:17) (cid:17) . Hence,(3.57) γ m = 8 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q mk =1 (cid:16)(cid:16) b (cid:17) + 8 c (cid:16) k − (cid:17) (cid:17) · q m · m m · [(2 m − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( q m ( m +1) )for m ≥
2. If n is odd, i.e., n = 2 m − m ∈ Z > , then(3.58) ( n + j ) + ( j − j ) + · · · + ( j k − j k − ) + ( n − j k ) = 2 n = 4 m − , HITTAKER-HILL 15 which implies that each walk from − m to 2 m has odd number of steps with length2. We have(3.59) P m − (cid:16) b (cid:17) = b m − Y k =1 (cid:16)(cid:16) b (cid:17) − y k (cid:17) , where y k , k = 1 , · · · , m −
1, depend on m . By Lemma 2.4, we deduce(3.60) P m − (cid:16) b (cid:17) = b m − Y k =1 (cid:16)(cid:16) b (cid:17) + 8 ck (cid:17) . Hence,(3.61) γ m − = 32 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b Q m − k =1 (cid:16)(cid:16) b (cid:17) + 8 ck (cid:17) · q m · (2 m − m · [(2 m − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( q m (2 m +1) )for m ≥
2. 4.
Proof of Theorem 1.4
Since V ( ±
2) = b q m and V ( ±
4) = c q m , thus,(4.1) k v k = 12 (cid:16) b q m + c q m (cid:17) . By Lemma 2.1, we get(4.2) γ n = 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ X B n ( x, z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) O (cid:18) n (cid:19)(cid:19) , where z = z n depends on n , but | z | < σ n = P ξ ∈ X + B n ( ξ,
0) := σ + n + σ − n , where σ ± n := P ξ : B n ( ξ, ≷ B n ( ξ, ξ ∈ X + ,(4.3) B n ( ξ,
0) = V ( x ) · · · V ( x ν +1 )( n − j ) · · · ( n − j ν ) , where x i = 2 or 4 for i = 1 , · · · , ν + 1.Note that X \ X + = S ξ ∈ X + X ξ , we choose disjoint sets X ′ ξ ⊂ X ξ so that(4.4) X \ X + = [ ξ ∈ X + X ′ ξ . Then(4.5) X x ∈ X \ X + B n ( x, z ) = X ξ ∈ X + X x ∈ X ′ ξ B n ( x, z ) , therefore, we have X x ∈ X B n ( x, z ) = X ξ ∈ X + B n ( ξ, z ) + X x ∈ X ′ ξ B n ( x, z ) = X ξ : B n ( ξ, > B n ( ξ, z ) + X x ∈ X ′ ξ B n ( x, z ) + X ξ : B n ( ξ, < B n ( ξ, z ) + X x ∈ X ′ ξ B n ( x, z ) : Σ = Σ + + Σ − , (4.6)where Σ ± := P ξ : B n ( ξ, ≷ (cid:16) B n ( ξ, z ) + P x ∈ X ′ ξ B n ( x, z ) (cid:17) .By Lemma 2.7 and Lemma 2.9, we get there exists a constant C > (cid:20) ∓ C log nn (cid:21) σ ± n ≤ Σ ± ≤ (cid:20) ± C log nn (cid:21) σ ± n , which is followed by(4.8) (cid:12)(cid:12)(cid:12)(cid:12) Σ σ n − (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | σ − n | + σ + n | σ n | · log nn . If ξ ∈ X + , then V ( x ) · · · V ( x ν +1 ) is a monomial in q of degree m · ( x + · · · + x ν +1 ) = m · n . From Case 3 of Theorem 1.3, we have(4.9) σ m = X ξ ∈ X + B m ( ξ,
0) = q m · m m − · [(2 m − · m Y k =1 (cid:18) b (cid:19) + 8 c (cid:18) k − (cid:19) ! and(4.10) σ m − = X ξ ∈ X + B m − ( ξ,
0) = q m (2 m − m − · [(2 m − · b · m − Y k =1 (cid:18) b (cid:19) + 8 ck ! . Moreover, σ m = 0 when b = 2 √− c · ( k − ) and σ m − = 0 when b = 2 √− c · k ,where c <
0. So(4.11) | σ − m | + σ +2 m | σ m | = Q mk =1 (cid:16) − b c (2 k − (cid:17)Q mk =1 (cid:12)(cid:12)(cid:12) b c (2 k − (cid:12)(cid:12)(cid:12) ≤ Q ∞ k =1 (cid:16) − b c (2 k − (cid:17)Q ∞ k =1 (cid:12)(cid:12)(cid:12) b c (2 k − (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cosh (cid:16) bπ √− c (cid:17) cos (cid:16) bπ √− c (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Similarly, we have(4.12) | σ − m − | + σ +2 m − | σ m − | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sinh (cid:16) bπ √− c (cid:17) sin (cid:16) bπ √− c (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By (4.8), we obtain(4.13) X x ∈ X B m ( x, z ) = σ m (cid:20) O (cid:18) log mm (cid:19)(cid:21) = X ξ ∈ X + B m ( ξ, (cid:20) O (cid:18) log mm (cid:19)(cid:21) HITTAKER-HILL 17 and(4.14) X x ∈ X B m − ( x, z ) = σ m − (cid:20) O (cid:18) log mm (cid:19)(cid:21) = X ξ ∈ X + B m − ( ξ, (cid:20) O (cid:18) log mm (cid:19)(cid:21) . Notice that(4.15) cos (cid:18) bπ √− c (cid:19) = ∞ Y k =1 (cid:18) b c (2 k − (cid:19) and(4.16) sin (cid:18) bπ √− c (cid:19) = bπ √− c ∞ Y k =1 (cid:18) b c (2 k ) (cid:19) , then(4.17) cos (cid:18) bπ √− c (cid:19) = m Y k =1 (cid:18) b c (2 k − (cid:19) (cid:20) O (cid:18) m (cid:19)(cid:21) and(4.18) sin (cid:18) bπ √− c (cid:19) = bπ √− c m − Y k =1 (cid:18) b c (2 k ) (cid:19) (cid:20) O (cid:18) m (cid:19)(cid:21) . Hence,(4.19) X ξ ∈ X + B m ( ξ,
0) = q m · m · ( − m · c m m − · [(2 m − · cos (cid:18) bπ √− c (cid:19) · (cid:20) O (cid:18) m (cid:19)(cid:21) and(4.20) X ξ ∈ X + B m − ( ξ,
0) = q m (2 m − · ( − m − · c m − · √− c m − · [(2 m − · π · sin (cid:18) bπ √− c (cid:19) · (cid:20) O (cid:18) m (cid:19)(cid:21) . Combining (4.8), (4.11) and (4.12), we deduce X x ∈ X B m ( x, z ) = X ξ ∈ X + B m ( ξ, (cid:20) O (cid:18) log mm (cid:19)(cid:21) = q m · m · ( − m · c m m − · [(2 m − · cos (cid:18) bπ √− c (cid:19) · (cid:20) O (cid:18) log mm (cid:19)(cid:21) (4.21)and X x ∈ X B m − ( x, z ) = X ξ ∈ X + B m − ( ξ, (cid:20) O (cid:18) log mm (cid:19)(cid:21) = q m (2 m − · ( − m − · c m − · √− c m − · [(2 m − · π · sin (cid:18) bπ √− c (cid:19) · (cid:20) O (cid:18) log mm (cid:19)(cid:21) . (4.22)Therefore,(4.23) γ m = q m · m · | c | m m − · [(2 m − · (cid:12)(cid:12)(cid:12)(cid:12) cos (cid:18) bπ √− c (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:20) O (cid:18) log mm (cid:19)(cid:21) and(4.24) γ m − = q m (2 m − · | c | m − · √− c m − · [(2 m − · π · (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) bπ √− c (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:20) O (cid:18) log mm (cid:19)(cid:21) . References
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