Spectral analysis for the class of integral operators arising from well-posed boundary value problems of finite beam deflection on elastic foundation: characteristic equation
aa r X i v : . [ m a t h . C A ] J u l SPECTRAL ANALYSIS FOR THE CLASS OF INTEGRALOPERATORS ARISING FROM WELL-POSED BOUNDARYVALUE PROBLEMS OF FINITE BEAM DEFLECTION ONELASTIC FOUNDATION: CHARACTERISTIC EQUATION
SUNG WOO CHOI
Abstract.
We consider the boundary value problem for the deflection of afinite beam on an elastic foundation subject to vertical loading. We constructa one-to-one correspondence Γ from the set of equivalent well-posed two-pointboundary conditions to gl(4 , C ). Using Γ, we derive eigenconditions for theintegral operator K M for each well-posed two-point boundary condition repre-sented by M ∈ gl(4 , , C ). Special features of our eigenconditions include; (1)they isolate the effect of the boundary condition M on Spec K M , (2) they con-nect Spec K M to Spec K l,α,k whose structure has been well understood. Usingour eigenconditions, we show that, for each nonzero real λ Spec K l,α,k , thereexists a real well-posed boundary condition M such that λ ∈ Spec K M . Thisin particular shows that the integral operators K M arising from well-posedboundary conditions, may not be positive nor contractive in general, as op-posed to K l,α,k . Introduction
We consider the boundary value problem for the vertical deflection of a linear-shaped beam of finite length 2 l resting horizontally on an elastic foundation, whilethe beam is subject to a vertical loading. Due to its wide range of applica-tions, this problem has been one of the main topics in mechanical engineering fordecades [1,4–10,12,13]. By the classical Euler beam theory [12], the upward verticalbeam deflection u ( x ) satisfies the following linear fourth-order ordinary differentialequation. EI · u (4) ( x ) + k · u ( x ) = w ( x ) , x ∈ [ − l, l ] . (1.1)Here, k is the spring constant density of the elastic foundation, and w ( x ) is thedownward load density applied vertically on the beam. The constants E and I arethe Young’s modulus and the mass moment of inertia respectively, so that EI isthe flexural rigidity of the beam. Denoting α = p k/EI >
0, we transform (1.1)into the following equivalent form, which we call DE( w ).DE( w ) : u (4) + α u = α k · w. (1.2)Throughout this paper, we will assume that l , α , k are fixed positive constants.The homogeneous version of (1.2) isDE(0) : u (4) + α u = 0 . (1.3) Mathematics Subject Classification.
Key words and phrases. beam, deflection, Green’s function, eigenvalue, spectrum, integraloperator.
Let gl( m, n, C ) (respectively, gl( m, n, R )) be the set of m × n matrices withcomplex (respectively, real) entries. When m = n , we denote gl( n, C ) = gl( n, n, C )and gl( n, R ) = gl( n, n, R ). Define the following linear operator B : C [ − l, l ] → gl(8 , , C ) by B [ u ] = (cid:0) u ( − l ) u ′ ( − l ) u ′′ ( − l ) u (3) ( − l ) u ( l ) u ′ ( l ) u ′′ ( l ) u (3) ( l ) (cid:1) T , (1.4)where C n [ − l, l ] is the space of n times differentiable complex-valued functions onthe interval [ − l, l ]. Then any two-point boundary condition can formally be givenwith a 4 × M ∈ gl(4 , , C ) and a 4 × b ∈ gl(4 , , C ) as follows. M · B [ u ] = b . (1.5)For example, the boundary condition u ( − l ) = u − , u ′ ( − l ) = u ′− , u ( l ) = u + , u ′ ( l ) = u ′ + corresponds to the case when M = , b = u − u ′− u + u ′ + . The homogeneous boundary condition associated to (1.5), which we denote byBC( M ), is BC( M ) : M · B [ u ] = , (1.6)where = (cid:0) (cid:1) T . The boundary value problem consisting of the nonho-mogeneous equation DE( w ) and the boundary condition (1.5) is well-posed , if ithas a unique solution. In fact, it is easy to see that this boundary value problem iswell-posed for any fixed w and b , if and only if the boundary value problem consist-ing of the homogeneous equation DE(0) and the homogeneous boundary conditionBC( M ) is well-posed, in which case we will just call M ∈ gl(4 , , C ) well-posed . Wedenote the set of all well-posed matrices in gl(4 , , C ) by wp(4 , , C ).It is well-known from the classical Green’s function theory [11] that, for eachwell-posed M ∈ wp(4 , , C ), there exists a unique function G M ( x, ξ ) defined on[ − l, l ] × [ − l, l ], called the Green’s function corresponding to M , such that the uniquesolution of the boundary value problem consisting of DE( w ) and BC( M ) is givenby K M [ w ] = Z l − l G M ( x, ξ ) w ( ξ ) dξ for every continuous function w on [ − l, l ]. The integral operator K M becomes acompact linear operator on the Hilbert space L [ − l, l ] of complex-valued square-integrable functions on [ − l, l ]. Analyzing the structure of the spectrum Spec K M ,or the set of eigenvalues, of the operator K M , is of paramount importance forunderstanding the boundary value problem represented by given well-posed M ∈ wp(4 , , C ).We call M , N ∈ wp(4 , , C ) equivalent , and denote M ≈ N , when K M = K N ,or equivalently, when G M = G N . For given M ∈ wp(4 , , C ), denote by [ M ] theequivalence class with respect to ≈ containing M . The set of all these equivalenceclasses, which is the quotient set wp(4 , , C ) / ≈ of wp(4 , , C ) by the relation ≈ , isdenoted simply by wp( C ). PECTRAL ANALYSIS FOR WELL-POSED BEAM DEFLECTION 3
In [2, 3], Choi analyzed an integral operator K l,α,k = K l on L [ − l, l ] defined by K l,α,k [ w ]( x ) = Z l − l G ( x, ξ ) w ( ξ ) dξ, (1.7)where G ( x, ξ ) = α k exp (cid:18) − α √ | x − ξ | (cid:19) sin (cid:18) α √ | x − ξ | + π (cid:19) (1.8)is the Green’s function of the boundary value problem consisting of DE(0) and theboundary condition lim x →±∞ u ( x ) = 0 for an infinitely long beam. It turns outthat K l,α,k = K Q in our terminology, where Q = α −√ α √ α − α α √ α
10 0 0 0 −√ α − α (1.9)in wp(4 , , C ). What is special about this particular operator K Q is that its spec-trum is exceptionally well-understood. In Proposition 1 below, h is an explicitlydefined strictly increasing function from [0 , ∞ ) to itself such that h (0) = 0 andlim t →∞ h ( t ) /t = L , where L = 2 lα is the dimensionless constant called the intrin-sic length of the beam. For two nonnegative functions f , g defined either on [0 , ∞ )or on N , denote f ( t ) ∼ g ( t ), if there exists T > m ≤ f ( t ) /g ( t ) ≤ M forevery t > T for some constants 0 < m ≤ M < ∞ . Thus h − ( t ) ∼ t/L with thisnotation. Proposition 1 ( [3]) . The spectrum
Spec K Q of the operator K Q = K l,α,k is of theform n µ n k (cid:12)(cid:12)(cid:12) n = 1 , , , · · · o ∪ n ν n k (cid:12)(cid:12)(cid:12) n = 1 , , , · · · o ⊂ (cid:18) , k (cid:19) , where µ n and ν n for n = 1 , , , . . . depend only on the intrinsic length L of thebeam. µ n ∼ ν n ∼ n − , and
11 + (cid:8) h − (cid:0) πn + π (cid:1)(cid:9) < ν n <
11 + { h − (2 πn ) } < µ n <
11 + (cid:8) h − (cid:0) πn − π (cid:1)(cid:9) , n = 1 , , , . . . ,
11 + (cid:8) h − (cid:0) πn − π (cid:1)(cid:9) − µ n ∼ ν n −
11 + (cid:8) h − (cid:0) πn + π (cid:1)(cid:9) ∼ n − e − πn ,
11 + L (cid:0) π ( n − − π (cid:1) − µ n ∼
11 + L (cid:0) π ( n −
1) + π (cid:1) − ν n ∼ n − . In fact, numerical values of µ n and ν n can be computed with arbitrary precisionfor any given L >
0. See [3] for more details.In this paper, we will construct the Green’s function G M explicitly for every M ∈ wp(4 , , C ). As a result, we construct an explicit map wp(4 , , C ) → gl(4 , C ), M G M , in such a way that G M = G N , if and only if M ≈ N . This induces amap Γ : wp( C ) → gl(4 , C ), where Γ ([ M ]) = G M for M ∈ wp(4 , , C ). Especially,our construction of the map Γ has the following features.(Γ1) Γ is a one-to-one correspondence from wp( C ) to gl(4 , C ). S. W. CHOI (Γ2) Γ ([ Q ]) = O .(Γ3) Γ is constructive , in that Γ ([ M ]) can be computed explicitly for any given M ∈ wp(4 , , C ), and conversely, a representative of Γ − ( G ) in wp(4 , , C )can be computed explicitly for any given G ∈ gl(4 , C ).By (Γ1), Γ can be regarded as a faithful representation of wp( C ) by the algebragl(4 , C ). (Γ2) says that Γ is constructed to incorporate the special boundary con-dition Q . This will enable us in Theorem 1 and Corollaries 1, 2 below to obtainan eigencondition and characteristic equations for the operator K M , which connectSpec K M for general M ∈ wp(4 , , C ) to the well-understood Spec K Q in Proposi-tion 1. (Γ3) means that our eigencondition and characteristic equations for K M are constructed explicitly for each given M ∈ wp(4 , , C ). Conversely, wheneveryou find a class of matrices in gl(4 , C ) with which you can say something about thecorresponding eigencondition in Theorem 1, you can translate them back to thecorresponding boundary conditions explicitly. In fact, this is exactly what we doin Theorem 2 below.Among well-posed boundary conditions in wp(4 , , C ), ones with real entries areof particular interest. We denote by wp(4 , , R ), the set of well-posed matrices inwp(4 , , C ) with real entries. The set of all equivalence classes [ M ] in wp( C ) suchthat M ≈ N for some N ∈ wp(4 , , R ), is denoted by wp( R ). To characterizewp( R ), we introduce an R -algebra π (4) contained in gl(4 , C ). With π (4), we havea faithful representation of wp( R ), which is another feature of Γ.(Γ4) Γ (wp( R )) = π (4).The usefulness of π (4) is not just limited to characterizing wp( R ). The R -algebra π (4) is designed to measure an important symmetry of 4 × X λ ( x ) ∈ gl(4 , C )and y λ ( x ) ∈ gl(4 , , C ) will be defined explicitly for every λ ∈ C \ { } and x ∈ R inSection 6.2. Note that the second statement follows immediately from the first oneand (Γ2) above. Theorem 1.
Let M ∈ wp(4 , , C ) , = u ∈ L [ − l, l ] , and λ ∈ C . Then K M [ u ] = λ · u , if and only if λ = 0 and there exists = c ∈ gl(4 , , C ) such that u = y Tλ c and [ G M { X λ ( l ) − X λ ( − l ) } + X λ ( l )] c = . K Q [ u ] = λ · u , if and only if λ = 0 andthere exists = c ∈ gl(4 , , C ) such that u = y Tλ c and X λ ( l ) · c = . Thus, if we focus on the spectrum Spec K M , we have the following characteristicequation. Corollary 1.
Let M ∈ wp(4 , , C ) and λ ∈ C . Then λ ∈ Spec K M , if and only if λ = 0 and det [ G M { X λ ( l ) − X λ ( − l ) } + X λ ( l )] = 0 . λ ∈ Spec K Q , if and only if λ = 0 and det X λ ( l ) = 0 . Theorem 1 and Corollary 1 reveal an interesting connection between Spec K M for general M ∈ wp(4 , , C ) and the well-analyzed Spec K Q . The forms of theeigencondition and the characteristic equation for K M in them isolate the effect G M of the boundary condition M ∈ wp(4 , , C ), from the rest that is expressedessentially by the matrix X λ which is closely related to Spec K Q .By Corollary 1, X λ ( l ) is invertible for every 0 = λ Spec K Q . Thus we candefine Y λ ( l ) = X λ ( − l ) X λ ( l ) − − I ∈ gl(4 , C ) for every 0 = λ Spec K Q , where I is the 4 × PECTRAL ANALYSIS FOR WELL-POSED BEAM DEFLECTION 5
Corollary 2.
Let M ∈ wp(4 , , C ) . Suppose λ ∈ C \ Spec K Q . Then λ ∈ Spec K M ,if and only if λ = 0 and det { G M Y λ ( l ) − I } = 0 . Let M ∈ wp(4 , , C ). We call the dimensionless quantity k · kK M k the intrinsic L -norm of K M , where kK M k is the usual L -norm of K M , which is equal to the spectral radius max {| λ | : λ ∈ Spec K M } of K M . For each λ ∈ Spec K M , we call thedimensionless quantity k · λ an intrinsic eigenvalue . By Proposition 1, the operator K Q is positive in that all of its intrinsic eigenvalues are positive, and is contractive in that its intrinsic L -norm, which equals to its largest intrinsic eigenvalue µ , isless than 1. Since these properties of K Q are important in analyzing nonlinear non-uniform problem corresponding to DE( w ) in (1.2) [4, 5], one immediate question iswhether or not they are also shared by other general K M . Using Corollary 2, weprove the following negative answer. Theorem 2.
For each = λ ∈ R \ Spec K Q , there exists M ∈ wp(4 , , R ) such that λ ∈ Spec K M . Thus, one cannot expect K M to be positive nor contractive even for real M ∈ wp(4 , , R ). This result, which shows the diversity of general well-posed boundaryconditions, and might have been tricky to obtain otherwise, demonstrates the use-fulness of our presentation of the eigencondition and the characteristic equationsfor operators K M with general M ∈ wp(4 , , C ).The rest of the paper is organized as follows. In Section 2, we present basicmathematical terminologies we use, and introduce some specific matrices useful toour problems. In Section 3, the Green’s function G M is explicitly constructed,and an initial form of eigencondition for the operator K M is presented for eachwell-posed M ∈ wp(4 , , C ). Using the results in Section 3, the two intermedi-ate representations Γ − , Γ + of wp( C ) are constructed and analyzed in Section 4.1.In Section 4.2, the R -algebra π ( n ) is introduced, and π (4) is used to character-ize the real boundary conditions wp( R ). In Section 5, explicit computations onthe boundary condition Q in (1.9) are performed, resulting in explicit forms ofΓ − ([ Q ]) = G − Q and Γ + ([ Q ]) = G + Q . In Section 6.1, the representation Γ is con-structed, and is shown to have the features (Γ1), (Γ2), (Γ3), and (Γ4) above. InSection 6.2, the matrices X λ ( x ) and y λ ( x ) are defined explicitly, and Theorem 1is proved. In Section 7, some of the symmetries of X λ ( x ), Y λ ( x ) are explored,and in particular, we show that Y λ ( l ) ∈ π (4) for every 0 = λ ∈ R \ Spec K Q and l >
0. Using the results in Section 7, we prove Theorem 2 in Section 8. Finally,brief comments on future directions are given in Section 9.2.
Preliminaries
Terminologies.
We denote i = √−
1. When the ( i, j )th entry of A ∈ gl( m, n, C ) is a i,j , 1 ≤ i ≤ m , 1 ≤ j ≤ n , we write A = ( a i,j ) ≤ i ≤ m, ≤ j ≤ n . Incase m = n , we also write A = ( a i,j ) ≤ i,j ≤ n . For A ∈ gl( m, n, C ), we denote the( i, j )th entry of A by A i,j . The complex conjugate, the transpose, and the con-jugate transpose of A ∈ gl( m, n, C ) are denoted respectively by A , A T , A ∗ . For A ∈ gl( n, C ), adj A is the classical adjoint of A , so that, if A is invertible, then A − = adj A / det A . For n ∈ N , let GL ( n, C ) (respectively, GL ( n, R )) be the set of invertible matricesin gl( n, C ) (respectively, in gl( n, R )). A ∈ GL ( n, C ) is orthogonal , if A − = A T , S. W. CHOI and is unitary , if A − = A ∗ . For n ∈ N , let O ( n ) and U ( n ) be the set of orthogo-nal matrices and the set of unitary matrices in GL ( n, C ) respectively. Regardlessof their sizes, we denote by I and O , the identity matrix and the zero matrix re-spectively. In case of possible confusion with size, we denote I = I n ∈ gl( n, C ), O = O mn ∈ gl( m, n, C ), O = O n ∈ gl( n, C ). In particular, we denote the zerocolumn vector by = n = O n ∈ gl( n, , C ). The diagonal matrix with entries c , c , · · · , c n is denoted by diag ( c , c , · · · , c n ).2.2. Frequently used matrices.
Here, we introduce some special matrices whichwill be used extensively in this paper. They are useful for dealing with varioussymmetries in our problem, and readers are recommended to be acquainted withtheir properties.
Definition 2.1.
Denote ω j = e i π (2 j − for j ∈ Z , Ω = diag ( ω , ω , ω , ω ), and W = (cid:0) ω i − j (cid:1) ≤ i,j ≤ . ω , ω , ω , ω are the primitive 4th roots of −
1, and ω j +4 = ω j , j ∈ Z , hence ω j = − , ω j = ω − j , i ω j = ω j +1 , j ∈ Z , (2.1) ω = ω , ω = ω , (2.2) ω = − ω , ω = − ω , (2.3) Ω = − I , Ω = Ω − . (2.4) Definition 2.2.
Let ǫ = ǫ = 1, ǫ = ǫ = −
1, and ǫ j +4 = ǫ j , j ∈ Z . Denote E = diag ( ǫ , ǫ , ǫ , ǫ ) = diag(1 , − , − , ω j = ǫ j √ , Im ω j = ǫ j − √ , j ∈ Z . (2.5) Definition 2.3.
Denote y j ( x ) = e ω j αx , j = 1 , , ,
4, and y ( x ) = (cid:0) y ( x ) y ( x ) y ( x ) y ( x ) (cid:1) T = (cid:0) e ω αx e ω αx e ω αx e ω αx (cid:1) T . Denote the Wronskian matrix corresponding to y ( x ), y ( x ), y ( x ), y ( x ) by W ( x ) = y ( x ) y ( x ) y ( x ) y ( x ) y ′ ( x ) y ′ ( x ) y ′ ( x ) y ′ ( x ) y ′′ ( x ) y ′′ ( x ) y ′′ ( x ) y ′′ ( x ) y ′′′ ( x ) y ′′′ ( x ) y ′′′ ( x ) y ′′′ ( x ) = y ( x ) T y ′ ( x ) T y ′′ ( x ) T y ′′′ ( x ) T . Note that y ′ ( x ) = ddx e ω αx e ω αx e ω αx e ω αx = ω α · e ω αx ω α · e ω αx ω α · e ω αx ω α · e ω αx = α Ω · y ( x ) . (2.6)By Definitions 2.1 and 2.3, W ( x ) = (cid:16) ( ω j α ) i − e ω j αx (cid:17) ≤ i,j ≤ = diag (cid:0) , α, α , α (cid:1) · (cid:0) ω i − j (cid:1) ≤ i,j ≤ · diag ( e ω αx , e ω αx , e ω αx , e ω αx )= diag (cid:0) , α, α , α (cid:1) · W e Ω αx . (2.7) PECTRAL ANALYSIS FOR WELL-POSED BEAM DEFLECTION 7
By (2.1), W ∗ = (cid:16) ω j − i (cid:17) ≤ i,j ≤ = (cid:0) ω ij − (cid:1) ≤ i,j ≤ = (cid:16) ω − ji (cid:17) ≤ i,j ≤ . (2.8) Lemma 2.1. W − = W ∗ .Proof. By (2.8), W ∗ W = (cid:16) ω − ji (cid:17) ≤ i,j ≤ · (cid:0) ω i − j (cid:1) ≤ i,j ≤ = X r =1 ω − ri · ω r − j ! ≤ i,j ≤ = X r =1 (cid:18) ω j ω i (cid:19) r − ! ≤ i,j ≤ , hence ( W ∗ W ) i,j = , if i = j, − (cid:16) ωjωi (cid:17) − (cid:16) ωjωi (cid:17) , if i = j. If i = j , then 1 − ( ω j /ω i ) = 1 − ω j /ω i = 1 − ( − / ( −
1) = 0 by (2.1). Thus W ∗ W = 4 I , from which the result follows. (cid:3) The inverse W ( x ) − of W ( x ) is well-defined for every x ∈ R , and by (2.7) andLemma 2.1, W ( x ) − = e − Ω αx W − · diag (cid:0) , α, α , α (cid:1) − = 14 e − Ω αx W ∗ · diag (cid:0) , α, α , α (cid:1) − . (2.9) Definition 2.4.
Regardless of size, we denote R = ...
00 1 0 01 0 0 0 ∈ O ( n ) . In case of possible confusion, we denote R = R n ∈ O ( n ).Note that R T = R ∗ = R = R − . When multiplied to the left (respectively, tothe right) of a matrix, R reverses the order of the rows (respectively, the columns)of that matrix. Hence by (2.2), RΩ = ΩR , W R = W , R · y ( x ) = y ( x ) , W ( x ) R = W ( x ) . (2.10) Definition 2.5.
Denote L = ∈ O (4) . Note that, when multiplied to the left (respectively, to the right) of a matrix, L lifts up the rows cyclically by one row (respectively, moves the columns to the right S. W. CHOI cyclically by one column). Thus by (2.1),
LΩL − = ω ω
00 0 0 ω ω L − = diag ( ω , ω , ω , ω ) = i Ω , (2.11) W L − = (cid:0) ω i − j +1 (cid:1) ≤ i,j ≤ = (cid:16) ( i ω j ) i − (cid:17) ≤ i,j ≤ = diag (cid:0) , i , i , i (cid:1) · W . (2.12)In particular, L = ∈ O (4)is also frequently used. Note that (cid:0) L (cid:1) T = (cid:0) L (cid:1) ∗ = L = (cid:0) L (cid:1) − . By (2.3), L Ω = − ΩL , L · y ( x ) = y ( − x ) . (2.13) Definition 2.6.
Define B − , B + : C [ − l, l ] → gl(4 , , C ) by B − [ u ] = (cid:0) u ( − l ) u ′ ( − l ) u ′′ ( − l ) u ′′′ ( − l ) (cid:1) T , B + [ u ] = (cid:0) u ( l ) u ′ ( l ) u ′′ ( l ) u ′′′ ( l ) (cid:1) T . Let u ∈ C [ − l, l ]. Note that B [ u ] = (cid:18) B − [ u ] B + [ u ] (cid:19) , (2.14)where B is defined by (1.4). Let M − , M + ∈ gl(4 , C ), so that M = (cid:0) M − M + (cid:1) ∈ gl(4 , , C ). Then by (2.14), M · B [ u ] = (cid:0) M − M + (cid:1) (cid:18) B − [ u ] B + [ u ] (cid:19) = M − · B − [ u ] + M + · B + [ u ] . (2.15)By (2.1), the functions y , y , y , y in Definition 2.3 form a fundamental set ofsolutions of the linear homogeneous equation DE(0): u (4) + α u = 0 in (1.3). Thus u ∈ L [ − l, l ] is a solution of DE(0), if and only if u ( x ) = P j =1 c j · y j ( x ) = y ( x ) T c for some c = (cid:0) c c c c (cid:1) T ∈ gl(4 , , C ), in which case we have B ± [ u ] = B ± X j =1 c j · y j = X j =1 c j · B ± [ y j ] = X j =1 y j ( ± l ) y ′ j ( ± l ) y ′′ j ( ± l ) y ′′′ j ( ± l ) · c j = y ( ± l ) T y ′ ( ± l ) T y ′′ ( ± l ) T y ′′′ ( ± l ) T c = W ( ± l ) c . (2.16)3. Green’s functions for well-posed boundary conditions
Definition 3.1. M ∈ gl(4 , , C ) is called well-posed , if the boundary value problemconsisting of DE(0): u (4) + α u = 0 and BC( M ): M · B [ u ] = in (1.6), has theunique trivial solution u = 0 in L [ − l, l ]. The set of well-posed matrices in gl(4 , , C )is denoted by wp(4 , , C ). PECTRAL ANALYSIS FOR WELL-POSED BEAM DEFLECTION 9
Definition 3.2.
For M ∈ gl(4 , , C ), we denote f M − = M − W ( − l ), f M + = M + W ( l ),and f M = f M − + f M + ∈ gl(4 , C ), where M − , M + ∈ gl(4 , C ) are the 4 × M such that M = (cid:0) M − M + (cid:1) . Lemma 3.1.
Let M ∈ gl(4 , , C ) . Then M ∈ wp(4 , , C ) , if and only if det f M = 0 .Proof. Let M − , M + ∈ gl(4 , C ) be the 4 × M such that M = (cid:0) M − M + (cid:1) .Suppose u ∈ L [ − l, l ] is a solution of DE(0), so that u ( x ) = y ( x ) T c for some c ∈ gl(4 , , C ). Then by (2.15), (2.16), and Definition 3.2, the boundary conditionBC( M ) becomes = M · B [ u ] = M − · W ( − l ) c + M + · W ( l ) c = (cid:16)f M − + f M + (cid:17) c = f Mc . (3.1)By Definition 3.1, M ∈ wp(4 , , C ), if and only if c = is the only solution ingl(4 , , C ) satisfying (3.1), which is equivalent to det f M = 0. (cid:3) Since f M is invertible for every M ∈ wp(4 , , C ), the following is well-defined. Definition 3.3.
For M ∈ wp(4 , , C ), we denote G − M = f M − f M − ΩL and G + M = f M − f M + ΩL .By Definition 3.2, G − M + G + M = f M − f M − ΩL + f M − f M + ΩL = f M − f MΩL ,hence we have G − M + G + M = ΩL , M ∈ wp(4 , , C ) . (3.2) Definition 3.4.
Let M ∈ wp(4 , , C ). Define the function G M : [ − l, l ] × [ − l, l ] → C ,called the Green’s function corresponding to M , by G M ( x, ξ ) = α k · (cid:26) y ( x ) T · G + M · y ( ξ ) , if x ≤ ξ, − y ( x ) T · G − M · y ( ξ ) , if ξ ≤ x. Define the operator K M : L [ − l, l ] → L [ − l, l ] by K M [ w ]( x ) = Z l − l G M ( x, ξ ) w ( ξ ) dξ, w ∈ L [ − l, l ] , x ∈ [ − l, l ] . Note that G M is bounded on [ − l, l ] × [ − l, l ]. It is well-known [11] that anintegral operator of the form w R l − l g ( x, ξ ) w ( ξ ) dx with bounded kernel g ( x, ξ ),is a compact linear operator on L [ − l, l ]. Thus K M is a well-defined compact linearoperator on L [ − l, l ]. Note from Definitions 3.3 and 3.4 that the function G M is defined constructively in terms of given M ∈ wp(4 , , C ). Lemma 3.2 below,whose proof is in Appendix A, shows that G M is the usual Green’s function for theboundary value problem consisting of DE(0) and BC( M ). Lemma 3.2.
Let M ∈ wp(4 , , C ) and w ∈ L [ − l, l ] . Then K M [ w ] is the uniquesolution of the boundary value problem consisting of DE( w ) and BC( M ) . By Lemma 3.2, we have K M [ u ] (4) + α · K M [ u ] = α k · u, M ∈ wp(4 , , C ) , u ∈ L [ − l, l ] , (3.3) M · B [ K M [ u ]] = , M ∈ wp(4 , , C ) , u ∈ L [ − l, l ] . (3.4)Note that (3.3) in particular implies that the linear operator K M is one-to-one,or injective , for every M ∈ wp(4 , , C ). Definition 3.5.
For 0 = λ ∈ C , we denote by EDE( λ ), the homogeneous equationEDE( λ ) : u (4) + (cid:18) − λk (cid:19) α u = 0 . Note that 1 − / ( λk ) = 1 for any λ ∈ C . In fact, the homogeneous equationDE(0): u (4) + α u = 0 can be regarded as the limiting case EDE( ∞ ). In terms ofgiven M ∈ wp(4 , , C ), K M has the following eigencondition. Lemma 3.3.
Let M ∈ wp(4 , , C ) , = u ∈ L [ − l, l ] , and λ ∈ C . Then K M [ u ] = λ · u , if and only if λ = 0 and u satisfies EDE( λ ) and BC( M ) .Proof. Let u = 0 ∈ L [ − l, l ], λ ∈ C . Suppose K M [ u ] = λ · u . If λ = 0, then K M [ u ] = 0 · u = 0, hence u = 0 by (3.3), contradicting the assumption u = 0. Thus λ = 0. By (3.3), we have λ · u (4) = ( λ · u ) (4) = K M [ u ] (4) = − α · K M [ u ] + α k · u = − α · ( λ · u ) + α k · u = − (cid:18) λ − k (cid:19) α · u, which shows that u satisfies EDE( λ ). By (3.4), we have M · B [ u ] = (1 /λ ) · M ·B [ λ · u ] = (1 /λ ) · M · B [ K M [ u ]] = , hence u satisfies BC( M ).Conversely, suppose λ = 0 and u satisfies EDE( λ ) and BC( M ). Let ˆ u = K M [ u ] − λ · u . Then by (3.3), we haveˆ u (4) + α ˆ u = ( K M [ u ] − λ · u ) (4) + α ( K M [ u ] − λ · u )= (cid:16) K M [ u ] (4) + α · K M [ u ] (cid:17) − λ (cid:16) u (4) + α u (cid:17) = α k · u − λ (cid:16) u (4) + α u (cid:17) = − λ (cid:26) u (4) + (cid:18) − λk (cid:19) α u (cid:27) , hence ˆ u (4) + α ˆ u = 0, since u satisfies EDE( λ ). By (3.4), we have M · B [ˆ u ] = M ·B [ K M [ u ] − λ · u ] = M ·B [ K M [ u ]] − λ M ·B [ u ] = − λ M ·B [ u ], hence M ·B [ˆ u ] = ,since u satisfies BC( M ). It follows that ˆ u = K M [ u ] − λ · u is the unique solutionof the boundary value problem consisting of DE(0) and BC( M ), which is 0 byDefinition 3.1. Thus we have K M [ u ] = λ · u , and the proof is complete. (cid:3) Representation of well-posed boundary conditions
The representations Γ − and Γ + .Definition 4.1. M , N ∈ wp(4 , , C ) are called equivalent , and denote M ≈ N ,if K M = K N . The quotient set wp(4 , , C )/ ≈ of wp(4 , , C ) with respect to theequivalence relation ≈ , is denoted by wp( C ). For M ∈ wp(4 , , C ), we denote by[ M ] ∈ wp( C ) the equivalence class with respect to ≈ containing M .Note from Definitions 3.4 and 4.1 that M ≈ N , if and only if G M = G N . Lemma 4.1.
For M , N ∈ wp(4 , , C ) , the following (a) , (b) , (c) , (d) are equivalent. (a) M ≈ N . (b) G + M = G + N . (c) G − M = G − N . (d) N = PM for some P ∈ GL (4 , C ) .Proof. The equivalence of (b) and (c) follows immediately, since G − M + G + M = ΩL = G − N + G + N by (3.2). Since the entries y , y , y , y of y in Definition 2.3are linearly independent, it follows from Definition 3.4 that G M = G N , if and only PECTRAL ANALYSIS FOR WELL-POSED BEAM DEFLECTION 11 if G − M = G − N and G + M = G + N . Thus (a), (b), (c) are equivalent, and hence it issufficient to show the equivalence of (b) and (d).Let M − , M + ∈ gl(4 , C ) and N − , N + ∈ gl(4 , C ) be the 4 × M and N respectively such that M = (cid:0) M − M + (cid:1) , N = (cid:0) N − N + (cid:1) . Suppose (d).Then (cid:0) N − N + (cid:1) = N = PM = P (cid:0) M − M + (cid:1) = (cid:0) PM − PM + (cid:1) for some P ∈ GL (4 , C ). So we have N − = PM − and N + = PM + , and hence by Definition 3.2, e N ± = N ± W ( ± l ) = PM ± W ( ± l ) = P f M ± and e N = e N − + e N + = P f M − + P f M + = P f M . Thus by Definition 3.3, G + N = e N − e N + ΩL = (cid:16) P f M (cid:17) − (cid:16) P f M + (cid:17) ΩL = f M − P − P f M + ΩL = f M − f M + ΩL = G + M , hence we have (b).Conversely, suppose (b), so that G + M = G + N . Since (b) and (c) are equivalent,we also have G − M = G − N . By Definition 3.3, we have f M − f M − = e N − e N − and f M − f M + = e N − e N + , since ΩL is invertible. So we have e N ± = e N f M − · f M ± , hence N ± W ( ± l ) = e N f M − · M ± W ( ± l ) by Definition 3.2. Since W ( ± l ) are invertible, wehave N ± = e N f M − · M ± , hence N = (cid:0) N − N + (cid:1) = (cid:16) e N f M − · M − e N f M − · M + (cid:17) = e N f M − · M . Thus we have (d), since e N f M − ∈ GL (4 , C ) by Lemma 3.1, and the proof is com-plete. (cid:3) Definition 4.2.
Define Γ − , Γ + : wp( C ) → gl(4 , C ) by Γ − ([ M ]) = G − M andΓ + ([ M ]) = G + M for M ∈ wp(4 , , C ).By Lemma 4.1, Γ − , Γ + are well-defined and one-to-one. Lemma 4.2 below showsthat Γ − , Γ + are also onto, and hence are one-to-one correspondences from wp( C )to gl(4 , C ). Lemma 4.2.
Suppose G − , G + ∈ gl(4 , C ) satisfy G − + G + = ΩL . Then thereexists M ∈ wp(4 , , C ) such that G − M = G − , G + M = G + . In particular, M can be taken by M = (cid:0) M − M + (cid:1) , where M − = G − (cid:0) ΩL (cid:1) − W ( − l ) − , M + = G + (cid:0) ΩL (cid:1) − W ( l ) − .Proof. Let M − = G − (cid:0) ΩL (cid:1) − W ( − l ) − , M + = G + (cid:0) ΩL (cid:1) − W ( l ) − , and let M = (cid:0) M − M + (cid:1) ∈ gl(4 , , C ). Then by Definition 3.2, we have f M ± = M ± W ( ± l ) = G ± (cid:0) ΩL (cid:1) − W ( ± l ) − · W ( ± l ) = G ± (cid:0) ΩL (cid:1) − , (4.1)hence f M = f M − + f M + = G − (cid:0) ΩL (cid:1) − + G + (cid:0) ΩL (cid:1) − = (cid:0) G − + G + (cid:1) (cid:0) ΩL (cid:1) − = ΩL · (cid:0) ΩL (cid:1) − = I . (4.2)Thus M ∈ wp(4 , , C ) by Lemma 3.1, since f M = I is invertible. By Definition 3.3and (4.1), (4.2), we have G ± M = f M − f M ± ΩL = I − · G ± (cid:0) ΩL (cid:1) − · ΩL = G ± ,hence the proof is complete. (cid:3) Note from Definitions 3.2, 3.3, and 4.2 that the maps Γ − and Γ + are constructive,in that Γ − ([ M ]) = G − M , Γ + ([ M ]) = G + M can be computed explicitly in terms ofgiven M ∈ wp(4 , , C ). In fact, the inverses (Γ − ) − , (Γ + ) − are also constructive.Lemma 4.2 implies that (cid:0) Γ − (cid:1) − ( G )= h(cid:16) G (cid:0) ΩL (cid:1) − W ( − l ) − (cid:0) ΩL − G (cid:1) (cid:0) ΩL (cid:1) − W ( l ) − (cid:17)i , (4.3) (cid:0) Γ + (cid:1) − ( G )= h(cid:16) (cid:0) ΩL − G (cid:1) (cid:0) ΩL (cid:1) − W ( − l ) − G (cid:0) ΩL (cid:1) − W ( l ) − (cid:17)i (4.4)for every G ∈ gl(4 , C ).4.2. Real boundary conditions and the algebra π (4) . Of particular interestamong boundary conditions in wp(4 , , C ) are those with real entries. We charac-terize this important class of real boundary conditions in terms of the maps Γ − andΓ + in Definition 4.2. Definition 4.3.
Denote wp(4 , , R ) = wp(4 , , C ) ∩ gl(4 , , R ) andwp( R ) = { [ M ] ∈ wp( C ) | M ∈ wp(4 , , R ) } ⊂ wp( C ) . Let M ∈ wp(4 , , R ). By Lemma 3.2, it is clear that K M [ w ]( x ) is real-valuedfor every real-valued w ∈ L [ − l, l ]. Thus it follows that G M ( x, ξ ) is real-valued forevery M ∈ wp(4 , , R ). Lemma 4.3. RG − M R = G − M and RG + M R = G + M for every M ∈ wp(4 , , R ) .Proof. Let M ∈ wp(4 , , R ). Since G M ( x, ξ ) is real-valued, we have G M ( x, ξ ) = G M ( x, ξ ) for ( x, ξ ) ∈ [ − l, l ] × [ − l, l ]. By Definition 3.4 and (2.10), we have G M ( x, ξ ) = α k · ( y ( x ) T · G + M · y ( ξ ) , if x ≤ ξ, − y ( x ) T · G − M · y ( ξ ) , if ξ ≤ x = α k · ( { R · y ( x ) } T · G + M · { R · y ( ξ ) } , if x ≤ ξ, − { R · y ( x ) } T · G − M · { R · y ( ξ ) } , if ξ ≤ x = α k · ( y ( x ) T · RG + M R · y ( ξ ) , if x ≤ ξ, − y ( x ) T · RG − M R · y ( ξ ) , if ξ ≤ x, hence0 = 4 kα n G M ( x, ξ ) − G M ( x, ξ ) o = y ( x ) T · (cid:16) RG + M R − G + M (cid:17) · y ( ξ ) , if x ≤ ξ, − y ( x ) T · (cid:16) RG − M R − G − M (cid:17) · y ( ξ ) , if ξ ≤ x, which is equivalent to RG + M R − G + M = RG − M R − G − M = O , since the entries y , y , y , y of y in Definition 2.3 are linearly independent. (cid:3) Lemma 4.3 leads us to the following definition.
Definition 4.4.
For n ∈ N , we denote π ( n ) = (cid:8) A ∈ gl( n, C ) | RAR = A (cid:9) . PECTRAL ANALYSIS FOR WELL-POSED BEAM DEFLECTION 13
Note that π ( n ) is the set of matrices in gl( n, C ) invariant under the transfor-mation A RAR , which is the complex conjugation with the 180 ◦ rotation ofmatrix entries. Lemma 4.4 below, whose proof is immediate from Definition 4.4,shows in particular that π ( n ) forms an R -algebra. Lemma 4.4.
For n ∈ N , we have the following. (a) If A , B ∈ π ( n ) , then a A + b B ∈ π ( n ) for every a, b ∈ R . (b) If A , B ∈ π ( n ) , then AB ∈ π ( n ) . (c) If A ∈ π ( n ) is invertible, then A − ∈ π ( n )(d) If A ∈ π ( n ) , then A T ∈ π ( n )(e) O n , I n , R n ∈ π ( n ) and Ω , L , E ∈ π (4) . In particular, we have π (4) = a a a a a a a a a a a a a a a a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ij ∈ C , , , j = 1 , , , , which shows that the dimension of π (4) as an R -algebra is 16. In fact, it will beshown in Section 8 that π (2 n ) is isomorphic to gl(2 n, R ) for n ∈ N .Lemma 4.3 shows that the images of wp( R ) under Γ − and Γ + in Definition 4.2 aresubsets of π (4). Lemma 4.5 below shows that, in fact, Γ − (wp( R )) = Γ + (wp( R )) = π (4), by constructing representatives in wp(4 , , R ) of the inverses (Γ − ) − ( G ),(Γ + ) − ( G ) of G ∈ π (4). Denote U = 1 √ i − i i − i = (cid:18) I R i R − i I (cid:19) ∈ U (4) . (4.5)Note that U = UR . (4.6) Lemma 4.5.
Suppose G − , G + ∈ π (4) satisfy G − + G + = ΩL . Then thereexists M ∈ wp(4 , , R ) such that G − M = G − , G + M = G + . In particular, M canbe taken by M = (cid:0) M − M + (cid:1) , where M − = UG − (cid:0) ΩL (cid:1) − W ( − l ) − , M + = UG + (cid:0) ΩL (cid:1) − W ( l ) − .Proof. Let M − = UG − (cid:0) ΩL (cid:1) − W ( − l ) − , M + = UG + (cid:0) ΩL (cid:1) − W ( l ) − , and M = (cid:0) M − M + (cid:1) . Letˆ M = (cid:16) G − (cid:0) ΩL (cid:1) − W ( − l ) − G + (cid:0) ΩL (cid:1) − W ( l ) − (cid:17) . Then M = U ˆ M , hence M ≈ ˆ M by Lemma 4.1, since U is invertible. So byDefinition 4.2 and Lemma 4.2, we have G ± M = Γ ± ([ M ]) = Γ ± (cid:16)h ˆ M i(cid:17) = G ± ˆ M = G ± , since G − , G + ∈ gl(4 , C ). Thus it is sufficient to show that M − , M + ∈ gl(4 , R ).By Definition 4.4, G ± ( ΩL ) − = R · G ± (cid:0) ΩL (cid:1) − · R , (4.7) since G − (cid:0) ΩL (cid:1) − , G + (cid:0) ΩL (cid:1) − ∈ π (4) by Lemma 4.4. Since W ( x ) = W ( x ) R by(2.10), we have W ( ± l ) − = n W ( ± l ) o − = { W ( ± l ) R } − = R · W ( ± l ) − . (4.8)Thus by (4.6), (4.7), (4.8), we have M ± = U · G ± ( ΩL ) − · W ( ± l ) − = UR · n R · G ± (cid:0) ΩL (cid:1) − · R o · (cid:8) R · W ( ± l ) − (cid:9) = UG ± (cid:0) ΩL (cid:1) − W ( ± l ) − = M ± . This shows M − , M + ∈ gl(4 , R ), and the proof is complete. (cid:3) The boundary condition Q and the operator K Q = K l,α,k Let Q − = α −√ α √ α − α , Q + = α √ α −√ α − α , so that (cid:0) Q − Q + (cid:1) = Q in (1.9). In this section, we apply Definitions 3.2 and 3.3to Q to obtain explicit forms of G − Q = Γ − ([ Q ]) and G + Q = Γ + ([ Q ]). In additionto being needed to construct the map Γ in Section 6.1, this will also serve as aconcrete example of computing Γ − and Γ + . We also show in Lemma 5.1 belowthat K Q = K l,α,k , where K l,α,k is the integral operator defined in (1.7).By Definition 2.1, we have Q − · diag (cid:0) , α, α , α (cid:1) · W = α −√ √ − (cid:0) ω i − j (cid:1) ≤ i,j ≤ = α a − a − a − a − b − b − b − b − , (5.1) Q + · diag (cid:0) , α, α , α (cid:1) · W = α √ −√ − (cid:0) ω i − j (cid:1) ≤ i,j ≤ = α a +1 a +2 a +3 a +4 b +1 b +2 b +3 b +4 , (5.2)where we put a ± j = ω j ± √ ω j + ω j , b ± j = ∓√ − ω j + ω j , j = 1 , , ,
4. Note that ω j = (cid:8) e i π (2 j − (cid:9) = i j − = ( − j +1 i by Definition 2.1, and ω j + ω j = ω j − ω j =2 i Im ω j = √ ǫ j − i , − ω j + ω j = − ω j − ω j = − ω j = −√ ǫ j by (2.1), (2.5).Hence, for j = 1 , , ,
4, we have a ± j = ω j ± √ ω j + ω j = √ ǫ j − i ± √ · ( − j +1 i = 2 √ · ǫ j − ± ( − j +1 i , (5.3) b ± j = ∓√ − ω j + ω j = −√ ǫ j ∓ √ √ · − ǫ j ∓ . (5.4) PECTRAL ANALYSIS FOR WELL-POSED BEAM DEFLECTION 15
By (5.1), (5.2), (5.3), (5.4), we have Q − · diag (cid:0) , α, α , α (cid:1) · W = 2 √ α − i i − − i − i − − = 2 √ α i − i
00 1 1 00 0 0 00 0 0 0 , Q + · diag (cid:0) , α, α , α (cid:1) · W = 2 √ α i − i − i − − i − −
12 1 −
12 1 − − − = 2 √ α i − i − − , hence by Definition 3.2 and (2.7), e Q − = Q − W ( − l ) = Q − · diag (cid:0) , α, α , α (cid:1) · W e − Ω αl = 2 √ α i − i
00 1 1 00 0 0 00 0 0 0 e − Ω αl = 2 √ α i e − ω αl − i e − ω αl e − ω αl e − ω αl
00 0 0 00 0 0 0 , (5.5) e Q + = Q + W ( l ) = Q + · diag (cid:0) , α, α , α (cid:1) · W e Ω αl = 2 √ α i − i − − e Ω αl = 2 √ α i e ω αl − i e ω αl − e ω αl − e ω αl , (5.6) e Q = e Q − + e Q + = 2 √ α i e − ω αl − i e − ω αl e − ω αl e − ω αl i e ω αl − i e ω αl − e ω αl − e ω αl . (5.7)Let ˆ U = 1 √ i − i
00 1 1 0 i − i − − ∈ U (4) . Then by (5.7) and Definition 2.2, e Q = 4 α · √ i − i
00 1 1 0 i − i − − · diag (cid:0) e ω αl , e − ω αl , e − ω αl , e ω αl (cid:1) = 4 α ˆ U e E Ω αl , hence we have e Q − = 14 α e −E Ω αl ˆ U − = 14 α e −E Ω αl ˆ U ∗ , (5.8)since ˆ U is unitary. Note that this in particular shows that Q is well-posed byLemma 3.1. Let ˆ U − = 1 √ i − i
00 1 1 00 0 0 00 0 0 0 , ˆ U + = 1 √ i − i − − . Then by (5.5), (5.6), e Q − = 4 α · √ i − i
00 1 1 00 0 0 00 0 0 0 · diag (cid:0) e ω αl , e − ω αl , e − ω αl , e ω αl (cid:1) = 4 α ˆ U − e E Ω αl , (5.9) e Q + = 4 α · √ i − i − − · diag (cid:0) e ω αl , e − ω αl , e − ω αl , e ω αl (cid:1) = 4 α ˆ U + e E Ω αl . (5.10)By (5.8), (5.9), (5.10), we have e Q − e Q ± = 14 α e −E Ω αl ˆ U ∗ · α ˆ U ± e E Ω αl = e −E Ω αl ˆ U ∗ ˆ U ± e E Ω αl . (5.11)Note thatˆ U ∗ ˆ U − = 1 √ − i − − i i i − · √ i − i
00 1 1 00 0 0 00 0 0 0 = diag(0 , , , , ˆ U ∗ ˆ U + = 1 √ − i − − i i i − · √ i − i − − = diag(1 , , , , hence by (5.11), e Q − e Q − = e −E Ω αl · diag(0 , , , · e E Ω αl = diag(0 , , , , (5.12) e Q − e Q + = e −E Ω αl · diag(1 , , , · e E Ω αl = diag(1 , , , . (5.13)Thus by Definition 3.3, we finally have G − Q = e Q − e Q − ΩL = diag(0 , , , · ω
00 0 0 ω ω ω = ω ω , (5.14) G + Q = e Q − e Q + ΩL = diag(1 , , , · ω
00 0 0 ω ω ω = ω
00 0 0 00 0 0 00 ω . (5.15) PECTRAL ANALYSIS FOR WELL-POSED BEAM DEFLECTION 17
Note that Q ∈ wp(4 , , R ) and G − Q , G + Q ∈ π (4), satisfying Lemma 4.3. Lemma 5.1. K Q = K l,α,k .Proof. By (1.7) and Definition 3.4, it is sufficient to show that G Q = G , where G is defined by (1.8). By Definition 2.3, and (2.3), (5.14), (5.15), we have y ( x ) T · G − Q · y ( ξ ) = ω e ω αx e ω αξ + ω e ω αx e ω αξ = − ω e − ω α ( x − ξ ) − ω e − ω α ( x − ξ ) , y ( x ) T · G + Q · y ( ξ ) = ω e ω αx e ω αξ + ω e ω αx e ω αξ = ω e − ω α ( ξ − x ) + ω e − ω α ( ξ − x ) , hence by Definitions 2.1, 3.4, and (2.2), G Q ( x, ξ ) = α k · (cid:26) ω e − ω α ( ξ − x ) + ω e − ω α ( ξ − x ) , x ≤ ξω e − ω α ( x − ξ ) + ω e − ω α ( x − ξ ) , ξ ≤ x = α k (cid:16) ω e − ω α | x − ξ | + ω e − ω α | x − ξ | (cid:17) = α k · (cid:16) ω e − ω α | x − ξ | (cid:17) = α k Re (cid:18) e i π e − (cid:16) √ + i √ (cid:17) α | x − ξ | (cid:19) = α k Re e − α √ | x − ξ | + i (cid:16) π − α √ | x − ξ | (cid:17) = α k e − α √ | x − ξ | cos (cid:18) π − α √ | x − ξ | (cid:19) = α k e − α √ | x − ξ | sin (cid:18) α √ | x − ξ | + π (cid:19) , which is identical to G ( x, ξ ) in (1.8). Thus we have the proof. (cid:3) The representation Γ and proof of Theorem 1 The representation Γ .Definition 6.1. For M ∈ wp(4 , , C ), denote G M = (cid:16) G + M − G + Q (cid:17) (cid:0) ΩL (cid:1) − E .Define Γ : wp( C ) → gl(4 , C ) by Γ ([ M ]) = G M for M ∈ wp(4 , , C ).Readers should be cautious to distinguish the 4 × matrix G M in Definition 6.1from the Green’s function G M in Definition 3.4. Note that the map Γ is well-defined,sinceΓ ([ M ]) = (cid:16) G + M − G + Q (cid:17) (cid:0) ΩL (cid:1) − E = n Γ + ([ M ]) − G + Q o (cid:0) ΩL (cid:1) − E (6.1)by Definition 4.2. By (3.2), G M = n(cid:0) ΩL − G − M (cid:1) − (cid:16) ΩL − G − Q (cid:17)o (cid:0) ΩL (cid:1) − E = − (cid:16) G − M − G − Q (cid:17) (cid:0) ΩL (cid:1) − E , which could have been used for an alternative definition of G M . Note also that G Q = O . (6.2) Lemma 6.1.
Γ : wp( C ) → gl(4 , C ) is a one-to-one correspondence, and Γ (wp( R )) = π (4) .Proof. Since Γ + is a one-to-one correspondence and (cid:0) ΩL (cid:1) − E is invertible, itfollows from (6.1) that Γ also is a one-to-one correspondence. Since Γ + (wp(4 , R ))= π (4) and G + Q , (cid:0) ΩL (cid:1) − E ∈ π (4), we have Γ (wp(4 , R )) = π (4) by (6.1) andLemma 4.4. (cid:3) Thus we finally have our representation M G M , from the set of well-posedboundary conditions wp(4 , , C ) to the algebra gl(4 , C ), and from the set of well-posed real boundary conditions wp(4 , , R ) to the R -algebra π (4). Note that thisrepresentation is constructive in both directions, in that G M ∈ gl(4 , C ) is expressedexplicitly in terms of given M ∈ wp(4 , , C ), and conversely, M ∈ wp(4 , , C ) suchthat G M = G can be chosen explicitly in terms of given G ∈ gl(4 , C ). Especially,given M = (cid:0) M − M + (cid:1) ∈ wp(4 , , C ), M − , M + ∈ gl(4 , C ), we have G M = (cid:16) G + M − G + Q (cid:17) (cid:0) ΩL (cid:1) − E = (cid:16)f M − f M + ΩL − e Q − e Q + ΩL (cid:17) (cid:0) ΩL (cid:1) − E = (cid:16)f M − f M + − e Q − e Q + (cid:17) E = (cid:8) M − W ( − l ) + M + W ( l ) (cid:9) − M + W ( l ) E − diag(1 , , ,
1) (6.3)by combining Definitions 3.2, 3.3, 6.1, and (5.13). Conversely, suppose G ∈ gl(4 , C )is given. By (6.1), we have Γ − ( G ) = (Γ + ) − (cid:16) G E ΩL + G + Q (cid:17) , hence by (4.4),(5.13), and Definition 3.3, M = (cid:0) M − M + (cid:1) ∈ wp(4 , , C ) is a representative ofΓ − ( G ) ∈ wp( C ), where M − = n ΩL − (cid:16) G E ΩL + G + Q (cid:17)o (cid:0) ΩL (cid:1) − W ( − l ) − = (cid:16) ΩL − G E ΩL − e Q − e Q + ΩL (cid:17) (cid:0) ΩL (cid:1) − W ( − l ) − = { diag(0 , , , − G E} W ( − l ) − , M + = (cid:16) G E ΩL + G + Q (cid:17) (cid:0) ΩL (cid:1) − W ( l ) − = (cid:16) G E ΩL + e Q − e Q + ΩL (cid:17) (cid:0) ΩL (cid:1) − W ( l ) − = { diag(1 , , ,
1) + G E} W ( l ) − . Thus, given G ∈ gl(4 , C ), we haveΓ − ( G ) = (cid:2)(cid:0) { diag(0 , , , − G E} W ( − l ) − (cid:12)(cid:12)(cid:12)(cid:12) { diag(1 , , ,
1) + G E} W ( l ) − (cid:1)(cid:3) (6.4)in wp( C ). When G ∈ π (4), a representative of Γ − ( G ) in wp(4 , , R ) is U · (cid:0) { diag(0 , , , − G E} W ( − l ) − { diag(1 , , ,
1) + G E} W ( l ) − (cid:1) by Lemma 4.5, where U is defined by (4.5).The boundary condition BC( M ) in Lemma 3.3 is transformed into the followingequivalent condition which is expressed now in terms of G M . Lemma 6.2.
For M ∈ wp(4 , , C ) , the boundary condition BC( M ) : M · B [ u ] = is equivalent to = G M E (cid:8) W ( l ) − B + [ u ] − W ( − l ) − B − [ u ] (cid:9) + diag(0 , , , · W ( − l ) − B − [ u ] + diag(1 , , , · W ( l ) − B + [ u ] . Proof.
By (6.4), h ˆ M i = Γ − ( G M ) ∈ wp( C ), whereˆ M = (cid:0) { diag(0 , , , − G M E} W ( − l ) − (cid:12)(cid:12)(cid:12)(cid:12) { diag(1 , , ,
1) + G M E} W ( l ) − (cid:1) . (6.5) PECTRAL ANALYSIS FOR WELL-POSED BEAM DEFLECTION 19
So by Definition 6.1, [ M ] = Γ − ( G M ) = h ˆ M i , hence by Lemma 4.1, there exists P ∈ GL (4 , C ) such that M = P ˆ M . Thus the condition BC( M ): M · B [ u ] = isequivalent to ˆ M · B [ u ] = , since P is invertible. By (2.14), (6.5),ˆ M · B [ u ] = ˆ M · (cid:18) B − [ u ] B + [ u ] (cid:19) = { diag(0 , , , − G M E} W ( − l ) − · B − [ u ]+ { diag(1 , , ,
1) + G M E} W ( l ) − · B + [ u ]= G M E (cid:8) W ( l ) − B + [ u ] − W ( − l ) − B − [ u ] (cid:9) + diag(0 , , , · W ( − l ) − B − [ u ] + diag(1 , , , · W ( l ) − B + [ u ] , hence the lemma follows. (cid:3) Proof of Theorem 1.
Note that the solution space of the linear homogeneousequation EDE( λ ) in Definition 3.5 depends on the value λ = 0. In particular,depending on whether 1 − / ( λk ) = 0 or not, or equivalently, whether λ = 1 /k ornot, EDE( λ ) becomes as follows.(I) When λ = 1 /k : EDE( λ ) becomes u (4) = 0.(II) When λ = 1 /k : EDE( λ ) becomes u (4) + ( κα ) u = 0, where κ = χ ( λ ) = 0is defined in Definition 6.2 below. Definition 6.2.
For λ ∈ C \{ , /k } , define χ ( λ ) to be the unique complex numbersatisfying χ ( λ ) = 1 − / ( λk ) and 0 ≤ Arg χ ( λ ) < π/ χ ( λ ) = 0 and χ ( λ ) = 1 for λ ∈ C \ { , /k } . In fact, χ is a one-to-onecorrespondence from C \{ , /k } to (cid:8) κ ∈ C | ≤ Arg κ < π (cid:9) \{ , } , and its inverseis given by χ − ( κ ) = 1 / (cid:8) k (cid:0) − κ (cid:1)(cid:9) . Definition 6.3.
For 0 = λ ∈ C , denote y λ,j ( x ) = ( x j − ( j − , if λ = k ,e ω j καx , if λ = k , j = 1 , , , , where κ = χ ( λ ). Denote y λ ( x ) = (cid:0) y λ, ( x ) y λ, ( x ) y λ, ( x ) y λ, ( x ) (cid:1) T and W λ ( x ) = (cid:16) y ( i − λ,j ( x ) (cid:17) ≤ i,j ≤ = y λ ( x ) T y ′ λ ( x ) T y ′′ λ ( x ) T y ′′′ λ ( x ) T . Note that the functions y λ, , y λ, , y λ, , y λ, form a fundamental set of solutionsof EDE( λ ) for every 0 = λ ∈ C , and W λ ( x ) = (cid:16) H ( j − i ) x j − i ( j − i )! (cid:17) ≤ i,j ≤ = x x x x x x , λ = k , (cid:16) ( ω j κα ) i − e ω j καx (cid:17) ≤ i,j ≤ , λ = k , (6.6)where κ = χ ( λ ) and H ( t ) = (cid:26) , if t ≥ , , if t < . Let M ∈ wp(4 , , C ). By Lemma 3.3, λ ∈ C is an eigenvalue of K M and u = 0 isa corresponding eigenfunction, if and only if λ = 0 and u is a nontrivial solution ofEDE( λ ) satisfying the boundary condition BC( M ). Note that u is a nontrivial so-lution of EDE( λ ), if and only if there exists = c = (cid:0) c c c c (cid:1) T ∈ gl(4 , , C )such that u = P j =1 c j y λ,j = y Tλ c . If u = P j =1 c j y λ,j , then by Definitions 2.6 and6.3, B ± [ u ] = X j =1 c j · B ± [ y λ,j ] = X j =1 y λ,j ( ± l ) y ′ λ,j ( ± l ) y ′′ λ,j ( ± l ) y ′′′ λ,j ( ± l ) · c j = y λ ( ± l ) T y ′ λ (( ± l ) T y ′′ λ (( ± l ) T y ′′′ λ (( ± l ) T c = W λ ( ± l ) c . It follows from Lemma 6.2 that the condition BC( M ) is equivalent to = G M E (cid:8) W ( l ) − W λ ( l ) c − W ( − l ) − W λ ( − l ) c (cid:9) + diag(0 , , , · W ( − l ) − W λ ( − l ) c + diag(1 , , , · W ( l ) − W λ ( l ) c = (cid:2) G M E (cid:8) W ( l ) − W λ ( l ) − W ( − l ) − W λ ( − l ) (cid:9) + diag(0 , , , · W ( − l ) − W λ ( − l ) + diag(1 , , , · W ( l ) − W λ ( l ) (cid:3) c . (6.7)Thus Lemma 3.3 can be rephrased as the following. Lemma 6.3.
Let M ∈ wp(4 , , C ) , = u ∈ L [ − l, l ] , and λ ∈ C . Then K M [ u ] = λ · u , if and only if λ = 0 and u = y Tλ c for some = c ∈ gl(4 , , C ) which satisfies (6.7) . The following matrix X λ ( x ) will have a key role in our discussions. Definition 6.4.
For 0 = λ ∈ C and x ∈ R , we denote X λ ( x ) = diag(0 , , , · W ( − x ) − W λ ( − x ) + diag(1 , , , · W ( x ) − W λ ( x ) . Note from Definitions 2.3 and 6.3 that, for each 0 = λ ∈ C and x ∈ R , X λ ( x ) isa concrete 4 × not depend on M . By Definition 6.4, X λ ( x ) − X λ ( − x )= (cid:8) diag(0 , , , · W ( − x ) − W λ ( − x ) + diag(1 , , , · W ( x ) − W λ ( x ) (cid:9) − (cid:8) diag(0 , , , · W ( x ) − W λ ( x ) + diag(1 , , , · W ( − x ) − W λ ( − x ) (cid:9) = E (cid:8) W ( x ) − W λ ( x ) − W ( − x ) − W λ ( − x ) (cid:9) , = λ ∈ C , x ∈ R . (6.8)Now we are ready to prove Theorem 1. Proof of Theorem 1.
By Definition 6.4 and (6.8), the condition (6.7) is equivalentto [ G M { X λ ( l ) − X λ ( − l ) } + X λ ( l )] c = . Thus the first assertion follows fromLemma 6.3. The second assertion follows from the first one, since G Q = O by(6.2). (cid:3) Symmetries of X λ and Y λ As a consequence of Theorem 1 and Corollary 1, the matrix X λ ( l ) is invertiblefor every 0 = λ ∈ C which is not in Spec K Q . In fact, this is true for arbitrary l > Definition 7.1.
Denote Y λ ( x ) = X λ ( − x ) X λ ( x ) − − I for 0 = λ ∈ C and x > X λ ( x ) = 0. PECTRAL ANALYSIS FOR WELL-POSED BEAM DEFLECTION 21
In view of Theorem 1 and Corollaries 1, 2, it is now apparent that analysis onthe 4 × X λ ( x ) and Y λ ( x ) are important. It turns out that they havevarious symmetries, and some of them are explored in this section. In particular,we will obtain the following result, which is crucial in proving Theorem 2. Lemma 7.1. Y λ ( x ) ∈ π (4) for every = λ ∈ R and x > such that det X λ ( x ) = 0 . The proof of Lemma 7.1 will be given at the end of Section 7.2. To facilitate ouranalysis, we introduce the following change of variables z = αx, (7.1)which will be used extensively for the rest of the paper. By (2.7), (2.9), and (7.1), W ( x ) = diag (cid:0) , α, α , α (cid:1) · W e Ω z , (7.2) W ( x ) − = 14 e − Ω z W ∗ · diag (cid:0) , α, α , α (cid:1) − . (7.3)The following form of X λ ( x ) will be useful. Lemma 7.2.
For = λ ∈ C and x ∈ R , X λ ( x ) = 14 e −E Ω z n diag(0 , , , · W ∗ · diag (cid:0) , α, α , α (cid:1) − W λ ( − x )+ diag(1 , , , · W ∗ · diag (cid:0) , α, α , α (cid:1) − W λ ( x ) o . Proof.
By Definition 6.4 and (7.1), (7.3), we have X λ ( x ) = 14 n diag(0 , , , · e Ω z W ∗ · diag (cid:0) , α, α , α (cid:1) − W λ ( − x )+ diag(1 , , , · e − Ω z W ∗ · diag (cid:0) , α, α , α (cid:1) − W λ ( x ) o . (7.4)Note thatdiag(0 , , , · e Ω z = diag (0 , e ω z , e ω z , (cid:0) e − ω z , e ω z , e ω z , e − ω z (cid:1) · diag(0 , , , e −E Ω z · diag(0 , , , , (7.5)diag(1 , , , · e − Ω z = diag (cid:0) e − ω z , , , e − ω z (cid:1) = diag (cid:0) e − ω z , e ω z , e ω z , e − ω z (cid:1) · diag(1 , , , e −E Ω z · diag(1 , , , . (7.6)Now the lemma follows from (7.4), (7.5), (7.6). (cid:3) The case λ = 1 /k . In this section, we assume λ is a complex number suchthat λ = 0, λ = 1 /k . Let κ = χ ( λ ) as in Definition 6.2. By Definition 2.1, and(6.6), (7.1), we have W λ ( x ) = (cid:16) ( ω j κα ) i − e ω j καx (cid:17) ≤ i,j ≤ = (cid:0) α i − κ i − ω i − j e ω j κz (cid:1) ≤ i,j ≤ = diag (cid:0) , α, α , α (cid:1) · diag (cid:0) , κ, κ , κ (cid:1) · W e Ω κz , (7.7) hence diag (cid:0) , α, α , α (cid:1) − W λ ( x ) = diag (cid:0) , κ, κ , κ (cid:1) · W e Ω κz . Thus by (7.1) andLemma 7.2, we have X λ ( x ) = 14 e −E Ω z (cid:8) diag(0 , , , · W ∗ · diag (cid:0) , κ, κ , κ (cid:1) · W e − Ω κz + diag(1 , , , · W ∗ · diag (cid:0) , κ, κ , κ (cid:1) · W e Ω κz (cid:9) , (7.8)where κ = χ ( λ ). Note that (7.8) is well-defined for every z, κ ∈ C , though weoriginally restricted the domains of z, κ . Definition 7.2.
For z, κ ∈ C , define X ( z, κ ) = 14 e −E Ω z (cid:8) diag(0 , , , · W ∗ · diag (cid:0) , κ, κ , κ (cid:1) · W e − Ω κz + diag(1 , , , · W ∗ · diag (cid:0) , κ, κ , κ (cid:1) · W e Ω κz (cid:9) . By (7.1), (7.8), we have X λ ( x ) = X ( αx, χ ( λ )) , x ∈ R , λ ∈ C \ { , /k } . (7.9) Lemma 7.3. (a) X ( z, i κ ) = X ( z, κ ) · L − for every z, κ ∈ C . (b) RX ( z, κ ) R = X ( z, κ ) for every z ∈ R and κ ∈ C .Proof. By (2.11), (2.12), we havediag (cid:16) , i κ, ( i κ ) , ( i κ ) (cid:17) · W e ± Ω ( i κ ) z = diag (cid:0) , κ, κ , κ (cid:1) · diag (cid:0) , i , i , i (cid:1) · W e ± ( i Ω ) κz = diag (cid:0) , κ, κ , κ (cid:1) · W L − · e ± LΩL − κz = diag (cid:0) , κ, κ , κ (cid:1) · W L − · L e ± Ω κz L − = diag (cid:0) , κ, κ , κ (cid:1) · W e ± Ω κz · L − , which implies (a) by Definition 7.2. Suppose z ∈ R , and let , ˆ X ( z, κ ) = diag(0 , , , · W ∗ · diag (cid:0) , κ, κ , κ (cid:1) · W e − Ω κz + diag(1 , , , · W ∗ · diag (cid:0) , κ, κ , κ (cid:1) · W e Ω κz (7.10)so that by Definition 7.2, X ( z, κ ) = 14 e −E Ω z ˆ X ( z, κ ) . (7.11)Note that diag(0 , , , , diag(0 , , , ∈ π (4). Since z ∈ R , e −E Ω z ∈ π (4), and R e ± Ω κz R = R e ± Ω κz R = R e ± RΩR κz R = R · R e ± Ω κz R · R = e ± Ω κz by (2.10).Hence by (7.10), we have R ˆ X ( z, κ ) R = R { diag(0 , , , · W ∗ · diag (1 , κ, κ , κ ) · W · e − Ω κz } R + R { diag(1 , , , · W ∗ · diag (1 , κ, κ , κ ) · W · e Ω κz } R = R diag(0 , , , R · RW T · diag (1 , κ, κ , κ ) · W R · R e − Ω κz R + R diag(1 , , , R · RW T · diag (1 , κ, κ , κ ) · W R · R e Ω κz R = diag(0 , , , · ( W R ) T · diag (cid:0) , κ, κ , κ (cid:1) · W R · e − Ω κz + diag(1 , , , · ( W R ) T · diag (cid:0) , κ, κ , κ (cid:1) · W R · e Ω κz = ˆ X ( z, κ ) , PECTRAL ANALYSIS FOR WELL-POSED BEAM DEFLECTION 23 since ( W R ) T = W T = W ∗ and W R = W = W by (2.10). Thus by (7.11), RX ( z, κ ) R = R (cid:26) e −E Ω z ˆ X ( z, κ ) (cid:27) R = 14 R e −E Ω z R · R ˆ X ( z, κ ) R = 14 e −E Ω z ˆ X ( z, κ ) = X ( z, κ ) , since e −E Ω z ∈ π (4). This shows (b), and the proof is complete. (cid:3) Interpreting Lemma 7.3 in terms of X λ ( x ) and Y λ ( x ), we have the following. Lemma 7.4. (a)
For λ ∈ C \ { , /k } and x ∈ R , RX λ ( x ) R = (cid:26) X λ ( x ) , if λ ∈ ( −∞ , ∪ (cid:0) k , ∞ (cid:1) X λ ( x ) · L , otherwise . In particular, X λ ( x ) ∈ π (4) for every λ ∈ ( −∞ , ∪ (1 /k, ∞ ) and x ∈ R . (b) RY λ ( x ) R = Y λ ( x ) for every λ ∈ C \ { , /k } and x > such that det X λ ( x ) = 0 . In particular, Y λ ( x ) ∈ π (4) for every λ ∈ R \ { , /k } and x > such that det X λ ( x ) = 0 .Proof. Suppose λ ∈ C \ { , /k } and x ∈ R . Let re i θ = 1 − / ( λk ), r >
0, 0 ≤ θ < π . Then by Definition 6.2, χ ( λ ) = √ re i θ . Suppose θ = 0, which is equivalent to λ ∈ ( −∞ , ∪ (1 /k, ∞ ). Then χ ( λ ) = √ r = χ ( λ ), hence by (7.9) and Lemma 7.3(b), RX λ ( x ) R = RX ( αx, χ ( λ )) R = X (cid:16) αx, χ ( λ ) (cid:17) = X ( αx, χ ( λ )) = X λ ( x ). Sup-pose 0 < θ < π . Then by Definition 6.2, χ (cid:0) λ (cid:1) = χ (cid:0) re − i θ (cid:1) = √ re i ( π − θ ) = i · √ re − i θ = i · χ ( λ ), since 0 < Arg (cid:16) √ re i ( π − θ ) (cid:17) = π/ − θ/ < π/
2. Thus by (7.9)and Lemma 7.3 (a), X λ ( x ) = X (cid:0) αx, χ (cid:0) λ (cid:1)(cid:1) = X (cid:16) αx, i · χ ( λ ) (cid:17) = X (cid:16) αx, χ ( λ ) (cid:17) · L − , and hence by (7.9) and Lemma 7.3 (b), RX λ ( x ) R = RX ( αx, χ ( λ )) R = X (cid:16) αx, χ ( λ ) (cid:17) = X λ ( x ) · L . This shows (a).Suppose λ ∈ C \ { , /k } , x >
0, and det X λ ( x ) = 0. Suppose first that λ ∈ ( −∞ , ∪ (1 /k, ∞ ). Then by (a), X λ ( x ) , X λ ( − x ) ∈ π (4). Thus by Lemma 4.4and Definition 7.1, Y λ ( x ) = X λ ( − x ) X λ ( x ) − − I ∈ π (4), and hence RY λ ( x ) R = Y λ ( x ) = Y λ ( x ), since λ is real. Suppose λ ( −∞ , ∪ (1 /k, ∞ ). Then byDefinition 7.1 and (a), RY λ ( x ) R = R { X λ ( − x ) X λ ( x ) − − I } R = RX λ ( − x ) R · RX λ ( x ) − R − I = (cid:8) X λ ( − x ) L (cid:9) (cid:8) X λ ( x ) L (cid:9) − − I = X λ ( − x ) X λ ( x ) − − I = Y λ ( x ) . Thus we showed (b), and the proof is complete. (cid:3)
The case λ = 1 /k . By (6.6), (7.1), we have n diag (cid:0) , α, α , α (cid:1) − · W k ( x ) · diag (cid:0) , α, α , α (cid:1)o i,j = α − i · n W k ( x ) o i,j · α j − = α j − i · H ( j − i ) x j − i ( j − i )! = H ( j − i ) ( αx ) j − i ( j − i )!= H ( j − i ) z j − i ( j − i )! = n W k ( z ) o i,j , ≤ i, j ≤ , hence diag (cid:0) , α, α , α (cid:1) − · W k ( x ) = W k ( z ) · diag (cid:0) , α, α , α (cid:1) − . Thus byLemma 7.2, X k ( x ) = 14 e −E Ω z n diag(0 , , , · W ∗ W k ( − z ) · diag (cid:0) , α, α , α (cid:1) − + diag(1 , , , · W ∗ W k ( z ) · diag (cid:0) , α, α , α (cid:1) − o = 14 e −E Ω z n diag(0 , , , · W ∗ W k ( − z ) + diag(1 , , , · W ∗ W k ( z ) o ·· diag (cid:0) , α, α , α (cid:1) − . For z ∈ R , denote P ( z ) = diag(0 , , , · W ∗ W k ( − z ) + diag(1 , , , · W ∗ W k ( z ) , (7.12)so that X k ( x ) = 14 e −E Ω z P ( z ) · diag (cid:0) , α, α , α (cid:1) − . (7.13)See Appendix B for the proof of Lemma 7.5. Lemma 7.5. Y k ( x ) ∈ π (4) for every x > .Proof of Lemma 7.1. The statement follows from Lemmas 7.4 (b) and 7.5 respec-tively for the case λ = 1 /k and for the case λ = 1 /k . (cid:3) Proof of Theorem 2
Let U n = 1 √ (cid:18) I n R n i R n − i I n (cid:19) ∈ gl(2 n, C ) , n ∈ N . (8.1) U n is unitary, since U ∗ n U n = 1 √ (cid:18) I n − i R n R n i I n (cid:19) · √ (cid:18) I n R n i R n − i I n (cid:19) = (cid:18) I n O n O n I n (cid:19) = I . Note also that U n R = 1 √ (cid:18) I n R n i R n − i I n (cid:19) (cid:18) O n R n R n O n (cid:19) = 1 √ (cid:18) I n R n − i R n i I n (cid:19) = U n . (8.2) Lemma 8.1.
For n ∈ N , The map A U n AU T n is an R -algebra isomorphismfrom π (2 n ) to gl(2 n, R ) .Proof. Since U n is unitary, and hence U n − = U T n , it is clear that the map A U n AU T n is a C -algebra isomorphism from gl(2 n, C ) to gl(2 n, C ). So itis sufficient to show that A ∈ π (2 n ) if and only if U n AU T n ∈ gl(2 n, R ). Let B = U n AU T n . Suppose A ∈ π (2 n ). Then by Definition 4.4, A = RAR = RAR ,hence B = U n AU T n = U n AU ∗ n = U n · RAR · U ∗ n = ( U n R ) A ( U n R ) ∗ = U n AU T n = B by (8.2). Thus B ∈ gl(2 n, R ). Conversely, suppose B ∈ gl(2 n, R ).Since A = (cid:8) U n (cid:9) − B (cid:8) U T n (cid:9) − = U T n BU n , we have RAR = R (cid:0) U T n BU n (cid:1) R =( U n R ) ∗ B ( U n R ) = U T n BU n = A by (8.2). Thus A ∈ π (2 n ) by Definition 4.4,and the proof is complete. (cid:3) Note that U = U , where U is defined by (4.5). PECTRAL ANALYSIS FOR WELL-POSED BEAM DEFLECTION 25
Lemma 8.2.
For any O = G ∈ π (4) , there exists G ∈ π (4) such that det ( GG − I )= 0 .Proof. Let ˆ G = UG U T , which is not O , since G = O . By Lemma 8.1, ˆ G ∈ gl(4 , R ). So there exists = r ∈ gl(4 , , R ) such that ˆ G r = . It is clear that thereexists ˆ G ∈ gl(4 , R ) such that ˆ G · ˆ G r = r , since ˆ G r = . Take G = U T ˆ GU . Thenby Lemma 8.1, G ∈ π (4), and GG · U T r = U T ˆ GU · U T ˆ G U · U T r = U T · ˆ G ˆ G r = U T r . Since U T r = , it follows that 1 is an eigenvalue of GG , which is equivalentto det ( GG − I ) = 0. (cid:3) Lemma 8.3 below, which is the last ingredient for the proof of Theorem 2, showsthat Y λ ( x ), when defined, never becomes the zero matrix for x >
0. See Appen-dix C for its proof.
Lemma 8.3. Y λ ( x ) = O for every = λ ∈ C and x > such that det X λ ( x ) = 0 .Proof of Theorem 2. Suppose 0 = λ ∈ R \ Spec K Q . Then O = Y λ ( l ) ∈ π (4)by Lemmas 7.1 and 8.3. So by Lemma 8.2. there exists G ∈ π (4) such thatdet { G · Y λ ( l ) − I } = 0. By Definitions 4.3, 6.1, and Lemma 6.1, there exists M ∈ wp(4 , , R ) such that G M = G , since G ∈ π (4). Thus we have det { G M · Y λ ( l ) − I } = 0, which implies that λ ∈ Spec K M by Corollary 2. (cid:3) Discussion
The 4 × X λ ( x ) and Y λ ( x ) turn out to be rich in symmetries. In fact,only part of their symmetries are exploited to prove our results in this paper. Wehave also tried to refrain, as possible as we can, from resorting to more explicitforms of X λ ( x ) and Y λ ( x ), despite of their explicit nature. In view of what canbe done more in these respects, it is expected that we will have a clearer pictureof general well-posed boundary value problems for finite beam deflection, if wepursue closer investigations on X λ ( x ) and Y λ ( x ). Especially, detailed results suchas Proposition 1, which is for only one specific boundary condition Q , are expectedto be obtained for the class of all well-posed boundary conditions. Appendix A. Proof of Lemma 3.2
By Definition 3.4, we have K M [ w ]( x ) = − α k Z x − l y ( x ) T · G − M · y ( ξ ) w ( ξ ) dξ + α k Z lx y ( x ) T · G + M · y ( ξ ) w ( ξ ) dξ = α k · y ( x ) T ( − G − M Z x − l y ( ξ ) w ( ξ ) dξ + G + M Z lx y ( ξ ) w ( ξ ) dξ ) , (A.1)and by (3.2), ddx ( − G − M Z x − l y ( ξ ) w ( ξ ) dξ + G + M Z lx y ( ξ ) w ( ξ ) dξ ) = − G − M · ddx Z x − l y ( ξ ) w ( ξ ) dξ + G + M · ddx Z lx y ( ξ ) w ( ξ ) dξ = − G − M · y ( x ) w ( x ) − G + M · y ( x ) w ( x ) = − ΩL · y ( x ) w ( x ) . (A.2)Let f ( x ) = − G − M Z x − l y ( ξ ) w ( ξ ) dξ + G + M Z lx y ( ξ ) w ( ξ ) dξ, (A.3)so that K M [ w ]( x ) = α k · y ( x ) T · f ( x ) , (A.4)and f ′ ( x ) = − ΩL · y ( x ) w ( x ) by (A.1), (A.2). By (2.13), y ( x ) T · Ω n · f ′ ( x ) = − y ( x ) T · Ω n · ΩL · y ( x ) w ( x )= − w ( x ) (cid:8) y ( x ) T · Ω n +1 · y ( − x ) (cid:9) = − w ( x ) (cid:0) e ω αx e ω αx e ω αx e ω αx (cid:1) ·· diag (cid:0) ω n +11 , ω n +12 , ω n +13 , ω n +14 (cid:1) · e − ω αx e − ω αx e − ω αx e − ω αx = − w ( x ) X j =1 e ω j αx ω n +1 j e − ω j αx = − w ( x ) X j =1 ω n +1 j , n = 0 , , , . . . (A.5)By (2.1), P j =1 ω j = P j =1 ( −
1) = −
4, and P j =1 ω j = P j =1 (cid:0) i j − ω (cid:1) = ω ·· P j =1 ( − j − = 0. By (2.3), P j =1 ω j = 0, hence by (2.1), P j =1 ω j = P j =1 ( − ω j )= − P j =1 ω j = 0. So by (A.5), we have y ( x ) T · Ω n · f ′ ( x ) = 0 , n = 0 , , , y ( x ) T · Ω · f ′ ( x ) = 4 · w ( x ) . (A.6)By (2.6), y ′ ( x ) T = { α Ω · y ( x ) } T = α · y ( x ) T · Ω . So by (A.4), (A.6), we have K M [ w ] ′ ( x ) = α k (cid:8) y ′ ( x ) T · f ( x ) + y ( x ) T · f ′ ( x ) (cid:9) = α k · y ( x ) T · Ω · f ( x ) , (A.7) K M [ w ] ′′ ( x ) = α k (cid:8) y ′ ( x ) T · Ω · f ( x ) + y ( x ) T · Ω · f ′ ( x ) (cid:9) = α k · y ( x ) T · Ω · f ( x ) , (A.8) K M [ w ] ′′′ ( x ) = α k (cid:8) y ′ ( x ) T · Ω · f ( x ) + y ( x ) T · Ω · f ′ ( x ) (cid:9) = α k · y ( x ) T · Ω · f ( x ) , (A.9)hence by (2.4), (2.6), (A.4), (A.6), K M [ w ] (4) ( x ) = α k (cid:8) y ′ ( x ) T · Ω · f ( x ) + y ( x ) T · Ω · f ′ ( x ) (cid:9) = α k (cid:8) α · y ( x ) T · Ω · f ( x ) + 4 · w ( x ) (cid:9) = α k (cid:8) − α · y ( x ) T · f ( x ) + 4 · w ( x ) (cid:9) = α k (cid:26) − α · kα K M [ w ]( x ) + 4 · w ( x ) (cid:27) = − α · K M [ w ]( x ) + α k · w ( x ) . This shows that K M [ w ]( x ) satisfies DE( w ). PECTRAL ANALYSIS FOR WELL-POSED BEAM DEFLECTION 27
By (A.4), (A.7), (A.8), (A.9), and (2.6), we have K M [ w ] ( n ) ( x ) = α n +1 k · y ( x ) T · Ω n · f ( x ) = α k { α n Ω n · y ( x ) } T · f ( x )= α k · y ( n ) ( x ) T · f ( x ) , n = 0 , , , . (A.10)By (A.3), f ( ± l ) = ∓ G ∓ M R l − l y ( ξ ) w ( ξ ) dξ , hence by (A.10), K M [ w ] ( n ) ( ± l ) = ∓ α k · y ( n ) ( ± l ) T · G ∓ M Z l − l y ( ξ ) w ( ξ ) dξ, n = 0 , , , . So by Definitions 2.3 and 2.6, B ± [ K M [ w ]] = ∓ α k · W ( ± l ) · G ∓ M Z l − l y ( ξ ) w ( ξ ) dξ. (A.11)Let M − , M + ∈ gl(4 , C ) be the 4 × M such that M = (cid:0) M − M + (cid:1) . ByDefinitions 3.2, 3.3, and (A.11), M ± · B ± [ K M [ w ]] = ∓ α k · M ± W ( ± l ) · G ∓ M Z l − l y ( ξ ) w ( ξ ) dξ = ∓ α k · f M ± · f M − f M ∓ ΩL Z l − l y ( ξ ) w ( ξ ) dξ, hence by (2.15), M · B [ K M [ w ]] = M − · B − [ K M [ w ]] + M + · B + [ K M [ w ]]= α k f M − f M − f M + ΩL Z l − l y ( ξ ) w ( ξ ) dξ − α k f M + f M − f M − ΩL Z l − l y ( ξ ) w ( ξ ) dξ = α k (cid:16)f M − f M − f M + − f M + f M − f M − (cid:17) ΩL Z l − l y ( ξ ) w ( ξ ) dξ. Thus we have M · B [ K M [ w ]] = , since f M = f M − + f M + by Definition 3.2 and hence f M − f M − f M + − f M + f M − f M − = (cid:16)f M − f M + (cid:17) f M − f M + − f M + f M − (cid:16)f M − f M + (cid:17) = f M + − f M + f M − f M + − f M + + f M + f M − f M + = O . This shows that K M [ w ] satisfies BC( M ), and the proof is complete. Appendix B. Proof of Lemma 7.5
Denote p n,i ( z ) = n X r =0 ω n − ri r ! z r , n = 0 , , , , i ∈ Z , (B.1)where it is understood that 0 = 1. In particular, denote p n ( z ) = p n, ( z ) , n = 0 , , , . (B.2) By Definitions 2.1, 2.4, and (2.10), W ∗ = W T = ( W R ) T = RW T = R · (cid:16) ω j − i (cid:17) ≤ i,j ≤ = (cid:16) ω j − − i (cid:17) ≤ i,j ≤ , hence by (6.6), (B.1), we havediag(0 , , , · W ∗ W k ( − z )= ω ω ω ω ω ω ( − z ) ( − z ) ( − z ) ( − z ) − z ) ( − z ) ( − z ) − z ) ( − z ) − z ) = p , ( − z ) p , ( − z ) p , ( − z ) p , ( − z ) p , ( − z ) p , ( − z ) p , ( − z ) p , ( − z )0 0 0 0 , diag(1 , , , · W ∗ W k ( z ) = ω ω ω ω ω ω z z z z z z z z z z = p , ( z ) p , ( z ) p , ( z ) p , ( z )0 0 0 00 0 0 0 p , ( z ) p , ( z ) p , ( z ) p , ( z ) . Thus by (7.12), we have P ( z ) = p , ( z ) p , ( z ) p , ( z ) p , ( z ) p , ( − z ) p , ( − z ) p , ( − z ) p , ( − z ) p , ( − z ) p , ( − z ) p , ( − z ) p , ( − z ) p , ( z ) p , ( z ) p , ( z ) p , ( z ) . (B.3)Note from (2.2), (2.3), (B.1), (B.2) that, for n = 0 , , , p n, ( z ) = n X r =0 ω n − r r ! z r = n X r =0 ω n − r r ! z r = n X r =0 ω n − r r ! z r ! = p n, ( z ) = p n ( z ) ,p n, ( − z ) = n X r =0 ω n − r r ! ( − z ) r = n X r =0 ( − ω ) n − r r ! ( − z ) r = ( − n n X r =0 ω n − r r ! z r = ( − n p n, ( z ) = ( − n p n ( z ) ,p n, ( − z ) = n X r =0 ω n − r r ! ( − z ) r = n X r =0 ω n − r r ! ( − z ) r = ( n X r =0 ω n − r r ! ( − z ) r ) = p n, ( − z ) = ( − n p n ( z ) = ( − n p n ( z ) . Thus by (B.3), we have P ( z ) = p ( z ) p ( z ) p ( z ) p ( z ) p ( z ) − p ( z ) p ( z ) − p ( z ) p ( z ) − p ( z ) p ( z ) − p ( z ) p ( z ) p ( z ) p ( z ) p ( z ) . (B.4) PECTRAL ANALYSIS FOR WELL-POSED BEAM DEFLECTION 29
Denote P + ( z ) = (cid:18) p ( z ) p ( z ) p ( z ) p ( z ) (cid:19) , P − ( z ) = (cid:18) − p ( z ) − p ( z ) p ( z ) p ( z ) (cid:19) , (B.5) V = 1 √ (cid:18) I I − I I (cid:19) = 1 √ − − , ˆ V = . (B.6)Note that V , ˆ V ∈ O (4) anddet V = 1 , det ˆ V = − . (B.7) Lemma B1. V · P ( z ) · ˆ V = √ (cid:18) P + ( z ) OO P − ( z ) (cid:19) .Proof. By (B.4), (B.6), V · P ( z ) = 1 √ − − p ( z ) p ( z ) p ( z ) p ( z ) p ( z ) − p ( z ) p ( z ) − p ( z ) p ( z ) − p ( z ) p ( z ) − p ( z ) p ( z ) p ( z ) p ( z ) p ( z ) = √ p ( z ) 0 p ( z ) 0 p ( z ) 0 p ( z ) 00 − p ( z ) 0 − p ( z )0 p ( z ) 0 p ( z ) , hence the lemma follows, since multiplying ˆ V on the right amounts to interchangingthe second and the third columns. (cid:3) By Lemma B1, we have P ( z ) = V − · √ (cid:18) P + ( z ) OO P − ( z ) (cid:19) · ˆ V − = √ · V T (cid:18) P + ( z ) OO P − ( z ) (cid:19) ˆ V T , (B.8)since V , ˆ V are orthogonal. So by (B.7)det P ( z ) = √ · det V · det (cid:18) P + ( z ) OO P − ( z ) (cid:19) · det ˆ V = − · det (cid:18) P + ( z ) OO P − ( z ) (cid:19) = − · det P + ( z ) · det P − ( z ) , hence by (7.13),det X k ( x ) = (cid:18) (cid:19) · det e −E Ω z · det P ( z ) · det diag (cid:0) , α, α , α (cid:1) − = − e − √ z α · det P + ( z ) · det P − ( z ) , (B.9)since det diag (cid:0) , α, α , α (cid:1) − = 1 · α − · α − · α − = α − , and det e −E Ω z = e − ω z · e ω z · e ω z · e − ω z = e −{ ( ω − ω )+( ω − ω ) } z = e − ω + ω ) z = e − · ω · z = e − √ z by (2.2), (2.3), (2.5). Since det X k ( x ) = 0 for every x > Corollary 1, and Lemma 5.1, it follows from (7.1), (B.9) that det P + ( z ) = 0 anddet P − ( z ) = 0 for every z >
0. From (B.5), we havedet P + ( z ) = p ( z ) p ( z ) − p ( z ) p ( z ) = 2 i Im n p ( z ) p ( z ) o , (B.10)det P − ( z ) = p ( z ) p ( z ) − p ( z ) p ( z ) = 2 i Im n p ( z ) p ( z ) o . (B.11)Note from Definition 4.4 that (cid:18) a a a a (cid:19) ∈ π (2), if and only if a = a and a = a . Lemma B2. P + ( − z ) P + ( z ) − , P − ( − z ) P − ( z ) − ∈ π (2) for every z > .Proof. From (B.5), we have i P + ( − z ) · adj P + ( z ) = i (cid:18) p ( − z ) p ( − z ) p ( − z ) p ( − z ) (cid:19) (cid:18) p ( z ) − p ( z ) − p ( z ) p ( z ) (cid:19) = i n p ( − z ) p ( z ) − p ( − z ) p ( z ) o i n − p ( − z ) p ( z ) + p ( − z ) p ( z ) o i { p ( − z ) p ( z ) − p ( − z ) p ( z ) } i n − p ( − z ) p ( z ) + p ( − z ) p ( z ) o = i n − p ( − z ) p ( z ) + p ( − z ) p ( z ) o i { p ( − z ) p ( z ) − p ( − z ) p ( z ) } i { p ( − z ) p ( z ) − p ( − z ) p ( z ) } i n − p ( − z ) p ( z ) + p ( − z ) p ( z ) o , i P − ( − z ) · adj P − ( z ) = i (cid:18) − p ( − z ) − p ( − z ) p ( − z ) p ( − z ) (cid:19) (cid:18) p ( z ) p ( z ) − p ( z ) − p ( z ) (cid:19) = i n − p ( − z ) p ( z ) + p ( − z ) p ( z ) o i n − p ( − z ) p ( z ) + p ( − z ) p ( z ) o i { p ( − z ) p ( z ) − p ( − z ) p ( z ) } i n p ( − z ) p ( z ) − p ( − z ) p ( z ) o = i n p ( − z ) p ( z ) − p ( − z ) p ( z ) o i { p ( − z ) p ( z ) − p ( − z ) p ( z ) } i { p ( − z ) p ( z ) − p ( − z ) p ( z ) } i n p ( − z ) p ( z ) − p ( − z ) p ( z ) o , hence we have i P + ( − z ) · adj P + ( z ) , i P − ( − z ) · adj P − ( z ) ∈ π (2) by Definition 4.4.Thus by (B.10), (B.11), P + ( − z ) P + ( z ) − = P + ( − z ) · adj P + ( z )det P + ( z ) = − i P + ( − z ) · adj P + ( z )2 Im n p ( z ) p ( z ) o , P − ( − z ) P − ( z ) − = P − ( − z ) · adj P − ( z )det P − ( z ) = − i P − ( − z ) · adj P − ( z )2 Im n p ( z ) p ( z ) o , both of which are in π (2) by Lemma 4.4. (cid:3) Note from (B.8) that P ( − z ) P ( z ) − = (cid:26) √ · V T (cid:18) P + ( − z ) OO P − ( − z ) (cid:19) ˆ V − (cid:27) ·· (cid:26) √ · V T (cid:18) P + ( z ) OO P − ( z ) (cid:19) ˆ V − (cid:27) − = √ · V T (cid:18) P + ( − z ) OO P − ( − z ) (cid:19) ˆ V − · √ · ˆ V (cid:18) P + ( z ) OO P − ( z ) (cid:19) − V = V T (cid:18) P + ( − z ) OO P − ( − z ) (cid:19) (cid:18) P + ( z ) − OO P − ( z ) − (cid:19) V = V T (cid:18) P + ( − z ) P + ( z ) − OO P − ( − z ) P − ( z ) − (cid:19) V , z > . (B.12) Proof of Lemma 7.5.
By Definition 7.1 and Lemma 4.4, it is sufficient to show that X k ( − x ) X k ( x ) − ∈ π (4) for x >
0. By (7.1), (7.13), X k ( − x ) X k ( x ) − = (cid:26) e −E Ω ( − z ) P ( − z ) · diag (cid:0) , α, α , α (cid:1) − (cid:27) ·· (cid:26) e −E Ω z P ( z ) · diag (cid:0) , α, α , α (cid:1) − (cid:27) − = 14 e E Ω z P ( − z ) · diag (cid:0) , α, α , α (cid:1) − · (cid:0) , α, α , α (cid:1) · P ( z ) − e E Ω z = e E Ω z P ( − z ) P ( z ) − e E Ω z , hence it is sufficient to show that P ( − z ) P ( z ) − ∈ π (4) for z >
0, since e E Ω z ∈ π (4)for z ∈ R . By (B.6), (B.12), R { P ( − z ) P ( z ) − } R = R (cid:26) V T (cid:18) P + ( − z ) P + ( z ) − OO P − ( − z ) P − ( z ) − (cid:19) V (cid:27) R = RV T R · R (cid:18) P + ( − z ) P + ( z ) − OO P − ( − z ) P − ( z ) − (cid:19) R · RVR . (B.13)By Lemma B2, R (cid:18) P + ( − z ) P + ( z ) − OO P − ( − z ) P − ( z ) − (cid:19) R = (cid:18) O RR O (cid:19) (cid:18) P + ( − z ) P + ( z ) − OO P − ( − z ) P − ( z ) − (cid:19) (cid:18) O RR O (cid:19) = (cid:18) O RP − ( − z ) P − ( z ) − RP + ( − z ) P + ( z ) − O (cid:19) (cid:18) O RR O (cid:19) = (cid:18) RP − ( − z ) P − ( z ) − R OO RP + ( − z ) P + ( z ) − R (cid:19) = (cid:18) P − ( − z ) P − ( z ) − OO P + ( − z ) P + ( z ) − (cid:19) , z > . (B.14)By (B.6), RVR = (cid:18) O RR O (cid:19) · √ (cid:18) I I − I I (cid:19) · (cid:18) O RR O (cid:19) = 1 √ (cid:18) − R RR R (cid:19) (cid:18)
O RR O (cid:19) = 1 √ (cid:18) I − II I (cid:19) = V T , (B.15) V = 1 √ (cid:18) I I − I I (cid:19) · √ (cid:18) I I − I I (cid:19) = (cid:18) O I − I O (cid:19) . (B.16)So by (B.12), (B.13), (B.14), (B.15), (B.16), we have R { P ( − z ) P ( z ) − } R = ( RVR ) T · (cid:18) P − ( − z ) P − ( z ) − OO P + ( − z ) P + ( z ) − (cid:19) · RVR = V (cid:18) P − ( − z ) P − ( z ) − OO P + ( − z ) P + ( z ) − (cid:19) V T = V T V (cid:18) P − ( − z ) P − ( z ) − OO P + ( − z ) P + ( z ) − (cid:19) (cid:0) V (cid:1) T V = V T (cid:18) O I − I O (cid:19) (cid:18) P − ( − z ) P − ( z ) − OO P + ( − z ) P + ( z ) − (cid:19) (cid:18) O − II O (cid:19) V = V T (cid:18) O P + ( − z ) P + ( z ) − − P − ( − z ) P − ( z ) − O (cid:19) (cid:18) O − II O (cid:19) V = V T (cid:18) P + ( − z ) P + ( z ) − OO P − ( − z ) P − ( z ) − (cid:19) V = P ( − z ) P ( z ) − , z > . Thus P ( − z ) P ( z ) − ∈ π (4) for z >
0, and the proof is complete. (cid:3)
Appendix C. Proof of Lemma 8.3
Lemma C1.
Suppose t ∈ R satisfies P r =1 ω nr e ω r t = 0 for n = 1 , , . Then t = 0 .Proof. Let a = P r =1 e ω r t . Then the condition for t is equivalent to a = (cid:0) ω i − j (cid:1) ≤ i,j ≤ · e ω t e ω t e ω t e ω t = W e ω t e ω t e ω t e ω t , which, by Lemma 2.1 and (2.8), is equivalent again to e ω t e ω t e ω t e ω t = 14 W ∗ a = a (cid:16) ω − ji (cid:17) ≤ i,j ≤ · = a . It follows that e ω i t = e ω j t for every i, j ∈ Z . In particular, e ω t = e ω t , hence1 = e ω t /e ω t = e ( ω − ω ) t = e √ t by Definition 2.1, which implies that t =0. (cid:3) Proof of Lemma 8.3.
Suppose on the contrary that Y λ ( x ) = O for some 0 = λ ∈ C and x > X λ ( x ) = 0. Then by Definition 7.1, we have X λ ( − x ) · X λ ( x ) − − I = O , hence X λ ( − x ) − X λ ( x ) = O . So by (6.8), wehave W ( x ) − W λ ( x ) = W ( − x ) − W λ ( − x ), hence W λ ( − x ) W λ ( x ) − = W ( − x ) W ( x ) − . (C.1)Let z = αx >
0. By (7.2), (7.3), W ( − x ) W ( x ) − = n diag (cid:0) , α, α , α (cid:1) · W e Ω ( − z ) o (cid:26) e − Ω z W ∗ · diag (cid:0) , α, α , α (cid:1) − (cid:27) = 14 diag (cid:0) , α, α , α (cid:1) · W e − Ω z W ∗ · diag (cid:0) , α, α , α (cid:1) − . (C.2)By (2.8), W e − Ω z W ∗ (C.3)= (cid:0) ω i − j (cid:1) ≤ i,j ≤ · diag (cid:0) e − ω z , e − ω z , e − ω z , e − ω z (cid:1) · (cid:16) ω − ji (cid:17) ≤ i,j ≤ = (cid:0) ω i − j e − ω j z (cid:1) ≤ i,j ≤ · (cid:16) ω − ji (cid:17) ≤ i,j ≤ = X r =1 ω i − r e − ω r z ω − jr ! ≤ i,j ≤ = X r =1 ω i − jr e − ω r z ! ≤ i,j ≤ , (C.4)hence by (C.2), n W ( − x ) W ( x ) − o i,j = α i − · α − j · (cid:0) W e − Ω z W ∗ (cid:1) i,j = α i − j · X r =1 ω i − jr e − ω r z , ≤ i, j ≤ . (C.5)Suppose λ = 1 /k . Note from (6.6) that W k ( x ) is upper diagonal. So W k ( − x ) ·· W k ( x ) − is upper diagonal as well. Hence by (C.1), (C.5), we have P r =1 ω nr e − ω r z = P r =1 ω nr e ω r ( − αx ) = 0 for n = 1 , ,
3. This implies that x = 0 by Lemma C1,which contradicts the assumption that x >
0. Thus we conclude that λ = 1 /k .Let κ = χ ( λ ), where χ is as in Definition 6.2. Note that κ = 1. κ = 0, since λ = 1 /k . By (7.7) and Lemma 2.1, W λ ( − x ) W λ ( x ) − = n diag (cid:0) , α, α , α (cid:1) · diag (cid:0) , κ , κ , κ (cid:1) W e Ω κ ( − z ) o ·· (cid:26) e − Ω κ z W ∗ · diag (cid:0) , κ , κ , κ (cid:1) − · diag (cid:0) , α, α , α (cid:1) − (cid:27) = 14 diag (cid:0) , α, α , α (cid:1) · diag (cid:0) , κ , κ , κ (cid:1) · W e − Ω κ z W ∗ ·· diag (cid:0) , κ , κ , κ (cid:1) − · diag (cid:0) , α, α , α (cid:1) − , hence by (C.1), (C.2), we have W e − Ω z W ∗ = diag (cid:0) , κ , κ , κ (cid:1) · W e − Ω κ z W ∗ · diag (cid:0) , κ , κ , κ (cid:1) − . (C.6)Similarly to (C.4), we have W e − Ω κ z W ∗ = X r =1 ω i − jr e − ω r κ z ! ≤ i,j ≤ , hence by (C.4), (C.6), κ i − j P r =1 ω i − jr e − ω r κ z = P r =1 ω i − jr e − ω r z for 1 ≤ i, j ≤
4, or equivalently, κ n P r =1 ω nr e − ω r κ z = P r =1 ω nr e − ω r z for − ≤ n ≤
3. So by(2.1), we have X r =1 ω nr e − ω r z = κ n X r =1 ω nr e − ω r κ z = κ n X r =1 (cid:0) − ω n − r (cid:1) e − ω r κ z = − κ · κ n −
40 4 X r =1 ω n − r e − ω r κ z = − κ · X r =1 ω n − r e − ω r z = − κ · X r =1 ( − ω nr ) e − ω r z = κ · X r =1 ω nr e − ω r z , n = 1 , , . Since κ = 1, it follows that P r =1 ω nr e − ω r z = P r =1 ω nr e ω r ( − αx ) = 0 for n =1 , ,
3, which implies x = 0 by Lemma C1. This again contradicts the assumptionthat x >
0. Thus we conclude Y λ ( x ) = O for every 0 = λ ∈ C and x > X λ ( x ) = 0. (cid:3) References [1] F. W. Beaufait and P. W. Hoadley,
Analysis of elastic beams on nonlinear foundations ,Comput. Struct. (1980), no. 5, 669–676.[2] S. W. Choi, On positiveness and contractiveness of the integral operator arising from thebeam deflection problem on elastic foundation , Bull. Korean Math. Soc. (2015), no. 4,1225–1240.[3] S. W. Choi, Spectral analysis of the integral operator arising from the beam deflection problemon elastic foundation II: eigenvalues , Bound. Value Probl. , 6 (2015)[4] S. W. Choi,
Existence and uniqueness of finite beam deflection on nonlinear non-uniformelastic foundation with arbitrary well-posed boundary condition , Bound. Value Probl. ,113 (2020)[5] S. W. Choi, T. S. Jang,
Existence and uniqueness of nonlinear deflections of an infinite beamresting on a non-uniform non-linear elastic foundation , Bound. Value Probl. , 5 (2012)[6] M. Galewski,
On the nonlinear elastic simply supported beam equation , An. S¸tiint¸. Univ.‘Ovidius’ Constant¸a Ser. Mat. (2011), no. 1, 109–119.[7] M. Hetenyi, Beams on Elastic Foundation , The University of Michigan Press, Ann Arbor,Mich., 1946.[8] Y. H. Kuo and S. Y. Lee,
Deflection of nonuniform beams resting on a nonlinear elasticfoundation , Comput. Struct. (1994), no. 5, 513–519.[9] X. Ma, J. W. Butterworth, and G. C. Clifton, Static analysis of an infinite beam resting ona tensionless Pasternak foundation , Eur. J. Mech., A, Solids (2009), no. 4, 697–703.[10] C. Miranda and K. Nair, Finite beams on elastic foundation , ASCE. J. Struct. Div. (1966),131–142.[11] I. Stakgold and M. Holst, Green’s Functions and Boundary Value Problems , 3rd ed., Pureand Applied Mathematics, John Wiley & Sons, Inc., Hoboken, NJ, 2011.[12] S. Timoshenko,
History of strength of materials. With a brief account of the history of theoryof elasticity and theory of structures , McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953.[13] B. Y. Ting,
Finite beams on elastic foundation with restraints , ASCE. J. Struct. Div. (1982), 611–621.
Sung Woo Choi, Department of Mathematics, Duksung Women’s University, Seoul01369, Korea
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