On Ambarzumyan-type Inverse Problems of Vibrating String Equations
OOn Ambarzumyan-type Inverse Problems ofVibrating String Equations
Yuri Ashrafyan ∗ and Dominik L. Michels † Abstract
We consider the inverse spectral theory of vibrating string equa-tions. In this regard, first eigenvalue Ambarzumyan-type unique-ness theorems are stated and proved subject to separated, self-adjointboundary conditions. More precisely, it is shown that there is a curvein the boundary parameters’ domain on which no analog of it is possi-ble. Necessary conditions of the n -th eigenvalue are identified, whichallows to state the theorems. In addition, several properties of thefirst eigenvalue are examined. Lower and upper bounds are identified,and the areas are described in the boundary parameters’ domain onwhich the sign of the first eigenvalue remains unchanged. This pa-per contributes to inverse spectral theory as well as to direct spectraltheory. Keywords:
Ambarzumyan theorem, first eigenvalue, inverse problems,vibrating string equations.
MSC 2010:
When dealing with direct problems, one considers a physical model and cal-culates a specific output given a specific input. In contrast, inverse problemsare dealing with the inversion of this model based on measured or observedoutputs, i.e. we consider a mathematical framework which is used to obtain ∗ Yuri Ashrafyan: KAUST, CEMSE Division, Thuwal, 23955-6900, KSA, E-mail:[email protected] † Dominik L. Michels: KAUST, CEMSE Division, Thuwal, 23955-6900, KSA, E-mail:[email protected] a r X i v : . [ m a t h . SP ] A ug nformation about parameters of a system from observed measurements. Thisframework is often useful since it provides information about physical param-eters of a system that can usually not directly be observed. As a consequence,inverse problem theory is being developed intensively.The inverse spectral problems aiming for the reconstruction of operatorusing spectral data, such as the spectrum (set of eigenvalues), norming con-stants ( L -norms of eigenfunctions), nodal points (roots of eigenfunctions),and other quantities.Historically, Ambarzumyan did pioneer work [1] with respect to the theoryof inverse problems. It is easy to calculate (direct problem) that the eigen-values of the Sturm-Liouville boundary value problem − y (cid:48)(cid:48) = µy , y (cid:48) (0) = y (cid:48) ( π ) = 0 are n , n ≥
0. In 1929, Ambarzumyan proved the inverse asser-tion, if the eigenvalues of the boundary value problem − y (cid:48)(cid:48) + q ( x ) y = µy , y (cid:48) (0) = y (cid:48) ( π ) = 0 are n , n ≥
0, then q ( x ) = 0 a.e. on (0 , π ).Later, Borg [4] showed that with general boundary conditions more in-formation in addition to the spectrum is required in order to uniquely re-construct the function q ( x ). In the same work he showed that two spectra(for different fixed boundary conditions) are sufficient for the unique deter-mination of a problem. Other fundamental work was done by Marchenko[19], who showed that if two problems have the same set of eigenvalues andnorming constants, then they coincide. These results were very importantand they opened the door for further investigations in inverse spectral the-ory. Many other kinds of uniqueness theorems for inverse Sturm-Liouvilleproblems have been stated and developed, we just mention several of themhere [17, 13, 21, 8, 7, 9, 28, 2].Let L ( q, γ, δ ) denote the self-adjoint Sturm-Liouville operator generatedby the boundary value problem − y (cid:48)(cid:48) + q ( s ) y = µy, s ∈ (0 , π ) , µ ∈ C , (1.1) y (0) cos γ + y (cid:48) (0) sin γ = 0 , γ ∈ (0 , π ] , (1.2) y ( π ) cos δ + y (cid:48) ( π ) sin δ = 0 , δ ∈ [0 , π ) , (1.3)where the potential function q is real-valued and summable. It is known, thatthe spectrum of L ( q, γ, δ ) is discrete and consists of real, simple eigenvalues,which we denote by µ n = µ n ( q, γ, δ ), n ≥ µ ( q, γ, δ ) < µ ( q, γ, δ ) < . . . < µ n ( q, γ, δ ) < . . . , emphasizing the dependence of µ n on q , γ and δ (see [18, 20, 10]). Thereare many generalizations of Ambarzumyan’s theorem for the Sturm-Liouvilleproblem in various directions (see e.g. [7, 15, 5, 7, 31, 6, 29, 30, 3], and thereferences therein). 2articularly, in [7], it is shown that it is not necessary to specify the wholespectrum: information regarding the first eigenvalue is sufficient. Precisely,if µ = π (cid:82) π q ( x ) dx , then q ( x ) = µ , a.e. on (0 , π ). This type of problemsetting is called a first eigenvalue Ambarzumyan-type inverse problem.Let A and ˜ A be two operators. In what follows, if a certain symbol τ denotes an object related to the operator A , then ˜ τ will denote a similarobject related to ˜ A and ˆ τ := ˜ τ − τ , ˇ τ := ˜ τ /τ . We recall the inner product( y, z ) = (cid:82) y ( x ) z ( x ) dx .In [31], Yurko proved the following generalization of Ambarzumyan the-orem for Sturm-Liouville problem (it’s true for any self-adjoint boundaryconditions). Theorem 1.1 (Yurko [31]) . Let q, ˜ q ∈ L R (0 , π ) . If µ − ˜ µ = (ˆ q ˜ ϕ , ˜ ϕ )( ˜ ϕ , ˜ ϕ ) , where ˜ ϕ is an eigenfunction of ˜ L related to ˜ µ . Then q ( x ) = ˜ q ( x ) + µ − ˜ µ ,a.e. on (0 , π ) . Recently, one of the authors [3] has proved another generalization of thefirst eigenvalue Ambarzumyan-type inverse problem.
Theorem 1.2 ([3]) . Let q, ˜ q ∈ L R (0 , π ) . If µ − ˜ µ = ess inf ˆ q or µ − ˜ µ = ess sup ˆ q, then q ( x ) = ˜ q ( x ) + µ − ˜ µ a.e. on (0 , π ) . The main aim of this paper is to investigate first eigenvalue Ambarzumyan-type theorems for boundary value problems of vibrating string equations: − u (cid:48)(cid:48) = λp ( x ) u, x ∈ (0 , , λ ∈ C , (1.4) u (0) cos α + u (cid:48) (0) sin α = 0 , α ∈ (0 , π ] , (1.5) u (1) cos β + u (cid:48) (1) sin β = 0 , β ∈ [0 , π ) , (1.6)where the density function p is piecewise continuous, positive, bounded awayfrom 0, real-valued and λ is the spectral parameter. This equation describessmall transversal vibrations of a string of a linear density p on the interval[0 ,
1] (see e.g. [27]). By S ( p, α, β ) we denote the self-adjoint operator gen-erated by problem (1.4)–(1.6). It is well-known, that the spectrum of theoperator S ( p, α, β ) is discrete and consists of real, simple eigenvalues, whichwe denote by λ n = λ n ( p, α, β ), n ≥ λ ( p, α, β ) < λ ( p, α, β ) < . . . < λ n ( p, α, β ) < . . . , λ n on p , α and β .The classical Ambarzumyan’s theorem for the operator S ( p, π/ , π/
2) onthe interval [0 , a ] can be stated as follows: if it is known that the eigenvaluesare π n /a , the question is if we can conclude that p ( x ) = 1 on [0 , a ]. In[25], Shen constructed a counterexample to this assertion, and showed thatone needs additional information about the density function to formulate aninverse problem with one spectrum, such as the function p ( x ) being even on[0 , a ], or the values of p (cid:48) (0) and p (cid:48) ( a ) are known. Hence, in general, there isno analog to the Ambarzumyan theorem for string equations in a classicalform.In Section 2, it is shown that first eigenvalue Ambarzumyan-type the-orems for the S ( p, α, β ) operator are valid, but not for the whole domain α, β ∈ (0 , π ] × [0 , π ). It is explained why for some boundary conditionsfirst-eigenvalue inverse problems are impossible. In Section 3, bounds forthe lowest eigenvalue are given, and the areas in the boundary parametersdomain are described, on which it keeps its sign. Let ϕ ( x ) = ϕ ( x, λ, α, p ) be the solution of (1.4), which satisfies the initialconditions ϕ (0 , λ, α, p ) = sin α, ϕ (cid:48) (0 , λ, α, p ) = − cos α. The eigenvalues λ n are the solutions of the equation ϕ (1 , λ, α ) cos β + ϕ (cid:48) (1 , λ, α ) sin β = 0 . It is easy to see that the functions ϕ n ( x ) := ϕ ( x, λ n , α, p ), n ≥
0, are theeigenfunctions, corresponding to the eigenvalue λ n .Our first eigenvalue Ambarzumyan-type theorems for vibrating stringequation are as follows. Theorem 2.1.
Let α, β ∈ (0 , π ] × [0 , π ) and cos α cos β − sin( α − β ) (cid:54) = 0 , for α, β ∈ [ π/ , π ] × [0 , π/ . If λ = ˜ λ max x ∈ [0 , ˇ p ( x ) or λ = ˜ λ min x ∈ [0 , ˇ p ( x ) , then p ( x ) = ˜ λ /λ ˜ p ( x ) on [0 , .Proof. Write down the fact that ϕ ( x ) and ˜ ϕ ( x ) are the eigenfunctions ofthe operators S ( p, α, β ) and S (˜ p, α, β ) corresponding to the eigenvalues λ λ , respectively: − ϕ (cid:48)(cid:48) ( x ) + λ p ( x ) ϕ ( x ) = 0 , (2.1) − ˜ ϕ (cid:48)(cid:48) ( x ) + ˜ λ ˜ p ( x ) ˜ ϕ ( x ) = 0 . (2.2)Multiplying (2.1) by ˜ ϕ ( x ), (2.2) by ϕ ( x ) and subtracting from the firstequation the second one, we obtain − ddx [ ϕ (cid:48) ( x ) ˜ ϕ ( x ) − ϕ ( x ) ˜ ϕ (cid:48) ( x )] ++ ϕ ( x ) ˜ ϕ ( x ) (cid:104) λ p ( x ) − ˜ λ ˜ p ( x ) (cid:105) = 0 . Integrating the latter from 0 to 1, we obtain − ϕ (cid:48) (1) ˜ ϕ (1) + ϕ (1) ˜ ϕ (cid:48) (1)++ ϕ (cid:48) (0) ˜ ϕ (0) − ϕ (0) ˜ ϕ (cid:48) (0)++ (cid:90) ϕ ( x ) ˜ ϕ ( x ) (cid:104) λ p ( x ) − ˜ λ ˜ p ( x ) (cid:105) dx = 0 . Since ϕ ( x ) and ˜ ϕ ( x ) satisfy the same boundary conditions, thus the firstfour terms vanish, and we obtain (cid:90) ϕ ( x ) ˜ ϕ ( x ) (cid:104) λ p ( x ) − ˜ λ ˜ p ( x ) (cid:105) dx = 0 . In virtue of Sturm’s oscillation theorem (see e.g. [7, 12, 26, 11]), the product ϕ ( x ) ˜ ϕ ( x ) has no zeros in interval (0 , Theorem 2.2.
Let α, β ∈ (0 , π ] × [0 , π ) and cos α cos β − sin( α − β ) (cid:54) = 0 , for α, β ∈ [ π/ , π ] × [0 , π/ . If λ = ˜ λ (ˇ p ˜ ϕ , ˜ ϕ )( ˜ ϕ , ˜ ϕ ) then p ( x ) = ˜ λ /λ ˜ p ( x ) on [0 , .Proof. Taking into account the condition of the theorem, these simple rela-tions are held( − p − ˜ ϕ , ˜ ϕ )( ˜ ϕ , ˜ ϕ ) = ( − ˜ p − ˜ ϕ , ˇ p ˜ ϕ )( ˜ ϕ , ˜ ϕ ) = ˜ λ (ˇ p ˜ ϕ , ˜ ϕ )( ˜ ϕ , ˜ ϕ ) = λ from which follows that ˜ ϕ ( x ) is an eigenfunction of the operator S ( p, α, β )with the eigenvalue λ . Particularly, − ˜ ϕ (cid:48)(cid:48) ( x ) = λ p ( x ) ˜ ϕ ( x ), thus we obtain λ p ( x ) = ˜ λ ˜ p ( x ). This completes the proof.5 emark 1. Consider the curve cos α cos β − sin( α − β ) = 0 , for α, β ∈ [ π/ , π ] × [0 , π/ , see Figure 1. The first eigenvalue λ ( p, α, β ) on this curveis always zero (the proof of this assertion is given in Theorem 3.3). Thereforefirst-eigenvalue Ambarzumyan-type theorems are impossible for this α and β . π π π π π π π π α β Figure 1: Illustration of the curve describing λ ( p, α, β ) = 0. Example 1.
Consider the boundary value problem − u (cid:48)(cid:48) = λp ( x ) u,u (0) = u (1) = 0 . Let ˜ p ( x ) = 1 , then ˜ λ = π and ˜ ϕ ( x ) = sin πx .Theorem 2.1 implies the following assertion. Corollary 1. If λ ( p, π,
0) = π max p − ( x ) then p ( x ) = π /λ on [0 , .Theorem 2.2 implies the following assertion. Corollary 2. If λ ( p, π,
0) = 2 π (cid:90) p − ( x ) sin ( πx ) dx, then p ( x ) = π /λ on [0 , .
6s we saw that the lowest eigenvalue contains significant informationabout the equation, a reasonable question may arise: can one state a unique-ness theorem having information regarding an arbitrary eigenvalue? Theanswer is positive, but more information is demanded, than in the case ofthe lowest eigenvalue.
Theorem 2.3.
Let λ n (cid:54) = 0 , for a fixed n > . If λ n = ˜ λ n (ˇ p ˜ ϕ n , ˜ ϕ n )( ˜ ϕ n , ˜ ϕ n ) and λ n = ˜ λ n max x ∈ [0 , ˇ p ( x ) , then p ( x ) = ˜ λ n /λ n ˜ p ( x ) on [0 , .Proof. From the first condition we obtain (cid:90) ( λ n − ˜ λ n ˇ p ( x )) ˜ ϕ n ( x ) dx = 0 . In virtue of Sturm’s oscillation theorem, it is known that the n -th eigen-function has exactly n isolated zeros in the open interval (0 , λ n − ˜ λ n ˇ p ( x ) = 0. This completes the proof.For Sturm-Liouville problem similar result was proved in [16]. Remark 2.
It is obvious that in Theorem 2.3 the second condition can bereplaced with λ n = ˜ λ n min x ∈ [0 , ˇ p ( x ) . For the case sin γ (cid:54) = 0 and sin δ (cid:54) = 0, i.e. γ, δ ∈ (0 , π ), the boundary conditions(1.2)–(1.3) can be written as a y (0) + y (cid:48) (0) = 0 ,b y ( π ) + y (cid:48) ( π ) = 0 . In [14] the authors showed, that for a Sturm-Liouville problem with fixedboundary conditions ( a, b ) and a q ∈ L (0 , π ), the lowest eigenvalue has theproperty −∞ < µ ( q, a, b ) ≤ µ ( e, − ( a − b ) / , ( a − b ) / , where e is a known even function ( e ( x ) = e ( π − x )) from L (0 , π ), whichis uniquely defined by the eigenvalues µ n ( q, a, b ) , n ≥ y (0) = y ( π ) = 0.7n [22], for a q ∈ L ∞ (0 , π ) and the Dirichlet boundary conditions, thefollowing bounds for the lowest eigenvalue is proved: | µ ( q, π, − µ (0 , π, | ≤ (cid:107) q (cid:107) ∞ . (3.1)In [10], for a q from L R (0 , π ) (in particular L ∞ R (0 , π )) and γ ∈ (0 , π ], δ ∈ [0 , π ), a formula for µ n ( q, γ, δ ) is found, and particularly for n = 0: µ ( q, γ, δ ) = µ (0 , γ, δ ) + (cid:90) (cid:90) π q ( x ) h ( x, tq, γ, δ ) dx dt, (3.2)where h ( x, tq, γ, δ ) is a normalized eigenfunction ( (cid:82) π h ( x, tq, γ, δ ) dx = 1)of the problem L ( tq, γ, δ ) and where t is a real parameter. From (3.2) thegeneralization of (3.1) follows: | µ ( q, γ, δ ) − µ (0 , γ, δ ) | ≤ (cid:107) q (cid:107) ∞ , which can be written as − ess sup | q | ≤ µ ( q, γ, δ ) − µ (0 , γ, δ ) ≤ ess sup | q | . (3.3)In the same work it is shown that the lowest eigenvalue tends to −∞ :lim γ → µ ( q, γ, δ ) = −∞ , for fixed δ ∈ [0 , π ) , lim δ → π µ ( q, γ, δ ) = −∞ , for fixed γ ∈ (0 , π ] . We found narrower bounds for the lowest eigenvalue of the Sturm-Liouvilleoperator.
Theorem 3.1.
The lowest eigenvalue of the operator L ( q, γ, δ ) has the prop-erty ess inf q ≤ µ ( q, γ, δ ) − µ (0 , γ, δ ) ≤ ess sup q. Remark 3.
Theorem 3.1 is also true for arbitrary self-adjoint boundary con-ditions.
Theorem 3.1 is a corollary of Theorem 1.2, when we take ˜ q ( x ) ≡
0. Wecan see that our bounds do not contradict the properties mentioned aboveand are more narrow than in (3.3) because of − ess sup | q | ≤ ess inf q ≤ ess sup q ≤ ess sup | q | . For the first eigenvalue of the vibrating string equation we obtain thefollowing bounds. 8 a) µ (0 , γ, δ ) (b) λ (1 , α, β ) Figure 2: Illustration of the first eigenvalues of L (0 , γ, δ ) and S (1 , α, β ). Theorem 3.2.
The lowest eigenvalue of operator S ( p, α, β ) has the property λ (1 , α, β ) min x ∈ [0 ,π ] p − ( x ) ≤ λ ( p, α, β ) ≤ λ (1 , α, β ) max x ∈ [0 ,π ] p − ( x ) . Proof.
This is a corollary of Theorem 2.1.The graphs of eigenvalues µ (0 , γ, δ ) and λ (1 , α, β ) are in Figure 2. Therelation between eigenvalues of L ( q, γ, δ ) and S ( p, α, β ) can be found in theappendix. Example 2.
Consider the boundary value problem S ( x r + 1 , π, , for a fixed r ∈ R + , − u (cid:48)(cid:48) = λ ( x r + 1) u,u (0) = u (1) = 0 . Then λ (1 , π,
0) = π , and max x ∈ [0 ,π ] ( x r + 1) − = 1 , min x ∈ [0 ,π ] ( x r + 1) − = 1 / .Theorem 3.2 implies the following assertion. orollary 3. For any fixed r ∈ R + : π / ≤ λ ( x r + 1 , π, ≤ π . For instance, when r=1,2, the lowest eigenvalues are λ ( x + 1 , π, ≈ . and λ ( x + 1 , π, ≈ . . The next theorem shows, that for any density function p ( x ) the sign ofthe first eigenvalue λ ( p, α, β ) only depends on the boundary parameters. Theorem 3.3.
The lowest eigenvalue of the operator S ( p, α, β ) has the prop-erty λ ( p, α, β ) = 0 , cos α cos β − sin( α − β ) = 0 , α, β ∈ [ π/ , π ] × [0 , π/ ,> , cos α cos β − sin( α − β ) < , α, β ∈ ( π/ , π ] × [0 , π/ ,< , otherwise . Proof.
First, we will show that the lowest eigenvalue λ is zero for boundaryparameters satisfyingcos α cos β − sin( α − β ) = 0 , α, β ∈ [ π/ , π ] × [0 , π/ . (3.4)Let us check that λ = 0 is an eigenvalue. When λ = 0 the equation (1.4)becomes u (cid:48)(cid:48) ( x ) = 0 , hence, in order for u ( x ) to be an eigenfunction of S ( p, α, β ) related to aneigenvalue λ = 0, it should be linear u ( x ) = kx + c , where k + c (cid:54) = 0, andsatisfy boundary conditions, (1.5)–(1.6) u (0) cos α + u (cid:48) (0) sin α = 0 ,u (1) cos β + u (cid:48) (1) sin β = 0 . For u ( x ) = kx + c , we consider the system of linear equations c cos α + k sin α = 0 ,c cos β + k (sin β + cos β ) = 0 . In order to obtain a unique solution for k and c , the determinant should bezero: (cid:12)(cid:12)(cid:12)(cid:12) cos α sin α cos β sin β + cos β (cid:12)(cid:12)(cid:12)(cid:12) = cos α sin β + cos α cos β − sin α cos β which is zero by the condition. Therefore, λ = 0 is an eigenvalue, and itremains to show that this is the smallest eigenvalue. Let there be a number10 > λ n = 0. This means that there exists σ (cid:54) = 0, such that thesmallest eigenvalue λ = − σ . Thus, the equation (1.4) can be rewritten as u (cid:48)(cid:48) ( x ) = σ p ( x ) u ( x ) . (3.5)Now we separate the curve into following three cases:1. α ∈ [ π/ , π/ , β ∈ [0 , π/ α = β = π/ α ∈ ( π/ , π ] , β ∈ ( π/ , π/ u (0) + u (cid:48) (0)(tan β + 1) = 0 ,u (1) + u (cid:48) (1) tan β = 0 . Subtracting from the second condition the first one, we obtain( u (cid:48) (1) − u (cid:48) (0)) tan β + ( u (1) − u (0) − u (cid:48) (0)) = 0 . For simplicity, it can be written as a tan β + b = 0 . To find a and b , we integrate (3.5) from 0 to 0 < t ≤ u (cid:48) ( t ) − u (cid:48) (0) = σ (cid:90) t p ( x ) u ( x ) dx,u (cid:48) (1) − u (cid:48) (0) = σ (cid:90) p ( x ) u ( x ) dx = a,u (1) − u (0) − u (cid:48) (0) = σ (cid:90) (cid:90) t p ( x ) u ( x ) dxdt = b. Since the density function p ( x ) is positive and u ( x ) has no zeros, due to theoscillation theorem, in the whole interval (0 , a (cid:54) = 0, b (cid:54) = 0 andthey have the same sign. Now, please note that tan β ≥
0, for β ∈ [0 , π/ a tan β + b = 0 is impossible. Thus, the first eigenvalue cannotbe negative and λ = 0.Here we proof the second assertion of the theorem, that is λ ( p, α, β ) > α, β ∈ [ π/ , π ] × [0 , π/
4] andcos α cos β − sin( α − β ) < . (3.6)11he third one of the theorem can be handled in a similar way. Please note,that we can show this for λ (1 , α, β ), after it, in virtue of Theorem 3.2, itwill be spread on λ ( p, α, β ).Assume that λ (1 , α, β ) <
0, i.e. there exists σ (cid:54) = 0, such that λ = − σ .So, the boundary value problem (1.4)–1.6 for p ( x ) = 1 is being written as u (cid:48)(cid:48) ( x ) = σ u ( x ) ,u (0) cos α + u (cid:48) (0) sin α = 0 ,u (1) cos β + u (cid:48) (1) sin β = 0 . The solution of this problem has the following form: u ( x ) = Ae | σ | x + Be −| σ | x , where A + B (cid:54) = 0. Inserting this into the boundary conditions, we obtain A (cos α + | σ | sin α ) + B (cos α − | σ | sin α ) = 0 ,Ae | σ | (cos β + | σ | sin β ) + Be −| σ | (cos β − | σ | sin β ) = 0 . To have a unique solution for A and B , the determinant should be zero: D := (cid:12)(cid:12)(cid:12)(cid:12) cos α + | σ | sin α cos α − | σ | sin αe | σ | (cos β + | σ | sin β ) e −| σ | (cos β − | σ | sin β ) (cid:12)(cid:12)(cid:12)(cid:12) == cos α cos β ( e −| σ | − e | σ | ) + sin( α − β ) | σ | ( e −| σ | + e | σ | ) + sin α sin βσ ( e | σ | − e −| σ | ) == 2( − cos α cos β sinh | σ | + sin( α − β ) | σ | cosh | σ | + sin α sin βσ sinh | σ | ) . Taking into account (3.6), for D we obtain D > α cos β ( | σ | cosh | σ | − sinh | σ | ) + sin α sin βσ sinh | σ | ) , or D > α − β )( | σ | cosh | σ | − sinh | σ | ) + sin α sin βσ sinh | σ | ) . Please note that sin α sin βσ sinh | σ | ≥
0, for all α, β ∈ ( π/ , π ] × [0 , π/ | σ | cosh | σ | − sinh | σ | >
0. Consider f ( x ) := x cosh x − sinh x (3.7)for x >
0. Differentiating (3.7), we obtain f (cid:48) ( x ) = x sinh x. Since f (0) = 0 and f (cid:48) ( x ) >
0, for all x >
0, thus f ( x ) > α ∈ ( π/ , π/ , β ∈ [0 , π/ α ∈ ( π/ , π ] , β ∈ [0 , π/ α ∈ ( π/ , π ] , β ∈ [ π/ , π/ α cos β ≥
0, and for the second casesin( α − β ) ≥
0. Thus
D >
Remark 4.
Since on the curve cos α cos β − sin( α − β ) = 0 , for α, β ∈ [ π/ , π ] × [0 , π/ , the first eigenvalue is always zero, thus the inequality inTheorem 3.2 holds also for this case. This paper deals with the inverse spectral theory of vibrating string equationssubject to separated, self-adjoint boundary conditions. It is shown, that inspite of the fact that there is no classical Ambarzumyan’s theorem analogfor string equations, the first eigenvalue Ambarzumyan-type theorems arevalid for some domain in the boundary conditions (Theorems 2.1–2.2 andthe extension for the n -th eigenvalue in Theorem 2.3).The curve in the boundary conditions’ domain ( α, β ∈ (0 , π ] × [0 , π )),on which no analog of the first eigenvalue theorem is possible, is given bythe equation cos α cos β − sin( α − β ) = 0, for α, β ∈ [ π/ , π ] × [0 , π/
4] (seeFigure 1). The explanation simply follows from one property of the firsteigenvalue, as on that curve it is always zero, despite of the density function p ( x ). This property is given in Theorem 3.3 together with another property,which states that the sign of the first eigenvalue is always positive (negative)on the right (left) area from that curve in the boundary parameters’ domain.It is well known that the set of eigenvalues is bounded from below. InTheorem 3.2, the exact lower and upper bounds are found for the lowesteigenvalue depending on boundary parameters and the density function. Acknowledgements
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Appendix: Relation of S ( p, α, β ) and L ( q, γ, δ ) Let the density function be p ∈ C [0 , s ( x ) = πc (cid:90) x (cid:112) p ( t ) dt, y ( s ) = π c p / ( x ) u ( x ) ,q ( s ) = p − / ( x ) d ds ( p / ( x )) , where c = (cid:82) (cid:112) p ( t ) dt , to transform S ( p, α, β ) into L ( q, γ, δ ). Please notethat s (0) = 0 and s (1) = π , therefore s ∈ [0 , π ]. For boundary parameterswe obtain cot α = cπ (cid:18) cot γp / (0) + p (cid:48) (0)4 p / (0) (cid:19) , if γ ∈ (0 , π ) ,α = π, if γ = π, β = 0 , if δ = 0 , cot β = cπ (cid:18) cot δp / (1) + p (cid:48) (1)4 p / (1) (cid:19) , if δ ∈ (0 , π ) , and for the eigenvalues λ n ( p, α, β ) = π c µ n ( q, γ, δ ) . When we take p ( x ) ≡
1, then c = 1 and q ( x ) ≡
0, thus λ n (1 , α, β ) = π µ n (0 , γ, δ, γ, δ