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The finite spectrum of Sturm-Liouville problems with transmission conditions. (English) Zbl 1242.34042

The authors ask whether it is possible to obtain Sturm-Liouville problems (SLPs) with transmission conditions having a finite spectrum of eigenvalues. They consider the (SLP) given by \[ (-px')'+qx=\lambda w x \quad \text{ on } J=(a,c)\cup (c,b),~ -\infty <a<b<\infty, \] subject to the boundary conditions \[ AY(a)+BY(b)=0 \quad \text{with } Y^{t}=\left(y, py'\right), \] and the so-called transmission conditions \[ CY(c-)+DY(c+)=0, \] where \(A, B\) are complex valued matrices of order \(2\times 2\) and \(C, D\) are real valued matrices of the same order satisfying \(\det(C)>0, \det(D)>0;\) \(\lambda\) is the spectral parameter; and the coefficients \(r=\frac{1}{p}, q, w\) are complex valued functions which are Lebesgue integrable on \(J.\)
The answer is in affirmative, in fact, they prove (Corollary 3.2): “For every \(m\in\{1,2,\dots\}\) there exist a (SLP) (as described above) with piecewise constant coefficients which has exactly \(m\) eigenvalues. Furthermore, given any \(k\) disjoint open sets \(\mathcal{N}_i\) in the complex plane and any \(k\) integers \(n_i\), there exists a (SLP) with transmission conditions with piecewise constant coefficients having exactly \(n_i\) eigenvalues in \(\mathcal{N}_i\) for \(i=1,\dots,k.\) Given any \(k\) disjoint open real intervals \(J_i\) and any \(k\) integers \(n_i\) there exists a self-adjoint (SLP) with transmission conditions with piecewise constant coefficients having exactly \(n_i\) eigenvalues in the intervals \(J_i\), \(i=1,\dots,k\).”
The construction of these (SLPs) follows the strategy developed in [Q. Kong, H. Wu and A. Zettl, J. Math. Anal. Appl. 263, 748–762 (2001; Zbl 1001.34019)].

MSC:

34B24 Sturm-Liouville theory
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators

Citations:

Zbl 1001.34019
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References:

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