## 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

Zbl 1001.34019
Full Text:

### References:

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