On the number of bound states for the one-dimensional Schrödinger equation. (English) Zbl 0931.34069

Summary: Consider the one-dimensional Schrödinger equation \[ \psi''(k,x)+ k^2\psi(k,x)= V(x)\psi(k, x),\tag{1} \] where the potential \(V\) is real-valued and belongs to \(L^1_1(\mathbb{R})\), the class of measurable functions for which \(\int^\infty_{-\infty} dx(1+|x|)|V(x)|\) is finite. The prime denotes the derivative with respect to the spatial coordinate \(x\). Let a partition of the real axis be given as \(\mathbb{R}= \bigcup^p_{j=1} (x_{j-1}, x_j)\), with \(x_{j-1}< x_j\) for \(j= 1,\dots, p\). Here, the authors use the convention \(x_0= -\infty\) and \(x_p=+\infty\). They obtain a fragmentation of the potential by setting \(V(x)= \sum^p_{j= 1}V_j(x)\), where \[ V_j(x)= \begin{cases} V(x),\quad & x\in(x_{j-1}, x_j),\\ 0,\quad & \text{elsewhere}.\end{cases}\tag{2} \] The number of bound states of equation (1) is analyzed in terms of the number of bound states corresponding to fragments of the potential. When the potential is integrable and has a finite first moment, the sharp inequalities \(1-p+ \sum^p_{j=1} N_j\leq N\leq \sum^p_{j=1} N_j\) are proved, where \(p\) is the number of fragments, \(N\) is the total number of bound states, and \(N_j\) is the number of bound states for the \(j\)th fragment. When \(p=2\) the question of whether \(N= N_1+ N_2\) or \(N= N_1+ N_2-1\) is investigated in detail. An illustrative example is provided.


34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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[6] DOI: 10.1002/cpa.3160320202 · Zbl 0388.34005
[7] DOI: 10.1063/1.532271 · Zbl 1001.34074
[8] DOI: 10.1063/1.530089 · Zbl 0777.34056
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[10] DOI: 10.1088/0266-5611/4/2/013 · Zbl 0669.34030
[11] DOI: 10.1063/1.531754 · Zbl 0910.34069
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