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Asymptotic behavior of eigenvalues of hydrogen atom equation
Boundary Value Problems volume 2015, Article number: 87 (2015)
Abstract
We consider a spectral problem for a class of singular Sturm-Liouville operators on the unit interval with explicit singularity \(2/x\)-\(2/x^{2}\), related to the Schrödinger operator with radially symmetric potential. In particular, we give the asymptotic behavior of the eigenvalues of the hydrogen atom equation.
1 Introduction
The distribution of eigenvalues in differential operator’s spectral theory has an important place. This classic issue was first examined in a finite interval for second order operators in the 19th century by Sturm and Liouville. Later, where the regular boundary conditions were satisfied, the distribution of eigenvalues of differential operators in a finite interval in arbitrary order was also examined by Birkhoff in 1908 [1].
Especially, the distribution of eigenvalues of the operators with a discrete spectrum defined in the whole of space for quantum mechanics has great importance. Firstly, the formula for the distribution of the eigenvalues of the single-dimensional Sturm operator defined in the whole of the straight-line axis with increasing potential at infinity was given by Titchmarsh in 1946 [2]. Titchmarsh also has shown the distribution formula for the Schrödinger operator. In later years, Levitan and Gasymov improved the Titchmarsh method and found important asymptotic formulas for the eigenvalues of different differential operators [3, 4].
Two important methods have been dealt with to examine the asymptotic formula for eigenvalues. The first method, the variation method, is due to Courant and Hilbert [5]. Birman and Solomyak have improved this method in recent years [6]. The second method that is related with the resolvent of the operator in question was suggested by Carleman [7]. Another important method for examining the asymptotic of the eigenvalues in singular condition was suggested by Fedoryuk [8]. This method is very useful in that it ensures that the distribution of the eigenvalues of the operators with partial derivation are such that the coefficients are analytic functions. Later, many studies have been conducted to examine the eigenvalues [9–25]. Many mathematicians have examined the eigenvalues so far.
The spectral problem for the Sturm-Liouville operator with Dirichlet boundary condition is given in detail in [9] by Poeschel and Trubowitz. Guillot and Ralston have extended these results to the singular Sturm-Liouville operator
with domain {\(y\in L^{2} [ 0,1 ] :y, y^{\prime}\), absolutely continuous on \((0,1 ]\), \(Ly\in L^{2} [ 0,1 ]\) and \(y ( 1 ) =0\)} [16].
Later, this work was generalized by Carlson [10]. For real numbers b and real valued functions \(q(x)\in L^{2} [ 0,1 ] \), Carlson dealt with the operator
with domain \(\{y\in L^{2} [ 0,1 ] :y, y^{\prime}\), absolutely continuous on \((0,1 ]\), \(L(m,q)y\in L^{2} [ 0,1 ] \), \(\lim_{x\downarrow0}y ( x ) =0\) and \(y^{\prime} ( 1 ) +by ( 1 ) =0\)}. Similar features of the Sturm-Liouville operator were studied in [16–18].
Consider the Schrödinger equation for two particles in dimensionless variables,
If the potential function \(V(x,y,z)\) depends only on \(r=(x^{2}+y^{2}+z^{2})^{1/2}\), i.e. \(V(x,y,z)=V(r)\), then the variables in (1.1) can be separated by putting
where \(x=r\sin\theta\cos\varphi\), \(y= r\sin\theta\sin\psi \), \(z=r\cos\theta\), and \(Y_{m}^{l}(\theta,\varphi)\) are the spherical harmonics. This gives a differential equation of the form
for the function \(\varphi(r)\), where \(\lambda=l+1/2\) (\(l=0,1,2,\ldots\)). If the potential function \(V(r)\) satisfies the condition \(\int_{0}^{\infty}r \vert V(r)\vert \, dr<\infty\), then, for a solution of (1.2), which is regular at zero and normalized, the following asymptotic formula is satisfied:
for fixed λ, and k, and \(r\rightarrow\infty\).
In this formula, \(A(k,\lambda)\) is called the scattering amplitude and \(\delta(k,\lambda)\) the scattering phase or phase shift [4].
In quantum mechanics the study of the energy levels of the hydrogen atom leads to the equation [26–28]
The substitution \(R=y/r\) reduces this equation to the form
Our aim here is to find the asymptotic behavior of the eigenvalues of the problem
with domain {\(y\in L^{2} [ 0,1 ] :y,y^{\prime}\) that are absolutely continuous on \((0,1 ]\), \(Ly\in L^{2} [ 0,1 ] \)}. Here we have \(\lambda=\sqrt{-E}\), \(E<0\).
Spectral problems for the hydrogen atom equation were considered by many mathematicians. Particularly, the inverse problem was examined in Panakhov and Yilmazer’s papers [12, 13].
2 Basic properties
We consider the singular Sturm-Liouville equation
where the function \(q ( x ) \in L^{2} [ 0,1 ] \). Let us give the solutions of this equation by integral equation representations.
Lemma 1
The solutions of (2.1) have the following form:
and
where \(q(x)\in L^{2} [ 0,1 ] \).
Proof
Let us show that (2.2) is satisfied. The general solution of the equation
is
Let us apply the method of variation of parameters of (2.1),
Taking the second derivative of the equation
and substituting this into (2.1), we obtain
If we multiply the first equation by \(-1/x\) and combine with the second equation we have
Take the integral of this equation from 0 to x:
If we multiply the first equation by 2 and the second equation by x and combine these equations we have
Take the integral of this equation from 0 to x:
Then we get the equation
We use the above method to show (2.3). Take the integral of (2.4) from x to 1:
Take the integral of (2.5) from x to 1:
Then we get the equation
So we proved the theorem. □
Now we will show that these solutions are analytic by using the method of successive approximations. Addressing (2.2) first, let
Theorem 1
The sequence \(y_{n}(x)\) converges uniformly to a function \(\varphi(x,\lambda,q)\) satisfying (2.2) and (2.1). Moreover, \(\lim_{x\downarrow0}x^{-2}\varphi(x,\lambda,q)=1\) and the mapping \(( \lambda,q ) \rightarrow\varphi(x,\lambda,q)\) is analytic from \(\mathbb{C} \times L^{2} [ 0,1 ] \rightarrow \mathbb{C} [ 0,1 ] \).
Proof
Let us show that
by using the method of induction. For \(k=1\),
We have \(\vert t-\frac{t^{4}}{x^{3}}\vert \leq t\),
By the Cauchy-Schwarz inequality, we get
For \(k=2\),
Since \(\vert t-\frac{t^{4}}{x^{3}}\vert \leq t\),
By the Cauchy-Schwarz inequality, we get
Assume that the inequality is true for \(k=n\). Now we will show that the inequality holds for \(k=n+1\),
By the ratio test, the series converges. Then \(y_{n}(x)\) converges uniformly by the Weierstrass sufficiency theorem. □
Differentiation of (2.2) gives the formula
and
Turning to (2.3), let
Theorem 2
The sequence \(xy_{n}(x)\) converges uniformly for \(x\in(0,1]\) to a function \(x\psi(x,\lambda,q)\) where \(\psi(x,\lambda,q)\) satisfies (2.3) and (2.1). Moreover, \(\lim_{x\downarrow0}x\psi(x,\lambda,q)\) exists and the mapping \(( \lambda,q ) \rightarrow x\psi (x,\lambda,q)\) is analytic from \(\mathbb{C} \times L^{2} [ 0,1 ] \rightarrow \mathbb{C} [ 0,1 ] \).
Proof
Let us show that
by using the method of induction. For \(k=1\),
Since \(t\geq x\) we get \(\vert \frac{x^{2}}{t}-\frac{t^{2}}{x}\vert =\vert \frac{t^{2}}{x} ( \frac {x^{3}}{t^{3}}-1 ) \vert \leq\frac{t^{2}}{x}\) and
Because of \(t\leq1\), we have
By the Cauchy-Schwarz inequality, we get
Assume that the inequality is true for \(k=n\). Now we will show that the inequality holds for \(k=n+1\),
Since \(t\geq x\) we get \(\vert \frac{x^{2}}{t}-\frac{t^{2}}{x}\vert =\vert \frac{t^{2}}{x} ( \frac {x^{3}}{t^{3}}-1 ) \vert \leq\frac{t^{2}}{x}\). We have
Because of \(t\leq1\), we have
By the Cauchy-Schwarz inequality, we get
By the ratio test, the series converges. Then \(y_{n}(x)\) converges uniformly by the Weierstrass sufficiency theorem. □
3 Asymptotic behavior of eigenvalues
The main result of the paper is given by the following theorem.
Assume that \(0< x_{1}< x_{2}\leq1\).
Theorem 3
If y is a nontrivial solution of the equation
with \(y(x_{1})=y^{\prime}(x_{2})+by(x_{2})=0\), then
where \(b\in \mathbb{R} \) and \(q(x)\in L^{2} [ 0,1 ] \).
Proof
Multiplying (3.1) by y and integrating of this equation from \(x_{1}\) to \(x_{2}\) gives the formula
Since \(\int_{x_{1}}^{x_{2}}\frac{2}{x^{2}}y^{2}\,dx\geq0\) the remaining term will be negative or zero,
Integrating the first term by parts gives
So (3.2) is equal to
Moreover, we find \(\int_{x_{1}}^{x_{2}}q(x)y^{2}(x)\,dx\) and that equals
Then we have
Integrating \(\int_{x_{1}}^{x_{2}}2y(x)y^{\prime}(x)\,dx\) by parts gives
So we get the formula
Adding (3.3) to \(by^{2}(x_{2})\) and using the triangle inequality gives the formula
Since \(x\leq x_{2}\) we get
By the Cauchy-Schwarz inequality, we get
and, from the inequality of \(2A^{1/2}B^{1/2}\leq\varepsilon A+(1/\varepsilon)B\), we get
For any \(\varepsilon>\vert b\vert +2\int_{x_{1}}^{x_{2}}\vert q\vert \,dx\),
where \(C(\varepsilon)>0\). If we assume that
the equation \(\int_{x_{1}}^{x_{2}} [ -y^{\prime\prime }y+q ( x ) y^{2}-\lambda y^{2}-\frac{2}{x}y^{2} ] \,dx\) should be positive. It contradicts our assumption. So we proved the theorem. □
Theorem 4
If y is a nontrivial solution of the equation
with \(y(x_{1})=y(x_{2})=0\), then
where \(q(x)\in L^{2} [ 0,1 ] \).
Proof
Multiplying the equation by y and integrating of this equation from \(x_{1} \) to \(x_{2}\) gives the formula
Since \(\int_{x_{1}}^{x_{2}}\frac{2}{x^{2}}y^{2}\,dx\geq0\) the remaining term will be negative or zero,
Integrating the first term by parts gives
So (3.4) is equal to
Moreover, we find \(\int_{x_{1}}^{x_{2}}q(x)y^{2}(x)\,dx\) and that equals
Then we have
since \(x\leq x_{2}\) we get
By the Cauchy-Schwarz inequality, we get
and from the inequality of \(2A^{1/2}B^{1/2}\leq\varepsilon A+(1/\varepsilon)B\), we get
For any \(\varepsilon>2\int_{x_{1}}^{x_{2}}\vert q\vert \,dx \),
where \(C(\varepsilon)>0\). If we assume that
the equation \(\int_{x_{1}}^{x_{2}} [ -y^{\prime\prime }y+q ( x ) y^{2}-\lambda y^{2}-\frac{2}{x}y^{2} ] \,dx\) should be positive. It contradicts our assumption. So we get our result. □
Theorem 5
If y is a nontrivial solution of the equation
where \(q(x)\in L^{2} [ 0,1 ] \), and if \(y^{\prime }(x_{1})=y^{\prime}(x_{2})=0\), where \(0\leq x_{1}\leq x_{2}\leq1\), then
Proof
Multiplying (3.5) by y and integrating of this equation from \(x_{1}\) to \(x_{2}\) gives the formula
Since
we get
From the mean value theorem we can write
where \(x_{3}\in [ x_{1},x_{2} ] \). From (3.7) and integrating \(\int_{x_{1}}^{x_{2}}2y(x)y^{\prime}(x)\,dx\) by parts we have
Adding (3.6) to \(\int_{x_{1}}^{x_{2}}\frac{2y^{2}}{x}\,dx\) we have
From the triangle inequality we have the formula
By using the inequality
we get
Integrating \(\int_{x_{1}}^{x_{2}}-y^{\prime\prime}y\, dx\) by parts gives
From (3.9) and (3.8) and for any \(\varepsilon>\vert b\vert +2\int_{x_{1}}^{x_{2}}\vert q\vert \,dx\), there exists a number \(C(\varepsilon)>0\) such that
Let us assume that
In this case, we get
Then we have
This is a contradiction. So we proved the theorem. □
Conclusion
In the Carlson case, the potentials are in \(L^{2} [ 0,1 ] \), but in our paper, the potentials are not in \(L^{2} [ 0,1 ] \).
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The authors would like to thank the referees for valuable comments in improving the original paper.
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IU carried out the design of the study and performed the analysis. EP (adviser) participated in its design and coordination. All authors read and approved the final manuscript.
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Panakhov, E.S., Ulusoy, I. Asymptotic behavior of eigenvalues of hydrogen atom equation. Bound Value Probl 2015, 87 (2015). https://doi.org/10.1186/s13661-015-0347-z
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DOI: https://doi.org/10.1186/s13661-015-0347-z