Publications of Kwang C. Shin

1. (With S. H. Ahn and D. S. Kim) Submanifolds with constant mean curvature vector fields.
Honam Mathematical Journal, 17 (1): 49-55, 1995.

2. On the Eigenproblems of PT-Symmetric Oscillators.
Journal of Mathematical Physics, 42 (6): 2513-2530, 2001.

3. On the Reality of the Eigenvalues for a Class of PT-Symmetric Oscillators.
Communications in Mathematical Physics, 229 (3): 543-564, 2002.

4. New polynomials P for which f"+P(z)f=0 has a solution with almost all real zeros.
Annales Academiae Scientiarum Fennicae Mathematica, 27: 491-498, 2002

5. On half-line spectra for a class of non-self-adjoint Hill operators.
Mathematische Nachrichten, 261-262: 171-175, 2003

6. Trace Formulas for Non-Self-Adjoint Periodic Schr\"odinger Operators and some Applications.
Journal of Mathematical Analysis and Applications, 299(1): 19-39, 2004.

7. On the shape of spectra for non-self-adjoint periodic Schr\"odinger operators.
Journal of Physics A: Mathematical and General, 37: 8287-8291, 2004.

8. Eigenvalues of PT-symmetric oscillators with polynomial potentials.
Journal of Physics A: Mathematical and General, 38: 6147-6166, 2005.

9. The potential (iz)^m generates real eigenvalues only, under symmetric rapid decay conditions.
Journal of Mathematical Physics, 46: 082110 (17 pages), 2005.

10. Asymptotics of eigenvalues of non-self adjoint Schr\"odinger operators on a half-line.
Computational Methods and Function Theory, 10 (1): 111-133, 2010.

11. Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry.
Symmetry, Integrability and Geometry: Methods and Applications(SIGMA), 6: 015, 9 pages, 2010.

12. Anharmonic Oscillators in the Complex Plane, PT-Symmetry, and Real Eigenvalues.
Accepted for publication in Potential Analysis, 27 pages, 2010.

A list of Kwang C. Shin's papers from MathSciNet

Publications and Preprints with Abstracts

1. (With S. H. Ahn and D. S. Kim) Submanifolds with constant mean curvature vector fields.
Honam Mathematical Journal, 17 (1): 49-55, 1995.


2. On the Eigenproblems of PT-Symmetric Oscillators.
Journal of Mathematical Physics, 42 (6): 2513-2530, 2001.

We consider the non-Hermitian Hamiltonian H=-d²/dx² +P(x²) -(ix)²ⁿ⁺¹ on the real line, where P(x) is a polynomial of degree at most n > 0 with all nonnegative real coefficients (possibly P = 0). It is proved that the eigenvalues E must be in the sector | arg(E) | \leq Pi/(2n+3). Also for the case H=-d²/dx²-(ix)³, we establish a zero-free region of the eigenfunction u and its derivative u' and we find some other interesting properties of the eigenfunctions.
Note: For the case H=-d²/dx²-(ix)³, the result on location of eigenvalues was later improved in my unpublished preprint, 34 pages, 2001, pdf, or ps.


3. On the Reality of the Eigenvalues for a Class of PT-Symmetric Oscillators.
Communications in Mathematical Physics, 229 (3): 543-564, 2002.

We study the eigenvalue problem -u"(z)-[(iz)^m+P(iz)]u(z)=E u(z) with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays arg(z)=-Pi/2-2Pi/(m+2) and arg(z)=-Pi/2+2Pi/(m+2) in the complex plane, where integer m > 1 and P(z)=a₁ z^m-1+a₂ z^m-2+...+a_m-1 z is a real polynomial. We prove that if for some integer 0 < j< (m+1)/2, we have (j-k)a_k \geq 0 for all 0 < k < m , then the eigenvalues are all real and positive. We then sharpen this to a slightly larger class of polynomial potentials.
In particular, this implies that the eigenvalues are all real and positive for the potentials aiz³+bz²+ciz when a, b, c are real with a \not= 0 and ac \geq 0, and with the boundary conditions that u(z) decays to zero as z tends to infinity along the positive and negative real axes. This verifies a conjecture of Bessis and Zinn-Justin.


4. New polynomials P for which f"+P(z)f=0 has a solution with almost all real zeros.
Annales Academiae Scientiarum Fennicae Mathematica, 27: 491-498, 2002.

Let a, b, c be real numbers with a \not= 0 with ac \leq 0. We prove that there exists a sequence of positive real numbers E_k\to \infty such that for each k, the equation f"(z)+(a z³+bz²+cz-E_k)f(z)=0 admits a solution with infinitely many real zeros and at most finitely many non-real zeros. This gives a new class of cubic polynomials P for which f"+P(z)f=0 has a solution with almost all real zeros.
We also find new quartic examples: for each a > 0 and b real, there exists a sequence of real numbers E_k\to \infty such that for each k, f"(z)+(a z⁴ +bz²-E_k)f(z)=0 has a solution with almost all real zeros. The case a > 0 and b > 0 was discovered earlier by Gundersen.


5. On half-line spectra for a class of non-self-adjoint Hill operators.
Mathematische Nachrichten, 261-262: 171-175, 2003.

In 1980, Gasymov showed that non-self-adjoint Hill operators with the complex-valued periodic potentials of the type $V(x)=\sum_{k=1}^{\infty} a_k e^ikx$, with $\sum_{k=1}^{\infty}|a_k|<\infty$, have spectra $[0,\infty)$. In this note, we provide an alternative and elementary proof of this result.


6. Trace Formulas for Non-Self-Adjoint Periodic Schr\"odinger Operators and some Applications.
Journal of Mathematical Analysis and Applications, 299(1): 19-39, 2004.

Recently, a trace formula for non-self-adjoint periodic Schr\"odinger operators in L²(R) associated with Dirichlet eigenvalues was proved by Gesztesy. Here we prove a corresponding trace formula associated with Neumann eigenvalues.
In addition we investigate Dirichlet and Neumann eigenvalues of such operators. In particular, using the Dirichlet and Neumann trace formulas we provide detailed information on location of the Dirichlet and Neumann eigenvalues for the model operator with the potential Ke^2ix, where K is a complex number.


7. On the shape of spectra for non-self-adjoint periodic Schr\"odinger operators.
Journal of Physics A: Mathematical and General, 37: 8287-8291, 2004.

The spectra of the Schr\"odinger operators with periodic potentials are studied. When the potential is real and periodic, the spectrum consists of at most countably many line segments (energy bands) on the real line, while when the potential is complex and periodic, the spectrum consists of at most countably many analytic arcs in the complex plane.
In some recent papers, such operators with complex PT-symmetric periodic potentials are studied. In particular, the authors argued that some energy bands would appear and disappear under perturbations. Here, we show that appearance and disappearance of such energy bands imply existence of nonreal spectra. This is a consequence of a more general result, describing the local shape of the spectrum.


8. Eigenvalues of PT-symmetric oscillators with polynomial potentials.
Journal of Physics A: Mathematical and General, 38: 6147-6166, 2005.

We study the eigenvalue problem -u"(z)-[(iz)^m+P(iz)]u(z)=E u(z) with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays arg(z)=-Pi/2-2Pi/(m+2) and arg(z)=-Pi/2+2Pi/(m+2) in the complex plane, where P(z)=a₁ z^m-1+a₂ z^m-2+...+a_m-1 z is a polynomial and m > 2. We provide an asymptotic expansion of the eigenvalues E_n as n tends to infinity, and prove that for each real polynomial P of degree < m, all but finitely many eigenvalues are real and positive.

Note: This paper treats the case l=1 of Paper 10 below and provides the induction basis for the proof in that paper. The proof of the main theorem in Paper 10 is a result of investigating the asymptotics of entire functions (Stokes multipliers) whose zeros are the eigenvalues, where I used induction on l.


9. The potential (iz)^m generates real eigenvalues only, under symmetric rapid decay conditions.
Journal of Mathematical Physics, 46: 082110 (17 pages), 2005.

We consider the eigenvalue problems -u"(z)+(-1)^l (iz)^m u(z)=E u(z), for integers m > 2 and l=1, 2, under every rapid decay boundary condition that is symmetric with respect to the imaginary axis in the complex z-plane. We prove that the eigenvalues E are all real and positive.


10. Asymptotics of eigenvalues of non-self adjoint Schr\"odinger operators on a half-line. Computational Methods and Function Theory, 10 (1): 111-133, 2010.

We study the eigenvalues of the non-self adjoint problem $-y^\dd+V(x)y=E y$ on the half-line $0\leq x<+\infty$ under the Robin boundary condition at $x=0$, where $V$ is a monic polynomial of degree $\geq 3$. We obtain a Bohr-Sommerfeld-like asymptotic formula for $E_n$ that depends on the boundary conditions. Consequently, we solve certain inverse spectral problems, recovering the potential $V$ and boundary condition from the first $(m+2)$ terms of the asymptotic formula.
Also, we showed that the spectrum of such an operator is real if and only if it is self-adjoint.

11. Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry. Symmetry, Integrability and Geometry: Methods and Applications(SIGMA), 6: 015, 9 pages, 2010.

For integers m > 2, we study the eigenvalue problem -u"(z)-[(iz)^m+P(iz)]u(z)=E u(z) with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays arg(z)=-Pi/2-2Pi/(m+2) and arg(z)=-Pi/2+2Pi/(m+2) in the complex plane, where P(z)=a₁ z^m-1+a₂ z^m-2+...+a_m-1 z is a polynomial. We provide very accurate and explicit asymptotic expansions of the eigenvalues E_n. Then we apply these to the inverse spectral problem, reconstructing some coefficients of polynomial potentials from asymptotic expansions of the eigenvalues. Also, we show for arbitrary real polynomial P of degree < m and for all symmetric decaying boundary conditions that the eigenvalues are all real and positive, with only finitely many exceptions.
Also, we showed that the eigenvalue problem has infinitely many real eigenvalues if and only if P is real.

12. Anharmonic Oscillators in the Complex Plane, PT-Symmetry, and Real Eigenvalues.
Accepted for publication in Potential Analysis, 27 pages, 2010.

For integers m > 2 and 0 < l < m , we study the eigenvalue problem -u"(z)+[(-1)^l(iz)^m-P(iz)]u(z)=E u(z) with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays arg(z)=-Pi/2-Pi (l+1)/(m+2) and arg(z)=-Pi/2+Pi (l+1)/(m+2) in the complex plane, where P(z)=a₁ z^m-1+a₂ z^m-2+...+a_m$ is a polynomial. We provide asymptotic expansions of the eigenvalue E_n. Then, we show for arbitrary real polynomial P of degree < m and for all symmetric decaying boundary conditions that the eigenvalues are all real and positive, with only finitely many exceptions.
Moreover, when gcd(m,l)=1, we obtain very interesting results. The potential can be, up to translations, recovered from the first few terms of asymptotic expansions of the eigenvalues. Also, we prove that there are infinitely many real eigenvalues if and only if it is PT-symmetric or one of its translations is PT-symmetric.
If two such eigenvalue problems have the n^th eigenvalues whose distance is o(1), then two n^th eigenvalues are all the same.