l******r
发帖数: 18699 | 1 对了，这些eigenfunctions都是normalized,使得它们的L2-norm都是1.
在这种normalization之下，是否可以证明它们一致有界？请看一楼更正，谢谢。
当然不同的eigenvalue对应的eigenfunctions都是正交的，所以它们构成L2[0,1]空间
里的标准正交基。只不过eigenfunctions的Sobolev norm不是1，而是一个依赖于特征
值的常数，而且没有上界。
不好意思，这个问题一开始设计的不严谨。这貌似是微分方程里很重要的问题，可
是文献里却找不到。谢谢热心帮助！
problem you have, associated with each eigenvalue there are infinitely many
eigenfunctions, one differing from another just by a scalar multiple. So for
each $\lambda$ you can always pick $g_{\lambda}$ such that its sup-norm is
exactl... 阅读全帖 |
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B********e
发帖数: 10014 | 2 sobolev imbedding thm says, on whole space R^n, or on domain with nice
boundary, if we have a function with high enough sobolev order, say H^r, r>n
/2, then f is continuous and we have
||f|_sup <= C ||f||_{H^r}.
for 1-dim space, r=1 is good enough.
so you are right on that it's not enough to normalize eigenfunctions in L^2
norm.
see if your work needs to normalize the eigenfunctions in the configuration
functional space H^2 instead of L^2. Or at least in H^1.
if not so, i doubt about the conjec... 阅读全帖 |
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s*****e
发帖数: 115 | 3 Not so familiar with the Green's function approach, so I cannot comment on that.
However, I believe your conjecture is true for the following reasons:
What you want to prove is equivalent to that if the eigenfunctions are normalized with respect to sup-norm then their L2 norms are always bounded away from zero.
Clearly, this not true for general functions. For example, both exp(-kt) and exp(-k)*exp(kt) (k>0) on [0,1] have L2 norms approaching to zero as k-->+ infinity. On the other hand, one can... 阅读全帖 |
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o****u
发帖数: 1299 | 4 拜求答疑解惑：
How to find the eigenvalues and eigenfunctions of a complex matrix? In my
case, it will be a Hamiltonian, i.e., a Hermitian matrix. I am interested
in the numerical solution, or, how to program it. In computer codes, you
have only real numbers. How do you code the imaginary part?
Can anybody recommend some online articles or books?
And, what is the physical meaning of the imaginary part of a Hamiltonian? I
think it’s related somehow to time-evolution of the system. But I am not
I am work... 阅读全帖 |
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o****u
发帖数: 1299 | 5 How to find the eigenvalues and eigenfunctions of a complex matrix? In my
case, it will be a Hamiltonian, i.e., a Hermitian matrix. I am interested
in the numerical solution, or, how to program it. In computer codes, you
have only real numbers. How do you code the imaginary part?
Can anybody recommend some online articles or books?
And, what is the physical meaning of the imaginary part of a Hamiltonian? I
think it’s related somehow to time-evolution of the system. But I am not
I am working on u... 阅读全帖 |
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g****g
发帖数: 1828 | 6 In probability theory, the normal (or Gaussian) distribution, is a
continuous probability distribution that is often used as a first
approximation to describe real-valued random variables that tend to cluster
around a single mean value. The graph of the associated probability density
function is “bell”-shaped, and is known as the Gaussian function or bell
curve:[nb 1]
f(x) = \tfrac{1}{\sqrt{2\pi\sigma^2}}\; e^{ -\frac{(x-\mu)^2}{2\sigma^2}
},
where parameter μ is the mean (location of the pe... 阅读全帖 |
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q**x
发帖数: 1636 | 7 your linear algebra is not bad.eigenvalue, eigenfunction. |
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m********e
发帖数: 5088 | 9 本人用有限差分法算8个耦合的eigenvalue functions.
最后的矩阵实在打得惊人，大约100万*100万。要算其eigenvalues and eigenfunctions
Matlab肯定是不行了(感觉大于8000*8000就不行了）
请问大牛们这种情况怎么处理，我想到用Fortran做并行计算，不过package很难确定，网
上很多算大型稀疏矩阵的package不知道如何选择。大家有没有好的建议？
先谢谢了 |
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a**a
发帖数: 416 | 10 看数据量吧，total=10^6*10^6=10^12.
就算每个数据占一个字节，这个数据量也大得惊人。先找到足够的存储器再说。
另外就是用稀疏矩阵。如果稀疏度能达到10^-6级别，那还是可以计算的。
eigenfunctions
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f********r
发帖数: 50 | 11 指望靠好的硬件不太现实
如果不能找到一个理想的算法，这个问题是解决不了的
我做的不是有限差分，而是强相关量子系统，
所以如果要求系统的eigenvalue,矩阵的大小也是2^N *2^N
最少，N是系统的大小。精确求解根本不可能。
但是可以利用renormalization的思想，去掉那些不重要的本证态
事实证明，仅保留10个态，如果方法得当，可以得到1e-4的精度的eigenvalue.
关于稀疏矩阵的算法，有很多种，我想很常用的一种就是jacobi-davidson,
你应该可以在网上找到matlab和fortran的程序。
eigenfunctions
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y**i
发帖数: 86 | 12 用fortran，
用arpack包，不用存矩阵，
中间费时的是矩阵和矢量相乘，并行处理，
低精度和计算少量本征值的话用时很少。
eigenfunctions
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g****c
发帖数: 299 | 13 holy cow
eigenfunctions
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c*******e
发帖数: 8624 | 14 You may try multi-grid. You shouldn't have tried to solve
a linear system like this.
eigenfunctions
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r****y
发帖数: 1437 | 15 I coded similar b.c. and PDE before.
There are two ways, both make use C(x)=Const, i.e., C(x) only
has DC component, wavenumber0, no AC components.
First is to make sure of eigenfunctions for this B.C., note
the different treatments for AC and DC components. And you can easily prove
d^nC/dx^n=0 for n = 2i-1
Second, find the quantity conserved in your equation, dC/dx=0 means
integral over x at B.C. is a value never change with time. Then you can make
use of this to |
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b*****k
发帖数: 80 | 16 First,make a correction for 1):the input signal should be x(wt)
And the output is |H(jw)|x(wt+arg(H(jw))
The result 1) is the eigenfunction property of complex exponentials for
LTI systems. The input sinusoid wave starts from time -infinate,so there
is no Transient response.
When you make the comment like 2),you assume that the sinusoid wave starts from time zero,
i.e. input is x(t)u(t), it is a suddenly applied input,
so there is a transient response.
From the view point of 2),if u assume the i |
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j******n
发帖数: 21 | 17 题目是
Can λ=0 be an eigenvalue of a Sturm Liouville Problem?
Can y=0 be an eigenfunction of a Sturm Liouville Problem?
Explain your answer.
只是会用几个special case去证明第一问。但是觉得不是充分解释。
请问大家有没有什么好的办法说明这两个问题吗？ |
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W*****k
发帖数: 158 | 18 考虑 下面这个eigenvalue 问题，c表示eigenvalue
a(x) f'(x)+b(x)^2/2 f''(x)= -c f(x)
满足混合边界条件
f'(0)=0 和 f(1)=0
假设存在一个eigenvalue序列 0
那么在什么条件下，或者说对a(x)和b(x)加上什么条件可以使得
c_1对应的eigenfunction f_1(x) 在区间(0,1)上恒是正的
有什么reference可以推荐吗？ |
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s*****e
发帖数: 115 | 19 Still not able to see the point of the question. For the boundary value problem you have, associated with each eigenvalue there are infinitely many eigenfunctions, one differing from another just by a scalar multiple. So for each $\lambda$ you can always pick $g_{\lambda}$ such that its sup-norm is exactly 1, unless you have to pick $g_{\lambda}$ in curtain special way that is not described here. |
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l******r
发帖数: 18699 | 20 实际上eigenfunctions都是属于无穷次连续可导的空间的，
虽然我们考虑的是4阶微分方程，可是这个4阶方程加上boundary condition刚刚吻合2
阶sobolev空间。一般来说2m阶微分方程加上类似的boundary condition刚好吻合m阶
sobolev space，用的是integration by parts.我越来越认为这是一个research topic
，其答案不是一两句话能说的清楚的。要想给出完整解答，恐怕得写一个20页的paper
才行。下面是一个背景paper，其中讨论的是这里面q(t)=1的情形，发在journal of
approximation上面。
Boundary Effects on Convergence for Tikhonov Regularization (by FLORENCIO I.
UTRERAS). JOURNAL OF APPROXIMATION THEORY 54, 235-249 (1988) |
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l******r
发帖数: 18699 | 21 Thanks for nice solution!
Just have a quick question. Is the Sobolev Imbedding theorem saying
||f||_sup<=constant||f||_soblev_norm ?
The sobolev norm of the eigenfunctions are usually depending on the
eigenvalues, which is an increasing sequence. So, it seems that directly
using Sobolev embedding thm cannot give us the desired result.
I agree with your first suggestion, mimick the SL theory to get the
conclusion. Thx! |
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l******r
发帖数: 18699 | 22 The normaization is only under L^2 norm, i.e., all the eigenfunctions have L
^2 norm equal to one. This will make their H^r norms inflating with the
corresponding eigenvalues since the eigenvalues must approach +infty.
I agree with you about mimicking SL's theory, probably that is the right
track. So little about this in literature:-)
>n
2
configuration |
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e**********n
发帖数: 359 | 23 You need to identify the Hilbert space (of the wavefunctions) first, which
is all periodic functions with period 2pi in this case. The operator -i(d/dx
) is an operator defined on this Hilbert space. Its eigenvalues are all
integers and the eigenfunctions are exp(i nx) with n \in Z. |
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q*d
发帖数: 22178 | 24 Landau,QM第19页:
The coordinate q itself is an example of a quantity having
a continuous spectrum.
.........
The eigenfunction of this operator must be determined,
according to the usual rule,by the equation qΨ_q0=q0Ψ_q0,
........
it is clear that the egien functions which satisfy the
normalization condition are:
Ψ_q0=δ(q-q0) |
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w*******U
发帖数: 256 | 25 the fortran library 'lapack' or its C version 'lapacke' can do this job.
you can solve the eigenvalue/eigenfunction of a complex matrix with or
without hermite symmetry (the routine of the latter would run faster).
I
including |
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l**n
发帖数: 67 | 26
The standard way will be to solve for the secular equation for eigenvalue
and eigenfunction as described in Goldstein(1980) But you problem
seems has a strong symmetry. It's rotational inviriant along the direction
of vector a^i. So apparently, one of the principal direction will be along
that's direction (a_x,a_y,a_z) and the other two principal directions
are degenerate(any two orthogonal direction perpendicular to a^i) since you
don't have more information other than vector a
But what's inte |
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c********s
发帖数: 123 | 27 A is a tensor, X is a vector, a is a scaler
AX = aX
find the eigenvalue a and eigenfunction X.
The primary axiles of A is the X. |
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